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Theoretical vs. Experimental Probability: How do they differ?

Probability is the study of chances and is an important topic in mathematics. There are two types of probability: theoretical and experimental.

So, how to define theoretical and experimental probability? Theoretical probability is calculated using mathematical formulas, while experimental probability is based on results from experiments or surveys. In order words, theoretical probability represents how likely an event is to happen. On the other hand, experimental probability illustrates how frequently an event occurs in an experiment.

Read on to find out the differences between theoretical and experimental probability. If you wonder How to Understand Statistics Easily , I wrote a whole article where I share 9 helpful tips to help you Ace statistics.

Table of Contents

What Is Theoretical Probability?

Theoretical probability is calculated using mathematical formulas. In other words, a theoretical probability is a probability that is determined based on reasoning. It does not require any experiments to be conducted. Theoretical probability can be used to calculate the likelihood of an event occurring before it happens.

Keep in mind that theoretical probability doesn’t involve any experiments or surveys; instead, it relies on known information to calculate the chances of something happening.

For example, if you wanted to calculate the probability of flipping a coin and getting tails, you would use the formula for theoretical probability. You know that there are two possible outcomes—heads or tails—and that each outcome is equally likely, so you would calculate the probability as follows: 1/2, or 50%.

How Do You Calculate Theoretical Probability?

First, start by counting the number of possible outcomes of the event.
Second, count the number of desirable (favorable) outcomes of the event.
Third, divide the number of desirable (favorable) outcomes by the number of possible outcomes.
Finally, express this probability as a decimal or percentage.

The theoretical probability formula is defined as follows: Theoretical Probability = Number of favorable (desirable) outcomes divided by the Number of possible outcomes.

How Is Theoretical Probability Used in Real Life?

Probability plays a vital role in the day to day life. Here is how theoretical probability is used in real life:

Sports and gaming strategies
Analyzing political strategies.
Buying or selling insurance
Determining blood groups
Online shopping
Weather forecast
Online games

What Is Experimental Probability?

Experimental probability, on the other hand, is based on results from experiments or surveys. It is the ratio of the number of successful trials divided by the total number of trials conducted. Experimental probability can be used to calculate the likelihood of an event occurring after it happens.

For example, if you flipped a coin 20 times and got heads eight times, the experimental probability of obtaining heads would be 8/20, which is the same as 2/5, 0.4, or 40%.

How Do You Calculate Experimental Probability?

The formula for the experimental probability is as follows: Probability of an Event P(E) = Number of times an event happens divided by the Total Number of trials .

If you are interested in learning how to calculate experimental probability, I encourage you to watch the video below.

How Is Experimental Probability Used in Real Life?

Knowing experimental probability in real life provides powerful insights into probability’s nature. Here are a few examples of how experimental probability is used in real life:

Rolling dice
Selecting playing cards from a deck
Drawing marbles from a hat
Tossing coins

The main difference between theoretical and experimental probability is that theoretical probability expresses how likely an event is to occur, while experimental probability characterizes how frequently an event occurs in an experiment.

In general, the theoretical probability is more reliable than experimental because it doesn’t rely on a limited sample size; however, experimental probability can still give you a good idea of the chances of something happening.

The reason is that the theoretical probability of an event will invariably be the same, whereas the experimental probability is typically affected by chance; therefore, it can be different for different experiments.

Also, generally, the more trials you carry out, the more times you flip a coin, and the closer the experimental probability is likely to be to its theoretical probability.

Also, note that theoretical probability is calculated using mathematical formulas, while experimental probability is found by conducting experiments.

Wrapping Up

Theoretical and experimental probabilities are two ways of calculating the likelihood of an event occurring. Theoretical probability uses mathematical formulas, while experimental probability uses data from experiments. Both types of probability are useful in different situations.

I believe that both theoretical and experimental probabilities are important in mathematics. Theoretical probability uses mathematical formulas to calculate chances, while experimental probability relies on results from experiments or surveys.

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Mastering Theoretical and Experimental Probability Comparisons Dive into the world of probability with our comprehensive guide. Learn to calculate, compare, and apply theoretical and experimental probability in real-world scenarios. Enhance your statistical skills today!

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Experimental probability VS. Theoretical probability

Die Outcome	Experimental results
1	II
2	IIII
3	I
4	III
5	IIIII
6	I

Coin Outcome	Experimental Results
H, H	IIII
H, T	IIIII
T, H	IIIIII
T, T	IIIII

What is the experimental probability of both coins landing on heads?

Calculate the theoretical probability of both coins landing on heads.
Compare the theoretical probability and experimental probability.
What can Jessie do to decrease the difference between the theoretical probability and experimental probability?

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What is experimental probability?

In math, when we deal with probability , we may be asked for the experimental probability of an experiment. What this means is that they're looking for the probability of something happening based off the results of an actual experiment. This is the experimental probability definition.

So for example, if you're asked for the probability of getting heads after flipping a coin 10 times, the experimental probability will be the number of times you got heads after flipping a coin 10 times. Let's say that you got 6 heads out of your 10 throws. Then your experimental probability is 6/10, or 60%.

For theoretical probability, it doesn't require you to actually do the experiment and then look at the results. Instead, the theoretical probability is what you expect to happen in an experiment (the expected probability). This is the theoretical probability definition.

In the case of the coin flips, since there's 2 sides to a coin and there's an equal chance that either side will land when you flip it, the theoretical probability should be 1 2 \frac{1}{2} 2 1 or 50%.

Why is there a difference in theoretical and experimental probability? The relationship between the two is that you'll find if you do the experiment enough times, the experimental probability will get closer and closer to the theoretical probability's answer. You can try this out yourself with a coin. You likely won't get exactly 50% for both heads and tails from your first 10 throws, but as you throw a coin 50 times or even 100 times, you'll see the experimental probability's answer getting closer to 50%.

We'll now see how experimental and theoretical probability works with these questions.

Question 1a: Two coins are flipped 20 times to determine the experimental probability of landing on heads versus tails. The results are in the chart below:

We are looking for the experimental probability of both coins landing on heads. Looking at the table in the question, we know that there were 4 out of 20 trials in which both coins landed on heads. So the experimental probability is 4 20 \frac{4}{20} 20 4 , which equals to 1 5 \frac{1}{5} 5 1 (20%) after simplifying the fraction

Question 1b: Calculate the theoretical probability of both coins landing on heads.

Now, we are looking for the theoretical probability. First, there are 4 possible outcomes (H,H), (H, T), (T,H), (T, T). 1 out of the 4 possible outcomes has both coins land on heads. So, the theoretical probability is 1 4 \frac{1}{4} 4 1 or 25%

Question 1c: Compare the theoretical probability and experimental probability.

From the previous parts, we know that the experimental probability of both coins landing on head equals 20%, while in theory, there should be a 25% chance that both coins lands on head. Therefore, the theoretical probability is higher than the experimental probability.

Question 1d:

What can we do to reduce the difference between the experimental probability and theoretical probability? We can simply continue the experimental by flipping the coin for many more times —say, 20,000 times. When more trials are performed, the difference between experimental probability and theoretical probability will diminish. The experimental probability will gradually get closer to the value of the theoretical probability. In this case, the experimental probability will get closer to 25% as the coins is tossed over more times.

If you're looking for more experimental vs.theoretical probability examples, feel free to try out this question . It'll require you to do some hands-on experimentation!

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Theoretical Probability versus Experimental Probability

You've heard the terms, theoretical probability and experimental probability , but what do they mean?

Are they in anyway related? This is what we are going to discover in this lesson.

If you've completed the lessons on i ndependent and dependent probability , then you've already found the theoretical probability for numerous problems.

Theoretical Probability

Theoretical probability is the probability that is calculated using math formulas. This is the probability based on math theory.

Experimental Probability

Experimental probability is calculated when the actual situation or problem is performed as an experiment. In this case, you would perform the experiment, and use the actual results to determine the probability.

In order to accurately perform an experiment, you must:

Identify what constitutes a " trial ".
Perform a minimum of 25 trials
Set up an organizer (table or chart) to record your data.

Let's take a look at an example where we first calculate the theoretical probability, and then perform the experiment to determine the experimental probability.

It will be interesting to compare the theoretical probability and the experimental probability. Do you think the two calculations will be close?

Example 1 - Theoretical Versus Experimental

This problem is from Example 1 in the independent events lesson. We calculated the theoretical probability to be 1/12 or 8.3%. Take a look:

Since we know that the theoretical probability is 8.3% chance of flipping a head and rolling a 6, let's see what happens when we actually perform the experiment.

Identify a trial: A trial consists of flipping a coin once and rolling a die once.

Conduct 25 trials and record your data in the table below.

For each trial, I flipped the coin once and rolled the die. I recorded and H for heads and a T for tails in the row labeled "Coin."

I recorded the number on the die in the row labeled "Die".

In the last row I determined whether the trial completed the event of flipping a head and rolling a six.

In this experiment, there was only 1 trial (out of 25) where a head was flipped on the coin and a 6 was rolled on the die.

This means that the experimental probability is 1/25 or 4%.

Please note that everyone's experiment will be different; thus allowing the experimental probability to differ.

Also, the more trials that you conduct in your experiment, the closer your calculations will be for the experimental and theoretical probabilities.

Conclusions

The theoretical probability is 8.3% and the experimental probability is 4%. Although the experimental probability is slightly lower, this is not a significant difference.

In most experiments, the theoretical probability and experimental probability will not be equal; however, they should be relatively close.

If the calculations are not close, then there's a possibility that the experiment was conducted improperly or more trials need to be completed.

I hope this helps to give you a sense of how to set up an experiment in order to compare theoretical versus experimental probabilities.

Probability
Theoretical/Experimental Probability

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Theoretical vs. Experimental Probability

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Experimental Probability vs. Theoretical Probability

What's the difference.

Experimental probability is based on actual observations and data collected from experiments or real-life events. It is calculated by dividing the number of favorable outcomes by the total number of outcomes. On the other hand, theoretical probability is based on mathematical calculations and predictions. It is determined by analyzing the possible outcomes and their likelihood in a given situation. While experimental probability provides an estimate of the likelihood of an event based on real-world data, theoretical probability provides a more precise and accurate prediction based on mathematical principles.

Attribute	Experimental Probability	Theoretical Probability
Definition	Based on actual outcomes observed through experiments or trials.	Based on mathematical calculations and assumptions.
Calculation	Number of favorable outcomes divided by total number of outcomes.	Number of favorable outcomes divided by total number of equally likely outcomes.
Accuracy	May vary due to limited sample size or external factors.	Assumes ideal conditions and infinite sample size.
Application	Used when direct observation or experimentation is possible.	Used when the underlying probability distribution is known.
Real-world Examples	Flipping a coin, rolling a dice, conducting surveys.	Calculating the probability of drawing a specific card from a deck, predicting the outcome of a fair game.
Representation	Usually represented as a decimal or a fraction.	Usually represented as a decimal or a fraction.

Further Detail

Introduction.

Probability is a fundamental concept in mathematics and statistics that allows us to quantify the likelihood of an event occurring. It plays a crucial role in various fields, including science, finance, and everyday decision-making. When discussing probability, two important terms often come up: experimental probability and theoretical probability. While both concepts deal with the likelihood of events, they differ in their approach and application. In this article, we will explore the attributes of experimental probability and theoretical probability, highlighting their similarities and differences.

Experimental Probability

Experimental probability, also known as empirical probability, is based on observations and data collected from experiments or real-life events. It involves conducting experiments or observations to determine the likelihood of an event occurring. The experimental probability of an event is calculated by dividing the number of times the event occurs by the total number of trials or observations.

One of the key attributes of experimental probability is its reliance on real-world data. By conducting experiments or observations, we can gather empirical evidence to estimate the probability of an event. This makes experimental probability particularly useful when dealing with situations where theoretical calculations may be challenging or impractical.

Another attribute of experimental probability is its subjectivity. Since it is based on observed data, the results can vary depending on the specific experiments or observations conducted. The more trials or observations we perform, the more reliable the experimental probability becomes. However, it is important to note that experimental probability is still an estimation and may not always accurately reflect the true probability of an event.

Experimental probability is often used in fields such as psychology, biology, and social sciences, where controlled experiments or observations can provide valuable insights into the likelihood of certain outcomes. For example, in a psychology study, researchers may conduct experiments to determine the probability of a specific behavior occurring in response to certain stimuli.

Theoretical Probability

Theoretical probability, also known as classical probability, is based on mathematical principles and calculations. It involves analyzing the underlying structure of a given situation or event to determine the probability of specific outcomes. Theoretical probability relies on assumptions and mathematical models to make predictions about the likelihood of events.

One of the key attributes of theoretical probability is its objectivity. Since it is based on mathematical calculations, the results are not influenced by specific experiments or observations. Theoretical probability provides a systematic and consistent approach to quantifying probabilities, making it particularly useful in situations where empirical data may be limited or unavailable.

Another attribute of theoretical probability is its precision. By using mathematical formulas and principles, we can calculate the exact probability of an event occurring. This allows for precise predictions and analysis, which can be valuable in fields such as finance, engineering, and physics.

Theoretical probability is often used in situations where the outcomes are well-defined and the underlying probabilities can be determined with certainty. For example, in a fair six-sided die, the theoretical probability of rolling a specific number is 1/6, as there are six equally likely outcomes.

While experimental probability and theoretical probability differ in their approach and application, they share some common attributes. Both concepts deal with the likelihood of events and aim to quantify probabilities. Additionally, both experimental and theoretical probabilities range from 0 to 1, where 0 represents an impossible event and 1 represents a certain event.

However, there are also notable differences between experimental and theoretical probability. Experimental probability relies on observed data, making it subjective and dependent on the specific experiments or observations conducted. On the other hand, theoretical probability is objective and based on mathematical calculations, providing a more systematic and consistent approach.

Another difference lies in the precision of the probabilities. Experimental probability provides an estimation of the likelihood of an event based on observed data, which may not always accurately reflect the true probability. Theoretical probability, on the other hand, allows for precise calculations and predictions, assuming the underlying assumptions and mathematical models are accurate.

Furthermore, experimental probability is often used in situations where real-world data is available or when conducting experiments is feasible. It is particularly useful in fields such as social sciences, where controlled experiments or observations can provide valuable insights. Theoretical probability, on the other hand, is more suitable for situations where the underlying probabilities can be determined with certainty or when empirical data is limited or unavailable. It is commonly used in fields such as mathematics, finance, and physics.

Experimental probability and theoretical probability are two important concepts in the study of probability. While both aim to quantify the likelihood of events, they differ in their approach and application. Experimental probability relies on observed data and provides an estimation of probabilities based on experiments or observations. Theoretical probability, on the other hand, is based on mathematical calculations and provides precise predictions assuming the underlying assumptions and models are accurate.

Understanding the attributes of experimental and theoretical probability is crucial for making informed decisions and analyzing probabilities in various fields. By recognizing the strengths and limitations of each approach, we can effectively apply probability concepts to real-world situations and enhance our understanding of uncertain events.

Comparisons may contain inaccurate information about people, places, or facts. Please report any issues.

Theoretical and experimental probability

There are two different types of probability that we often talk about: theoretical probability and experimental probability.

Theoretical probability describes how likely an event is to occur. We know that a coin is equally likely to land heads or tails, so the theoretical probability of getting heads is 1/2.

Experimental probability describes how frequently an event actually occurred in an experiment. So if you tossed a coin 20 times and got heads 8 times, the experimental probability of getting heads would be 8/20, which is the same as 2/5, or 0.4, or 40%.

The theoretical probability of an event will always be the same, but the experimental probability is affected by chance, so it can be different for different experiments. The more trials you carry out (for example, the more times you toss the coin), the closer the experimental probability is likely to be to the theoretical probability.

Maybe you could try tossing a coin 20 times to see how close your experimental probability is to the theoretical probability.

Using Theoretical and Experimental Probability to Make Predictions

Introduction.

When you are working with the probability that an event could happen, that is called theoretical probability .

For example, when rolling a typical six-sided number cube, there is only one "6" on the cube and an equal chance of any number landing face up.

For example, if a dart is randomly thrown at a dartboard 40 times, data can be collected based on the numbered sector in which the dart lands.




–10
–15
–20

Working with both theoretical and experimental probability is important, since frequently in the real world, what should happen, and what really does happen, are quite different!

Comparing Theoretical and Experimental Probabilities

In this section, you will use a spinner simulator to conduct several "spins" in order to compare theoretical probabilities with experimental probabilities.

Use the "Spinner" below which has three equal-sized sections that are blue, red, and yellow.

Spin the spinner a total of 30 times
Record your results in the table.

Comparing Theoretical and Experimental Probabilities

Assume that the areas of each sector of the spinner are the same.

What is the theoretical probability of the spinner landing on each color?
Based on the theoretical probability of the spinner landing on each color, how many spins out of 30 spins should land on each color?
Did the actual results of the spins match the results predicted by theoretical probability? Why do you think that is the case?

Repeat the process with the same spinner for a total of 100 spins.

Based on the theoretical probability of the spinner landing on each color, how many spins out of 100 spins should land on each color?

Using Probabilities to Make Predictions

Because theoretical and experimental probabilities are ratios, we can use proportions with probabilities to make predictions.

The results of a random survey of 8th grade students at Summit Middle School showed that 32 out of 70 students prefer listening to country music. If there are 490 students in the 8th grade at Summit Middle School, approximately how many students would be expected to prefer listening to country music?

Solve the proportion that you set up in order to determine the answer to the original question, How many students would be expected to prefer listening to country music?

1. The probability that one wristband is defective is 1 20 \frac{1}{20} . In a case of 840 wristbands, how many would be defective?

2. Benita observed that 28 out of 55 patrons at the public library used the computer. The librarian told Benita that 660 patrons enter the library throughout the day. Based on Benita's observations, how many patrons during the day would be expected to use the computer?

3. Clayton selected number tiles out of a bag. He recorded his results in the table.

Based on these results, if Clayton draws 100 tiles, how many odd numbers should he expect to draw?

You compared two different types of probability: theoretical probability and experimental probability.

Theoretical probability is the ratio of the number of favorable outcomes to the number of total possible outcomes. Experimental probability is the ratio of the number of times an event occurs to the total number of trials.

In other words, theoretical probability is a ratio that describes what should happen, but experimental probability is a ratio that describes what actually happened.

Differences between Theoretical and Experimental Probabilities You can use theoretical and experimental probabilities to distinguish between the likelihood of something happening from a purely theoretical perspective and the chances of something happening based on actual results.

Spinner with three yellow sections, two red sections and one blue section

Making Predictions You can also use theoretical and experimental probabilities to make predictions from situations by treating them like proportion problems.

The results of a random survey of 8th grade students at Jamison Middle School showed that 15 out of 45 students like eating lunch at 10:30 a.m. If there are 330 students in the 8th grade at Jamison Middle School, approximately how many students would be expected to prefer eating lunch at 10:30 a.m.?

Multiply by 7 and one-third. 15 over 45 equals 110 over 330.

110 students would like eating lunch at 10:30 a.m.

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Difference Between Experimental and Theoretical Probability

• Categorized under Mathematics & Statistics , Science | Difference Between Experimental and Theoretical Probability

Let’s admit that not all people love math. We always think that geek people only love math plus science. Computations and formulas can always mess up our exams thus flunking is inevitable.

In statistics, computations are not only required but you are also required to interpret the data you have computed. Statistics can be a fun subject depending on your teacher or professor. He or she can teach the subject with ease. If the teacher is a terror, it will be more difficult for you to learn.

One component of statistics is the understanding of probability. Probability can be summed up to one word. The word is chance. Probability can be used in social sciences subjects such as economics, sociology, plus in behavioral sciences, and medicine.

Two components of probability are experimental probability and theoretical probability. Both have major differences, obviously, as implied by the name.

With experimental probability, the person is interested in finding the ratio of the outcome to the number of attempts or trial. For example, Brad Pitt flipped a coin five times. He got three heads and two tails on those five attempts. If asked what the experimental probability of getting heads is, the answer of Brad Pitt should be three out of five.

On the other hand, with theoretical probability, the person is interested in the ratio of the wanted or favorable outcome in conjunction with the possible outcome. This is written as a ratio (e.g. 1:3 or read as 1 is to 3). For example, Angelina Jolie has put ten pieces of chocolate in a jar with the following numbers: five white chocolates, three dark chocolates, and two hazelnut chocolates. Since her favorite chocolate is the hazelnut chocolate, what is the theoretical probability that she will get a hazelnut chocolate? The answer is 2:10 or two hazelnut chocolates over ten chocolates. In simplest form it is one is to five.

Experimental probability is frequently used in medical and scientific research. It can also be used in socioeconomic research. Theoretical probability is also used in certain research and businesses.

1.Experimental probability measures the ratio of outcome vs. attempt while theoretical probability measures the favorable or wanted outcome vs. the possible outcome. 2.Experimental probability is widely used in experimental research while theoretical probability is widely used in businesses.

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$explain the difference between theoretical and experimental values$

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Experimental and Theoretical Probability

Probability is a branch of math that studies the chance or likelihood of an event occurring. There are two types of prob ability for a particular event: experimental probability and theoretical probability. Learn the difference between the two types of probabilities and the steps involved in their calculation. ...Read More Read Less

Experimental and Theoretical Probability in Math

$explain the difference between theoretical and experimental values$

What is Probability?

Experimental Probability
Theoretical Probability
Solved Examples
Frequently Asked Questions

Th e chance of a happening is named as the probability of the event happening. It tells us how likely an occasion is going to happen; it doesn’t tell us what’s happening. There is a fair chance of it happening (happening/not happening). They’ll be written as decimals or fractions . The probability of occurrence A is below.

P (A) = $\frac{\text{Number of favorable outcomes}}{\text{Number of total possible outcomes}}$

$casino$

Following are two varieties of probability:

Experimental probability
Theoretical probability

What is Experimental Probability

Definition : Probability that’s supported by repeated trials of an experiment is named as experimental probability.

P (event) = $\frac{\text{Number of times that event occurs}}{\text{Total number of trails}}$

Example: The table shows the results of spinning a penny 62 times. What’s the probability of spinning heads?

23	39

$coin$

Solution: Heads were spun 23 times in a total of 23 + 39 = 62 spins.

P (heads) = $\frac{\text{23}}{\text{69}}$ = 0.37 or 37.09 %

What is Theoretical Probability

Experimental vs Theoretical Probability: Difference and Comparison

Sharing is caring!

An experiment can have several possibilities while listing many circumstances, although there are two types of probabilities – Experimental and Theoretical Probability. The term probability is a common term in maths.

Probability lists out events to observe what is likely to happen in an experiment without or after performing an experiment. In other words, it is the prospect of an event happening.

Key Takeaways Experimental probability is based on the observed outcomes of an experiment, while theoretical probability relies on mathematical analysis. Experimental probability can change with each trial, whereas theoretical probability remains constant. Theoretical probability assumes a fair and unbiased sample space, while experimental probability depends on an experiment’s specific conditions and results.

Experimental vs Theoretical Probability

Experimental probability is a probably based on possible outcomes arrived after gathering information through experiments. The result of the experiment gives the chances of an event happening. Theoretical probability is the possibility of an event happening without an experiment, so it is assumption-based.

Experimental probability is a probability based on possible outcomes after gathering information by experimenting. In other words, the chances of happening an event by gathering information or collecting data by experimenting.

Its foundation is on what happened during an event. And the formula is the ratio of possible outcomes of a specific event to the total number of trials.

Theoretical probability is the possibility of an event happening without experimenting. Also, its foundation is assumption-based. In this probability, the experiment does not follow.

However, assumptions are taken into consideration to find a certain probability of an event. And the formula is the ratio of the number of suitable outcomes to the number of plausible outcomes.

Comparison Table

Parameters of comparison	Experimental Probability	Theoretical Probability
Definition	The possibility of a specific event happening with actually performing an experiment is experimental probability.	The possibility of a specific event based on an assumption without performing an experiment is theoretical probability.
Experiment	In experimental probability, an experiment does occur.	In theoretical probability, an experiment does not occur.
Data	In experimental probability, the data is collected by experimenting repeatedly.	In theoretical probability, the data is collected by considering every possible outcome that has a chance to happen during an experiment without actually performing it.
Basis of probability	The probability in experimental probability is based on facts and data.	The probability in theoretical probability is based on assumption.
Consideration of outcomes	The experimental probability considers outcomes gathered through experimenting.	Theoretical probability considers outcomes that are likely to happen.
Approach	In experimental probability, its approach is based on what has happened.	In the theoretical experiment, its approach is based on what would happen considered possible outcomes.
Reliability	Experimental probability is reliable in batting averages, shooting percentages, and other similar data from sports; predicting the weather, sales figure of a movie or series; polls and surveys that collect opinions; and historical data.	Theoretical probability is reliable based on a physical relationship, where objects involved can be seen easily, are measurable, and don’t change over time.
Formula	The formula of experimental probability is the ratio of possible outcomes of a specific event to the total number of trials.	The formula of theoretical probability is the ratio of the number of suitable outcomes to the number of plausible outcomes.

What is Experimental Probability?

Experimental Probability is the probability of the occurrence of a specific event based on an experiment. It is also known as empirical probability.

Experimental probability is based on outcomes gathered by repeatedly experimenting. Moreover, it focuses on what happened during an experiment rather than what would happen.

Some specific outcomes are gathered before determining a certain event’s probability. Besides, an experiment is conducted repeatedly to collect desired outcomes.

Its basic approach differs from theoretical probability, although both find probability.

Probability based on data and experiment results is reliable as it’s more likely to happen than assumption-based probability.

However, both have their pros and cons. A probability is just a prediction of what is likely to happen in the future. So, it may result in an unexpected outcome.

Although, an experimental-based probability is more likely to happen due to the greater number of outcomes that lead closer to happening an event.

As we already know, that experimental probability is more reliable.

But, many factors affect the results of an event in many situations, such as batting averages, shooting percentage, and other similar data from sports; predicting the weather, sales figure of a movie or series; polls and surveys that collect opinions; and historical data.

Coming to the formula the formula of experimental probability is the ratio of possible outcomes of a specific event to the total number of trials.

What is Theoretical Probability?

Theoretical Probability is the probability of the possibility of a specific event based on an assumption without actually experimenting. It is the theory behind probability.

For theoretical probability, knowing about an event is necessary rather than experimenting. The chances of happening a specific event are considered rather than actual outcomes.

Moreover, it predicts what will happen in the future based on the possibility of an event.

It accounts for favourable outcomes for further prediction of the possibility of an event. Rather than relying on data and experiment results, it depends on assumed data. The approaches of both probabilities are different from each other.

Its approach is to predict the outcomes without actually performing an event.

Moreover, it is not considered as reliable as an experimental probability because it does not acknowledge facts and perform an experiment. Although, both probabilities can be proven wrong as other factors affect situations and change the result at last.

However, it is considered reliable in certain situations, such as a physical relationship based on theoretical probability where the object involved in an event can be seen, measurable, and does not change over time.

It includes coin flippers, spinners, several coins, etc.

Lastly, the fourth formula is the ratio of the number of suitable outcomes to the number of plausible outcomes.

Main Differences Between Experimental and Theoretical Probability

Probability is the chance of an event happening with or without experimenting. Its ideal approach is to predict what will happen in the future. But, seldom do certain factors affect an event, and the result of an experiment can change drastically.

There are two types of probabilities – Experimental and Theoretical probability. Both are reliable, yet in different circumstances.

Experimental probability is the possibility of a particular event happening with experimenting. Meanwhile, the theoretical probability is the possibility of a particular
In experimental probability, an experiment is performed. While the theoretical probability, an experiment does not.
In experimental probability, the data is gathered by experimenting repeatedly. While in theoretical probability, the data is collected by considering every possible outcome that has a chance to happen during an experiment without actually performing it.
In experimental probability, outcomes gathered through the experiment are considered for finding the possibility of an event. Meanwhile, theoretical probability considers outcomes that are likely to happen.
In experimental probability, its approach is based on what has happened, while, in the theoretical experiment, its approach is based on what would happen considered possible outcomes.
Experimental probability is reliable in batting averages, shooting percentages, and other similar data from sports; predicting the weather, sales figure of a movie or series; polls and surveys that collect opinions; and historical data. While the theoretical probability is reliable in a kind of probability based on a physical relationship where objects involved can be seen, measurable, and doesn’t change over time.
The experimental probability it’s based on data and facts. In contrast, the theoretical probability it’s based on assumption.
The formula of experimental probability is the ratio of possible outcomes of a specific event to the total number of trials. In comparison, the formula of theoretical probability is the ratio of the number of suitable outcomes to the number of plausible outcomes.
https://link.springer.com/content/pdf/10.1007/s11858-012-0469-z.pdf
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What is the Difference Between Theoretical and Experimental Probability?

Answer: theoretical probability relies on mathematical analysis, using the ratio of favorable outcomes to possible outcomes, whereas experimental probability is derived from observed outcomes in real-world trials., what is theoretical probability.

Theoretical probability is based on mathematical analysis and relies on the assumption of equally likely outcomes in a sample space. It is calculated using the following formula:

P ( E ) = Number of Favorable Outcomes/ Total Number of Possible Outcome

P ( E ) is the probability of event E.
The number of favorable outcomes is determined through mathematical reasoning.

What is Experimental Probability?

Experimental probability, on the other hand, is derived from actual observations or experiments. It involves conducting trials or experiments and recording the outcomes to determine the probability. The formula for experimental probability is:

P ( E ) = Number of Favorable Outcomes in Experiment /Total Number of Trials or Experiments

P ( E ) is the experimental probability of event E.
The number of favorable outcomes is observed through experimentation.

Difference Between Theoretical and Experimental Probability

The following table gives the tabular difference between Theoretical and Experimental Probability:

Feature	Theoretical Probability	Experimental Probability
Definition	Based on mathematical analysis and reasoning.	Based on observations and empirical data.
Calculation	Calculated using mathematical formulas.	Determined by conducting experiments or trials and observing outcomes.
Prediction	Provides an idealized prediction of probability.	Represents a real-world approximation of probability.
Formula	?(?)=Number of favorable outcomesTotal number of possible outcomes ( )=Total number of possible outcomesNumber of favorable outcomes	?(?)=Number of times event E occurredTotal number of trials ( )=Total number of trialsNumber of times event E occurred
Example	Flipping a fair coin: Theoretical probability of getting heads is 1221.	Rolling a fair six-sided die: Experimental probability of getting a 5 after 100 rolls is 0.17.
Application	Commonly used in theoretical mathematics and probability theory.	Commonly used in experimental sciences and real-world situations where outcomes can be observed.
Assumptions	Assumes all outcomes are equally likely.	May involve assumptions about randomness and the conditions of the experiments.
Accuracy	Perfectly accurate under ideal conditions.	May be subject to errors due to limitations in sample size or biases in the experiment.

Related Resources:

Probability Theory Experimental Probability Empirical Probability

FAQs on Difference Between Theoretical and Experimental Probability

How do theoretical and experimental probabilities compare.

Theoretical probabilities are based on expected outcomes under ideal conditions, while experimental probabilities are based on actual data. They often converge with a large number of trials, but they may differ due to randomness and sample size.

Why might experimental probability differ from theoretical probability?

Differences can arise due to small sample sizes, randomness, experimental errors, or biases in the experiment. As the number of trials increases, experimental probability tends to approximate theoretical probability more closely.

How can you use experimental probability to verify theoretical probability?

By conducting a large number of trials and comparing the experimental probability to the theoretical probability, you can validate the theoretical model. For accurate verification, a sufficiently large sample size is necessary.

Can theoretical probability be used if experimental data is not available?

Yes, theoretical probability can be used to predict the likelihood of events when experimental data is not available. It relies on known mathematical principles and assumptions.

How can you improve the accuracy of experimental probability?

Increase the number of trials to reduce the impact of random variation and obtain a more accurate estimate of the probability. Ensuring proper experimental design and minimizing biases also help improve accuracy.

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Percent Error / Percent Difference: Definition, Examples

Statistics Definitions >

Percent Error
Percent Difference

What is Percent Error?

Percent errors tells you how big your errors are when you measure something in an experiment. Smaller values mean that you are close to the accepted or real value. For example, a 1% error means that you got very close to the accepted value, while 45% means that you were quite a long way off from the true value. Measurement errors are mostly unavoidable: equipment can be imprecise, hands can shake, or your instruments just might not have the capability to measure accurately. Percent error will let you know how badly these unavoidable errors affected your results.

The formula is:

PE = (|accepted value – experimental value| \ accepted value) x 100%. Example question: The accepted distance to the moon is 238,855 miles.* You measure the distance as 249,200 miles. What is the percent error? Solution: Step 1: Insert your data into the formula: PE = (|accepted value – experimental value| \ accepted value) x 100% = ((|238,855 miles – 249,200|) \ 238,855 miles) x 100% = Step 2: Solve: (10345 \ 238,855 miles) x 100% = 0.0433 * 100% = 4.33%.

*That’s the average distance, but let’s assume it’s the distance on the day you’re taking the measurement!

Note : in some sciences, the absolute value sign is sometimes (but not always) omitted. You may want to refer to your textbook to see if the author is omitting the absolute value sign. If you aren’t sure, the most common form is with the absolute value sign.

Alternate Wording

Accepted value is sometimes called the “true” value or “theoretical” value, so you might see the formula written in slightly different ways:

PE = (|true value – experimental value| \ true value) x 100%.
PE = (|theoretical value – experimental value| \ theoretical value) x 100%.

All three versions of the formula mean the exact same thing — it’s just different wording.

Alternative Definition of Percent Error using Relative Error

The percentage error is sometimes reported as being 100% times the relative error . Be careful though, because there are actually two types of relative error : one for precision and one for accuracy (not sure of the difference between the two? See: Accuracy and Precision ). The definition “100% times the relative error” is only true if you are using the “accuracy” version of relative error:

RE accuracy = (Absolute error / “True” value) * 100%.

The definition does not work if you’re using the RE for precision:

RE precision = absolute error / measurement being taken.

What is Percent Difference?

E 1 is the first experimental measurement.
E 2 is the second experimental measurement.

Example question: You make two measurements in an experiment of 21 mL and 22 mL. What is the percent difference?

Comparing Lattice Enthalpies ( AQA A Level Chemistry )

Revision note.

Chemistry Lead

Comparing Lattice Enthalpies

How accurate are lattice enthalpies.

the geometry of the ionic solid
the charge on the ions
the distance between the ions
This has been calculated for a number of ionic solids and allows a comparison between theoretical lattice enthalpies and experimental lattice enthalpies obtained from Born-Haber cycles

Table comparing theoretical and experimental lattice enthalpies

Table of theoretical and experimental lattice enthalpies, downloadable AS & A Level Chemistry revision notes

You can see from the table that there is quite close agreement between the two values for the lattice enthalpy of sodium chloride
The calculation of the theoretical value is based on an assumption that the substance is a highly ionic compound with only electrostatic attraction between cations and anions

Ionic Lattice Structure - NaCl, downloadable IGCSE & GCSE Chemistry revision notes

The ionic model for sodium chloride

However, the difference between theoretical and experimental lattice enthalpy increases for zinc sulfide
This suggests that the bonding is not purely ionic and some covalent character is present
Zinc is a smaller ion with a greater charge (+2) than sodium(+1)
Zinc ions attract electron density towards themselves, distorting the electron cloud and making the bonding slightly covalent
Sulfide ions are larger ions than chloride ions(-1) with a greater negative charge (-2)
The electron cloud around sulfide ions is more easily distorted than in chloride ions leading to further covalent character

Covalent Character in ionic compounds, downloadable AS & A Level Chemistry revision notes

As you move left to right across the period table the lattices become less ionic and more covalent l eading to a discrepancy in the lattice enthalpy values
The result of these analyses provides strong evidence that supports the ionic model for some compounds like sodium chloride

The distortion of the electron clouds is known as polarisation and illustrates that bonding is not either pure ionic or covalent, but rather a continuum between the two extremes.

Factors affecting lattice enthalpy

The two key factors which affect lattice energy, Δ H latt ꝋ , are the charge and radius of the ions that make up the crystalline lattice

Ionic radius

The lattice energy becomes less exothermic as the ionic radius of the ions increases
This is because the charge on the ions is more spread out over the ion when the ions are larger
The attraction between ions is between the centres of the ions involved, so the bigger the ions the bigger the distance between the centre of the ions
Therefore, the electrostatic forces of attraction between the oppositely charged ions in the lattice are weaker
Since both compounds contain a fluoride (F - ) ion, the difference in lattice energy must be due to the caesium (Cs + ) ion in CsF and potassium (K + ) ion in KF
Potassium is a Group 1 and Period 4 element
Caesium is a Group 1 and Period 6 element
This means that the Cs + ion is larger than the K + ion
There are weaker electrostatic forces of attraction between the Cs + and F - ions compared to K + and F - ions
As a result, the lattice energy of CsF is less exothermic than that of KF

The lattice energies get less exothermic as the ionic radius of the ions increases

Ionic charge

The lattice energy gets more exothermic as the ionic charge of the ions increases
The greater the ionic charge, the higher the charge density
This results in stronger electrostatic attraction between the oppositely charged ions in the lattice
As a result, the lattice energy is more exothermic
Calcium oxide is an ionic compound which consists of calcium (Ca 2+ ) and oxide (O 2- ) ions
Potassium chloride is formed from potassium (K + ) and chloride (C l - ) ions
The ions in calcium oxide have a greater ionic charge than the ions in potassium chloride
This means that the electrostatic forces of attraction are stronger between the Ca 2+ and O 2- compared to the forces between K + and C l -
Therefore, the lattice energy of calcium oxide is more exothermic, as more energy is released upon its formation from its gaseous ions
Ca 2+ and O 2- are also smaller ions than K + and C l - , so this also adds to the value for the lattice energy being more exothermic

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Featured Answers

What is the difference between Accepted Value vs. Experimental Value?

#"Error" = "|experimental value - accepted value|"#

The difference is usually expressed as percent error .

#"% error" = "|experimental value - accepted value|"/"experimental value" × 100 %#

For example, suppose that you did an experiment to determine the boiling point of water and got a value of 99.3 °C.

Your experimental value is 99.3 °C.

The theoretical value is 100.0 °C.

The experimental error is #"|99.3 °C - 100.0 °C| = 0.7 °C"#

The percent error is #"|99.3 °C - 100.0 °C|"/"100.0 °C" = "0.7 °C"/"100.0 °C" × 100% = 0.7 %#

Impact of this question

Lattice enthalpy and lattice energy are commonly used as if they mean exactly the same thing - you will often find both terms used within the same textbook article or web site, including on university sites.

In fact, there is a difference between them which relates to the conditions under which they are calculated. However, the difference is small, and negligible compared with the differing values for lattice enthalpy that you will find from different data sources.

Unless you go on to do chemistry at degree level, the difference between the two terms isn't likely to worry you.

While I have been writing this section, the different values for the same piece of data from different data sources has driven me crazy, because there is no easy way of knowing which is the most recent or most accurate data.

In the Born-Haber cycles below, I have used numbers which give a consistent answer, but please don't assume that they are necessarily the most accurate ones. If you are doing a course for 16 - 18 year olds, none of this really matters - you just use the numbers you are given.

If you use my , you will find a slightly different set of numbers. These came from the Chemistry Data Book edited by Stark and Wallace, published by John Murray. Values from this now fairly old book often differ slightly from more recent sources.

Don't worry about this. It doesn't affect the principles in any way. Just don't assume that any bit of data you are given (even by me) is necessarily "right"!

There are two different ways of defining lattice enthalpy which directly contradict each other, and you will find both in common use. In fact, there is a simple way of sorting this out, but many sources don't use it.

I will explain how you can do this in a moment, but first let's look at how the problem arises.

Lattice enthalpy is a measure of the strength of the forces between the ions in an ionic solid. The greater the lattice enthalpy, the stronger the forces.

Those forces are only completely broken when the ions are present as gaseous ions, scattered so far apart that there is negligible attraction between them. You can show this on a simple enthalpy diagram.

For sodium chloride, the solid is more stable than the gaseous ions by 787 kJ mol , and that is a measure of the strength of the attractions between the ions in the solid. Remember that energy (in this case heat energy) is given out when bonds are made, and is needed to break bonds.

So lattice enthalpy could be described in either of two ways.

In the sodium chloride case, that would be -787 kJ mol .

In the sodium chloride case, that would be +787 kJ mol .

Both refer to the same enthalpy diagram, but one looks at it from the point of view of making the lattice, and the other from the point of view of breaking it up.

Unfortunately, both of these are often described as "lattice enthalpy".

This is an absurdly confusing situation which is easily resolved. I suggest that you never use the term "lattice enthalpy" without qualifying it.

For NaCl, the lattice dissociation enthalpy is +787 kJ mol .

For NaCl, the lattice formation enthalpy is -787 kJ mol .

That immediately removes any possibility of confusion.

So . . .

Note: Find out which of these versions your syllabus is likely to want you to know (even if they just call it "lattice enthalpy") and concentrate on that one, but be aware of the confusion!

Incidentally, if you are ever uncertain about which version is being used, you can tell from the sign of the enthalpy change being discussed. If the sign is positive, for example, it must refer to breaking bonds, and therefore to a lattice dissociation enthalpy.

Factors affecting lattice enthalpy

The two main factors affecting lattice enthalpy are the charges on the ions and the ionic radii (which affects the distance between the ions).

The charges on the ions

Sodium chloride and magnesium oxide have exactly the same arrangements of ions in the crystal lattice, but the lattice enthalpies are very different.

Note: In this diagram, and similar diagrams below, I am not interested in whether the lattice enthalpy is defined as a positive or a negative number - I am just interested in their relative sizes. Strictly speaking, because I haven't added a sign to the vertical axis, the values are for lattice dissociation enthalpies. If you prefer lattice formation enthalpies, just mentally put a negative sign in front of each number.

You can see that the lattice enthalpy of magnesium oxide is much greater than that of sodium chloride. That's because in magnesium oxide, 2+ ions are attracting 2- ions; in sodium chloride, the attraction is only between 1+ and 1- ions.

The radius of the ions

The lattice enthalpy of magnesium oxide is also increased relative to sodium chloride because magnesium ions are smaller than sodium ions, and oxide ions are smaller than chloride ions.

That means that the ions are closer together in the lattice, and that increases the strength of the attractions.

You can also see this effect of ion size on lattice enthalpy as you go down a Group in the Periodic Table.

For example, as you go down Group 7 of the Periodic Table from fluorine to iodine, you would expect the lattice enthalpies of their sodium salts to fall as the negative ions get bigger - and that is the case:

Attractions are governed by the distances between the centres of the oppositely charged ions, and that distance is obviously greater as the negative ion gets bigger.

And you can see exactly the same effect as you go down Group 1. The next bar chart shows the lattice enthalpies of the Group 1 chlorides.

Note: To save anyone the bother of getting in touch with me to point it out, it's not strictly fair to include caesium chloride in this list. Caesium chloride has a different packing arrangement of ions in its crystal, and that has a small effect on the lattice enthalpy. The effect is small enough that it doesn't actually affect the trend.

Calculating lattice enthalpy

It is impossible to measure the enthalpy change starting from a solid crystal and converting it into its scattered gaseous ions. It is even more difficult to imagine how you could do the reverse - start with scattered gaseous ions and measure the enthalpy change when these convert to a solid crystal.

Instead, lattice enthalpies always have to be calculated, and there are two entirely different ways in which this can be done.

You can can use a Hess's Law cycle (in this case called a Born-Haber cycle) involving enthalpy changes which can be measured. Lattice enthalpies calculated in this way are described as experimental values.

Or you can do physics-style calculations working out how much energy would be released, for example, when ions considered as point charges come together to make a lattice. These are described as theoretical values. In fact, in this case, what you are actually calculating are properly described as lattice energies .

Note: If you aren't confident about Hess's Law cycles , it is essential that you follow this link before you go on.

Experimental values - Born-Haber cycles

Standard atomisation enthalpies

Before we start talking about Born-Haber cycles, there is an extra term which we need to define. That is atomisation enthalpy , ΔH° a .

You are always going to have to supply energy to break an element into its separate gaseous atoms.

All of the following equations represent changes involving atomisation enthalpy:

Notice particularly that the "mol -1 " is per mole of atoms formed - NOT per mole of element that you start with. You will quite commonly have to write fractions into the left-hand side of the equation. Getting this wrong is a common mistake.

Born-Haber cycles

I am going to start by drawing a Born-Haber cycle for sodium chloride, and then talk it through carefully afterwards. You will see that I have arbitrarily decided to draw this for lattice formation enthalpy. If you wanted to draw it for lattice dissociation enthalpy, the red arrow would be reversed - pointing upwards.

Focus to start with on the higher of the two thicker horizontal lines. We are starting here with the elements sodium and chlorine in their standard states. Notice that we only need half a mole of chlorine gas in order to end up with 1 mole of NaCl.

The arrow pointing down from this to the lower thick line represents the enthalpy change of formation of sodium chloride.

The Born-Haber cycle now imagines this formation of sodium chloride as happening in a whole set of small changes, most of which we know the enthalpy changes for - except, of course, for the lattice enthalpy that we want to calculate.

The +107 is the atomisation enthalpy of sodium. We have to produce gaseous atoms so that we can use the next stage in the cycle.

The +496 is the first ionisation energy of sodium. Remember that first ionisation energies go from gaseous atoms to gaseous singly charged positive ions.

The +122 is the atomisation enthalpy of chlorine. Again, we have to produce gaseous atoms so that we can use the next stage in the cycle.

The -349 is the first electron affinity of chlorine. Remember that first electron affinities go from gaseous atoms to gaseous singly charged negative ions.

And finally, we have the positive and negative gaseous ions that we can convert into the solid sodium chloride using the lattice formation enthalpy.

Note: If you have forgotten about ionisation energies or electron affinities follow these links before you go on.

Now we can use Hess's Law and find two different routes around the diagram which we can equate.

As I have drawn it, the two routes are obvious. The diagram is set up to provide two different routes between the thick lines.

So, here is the cycle again, with the calculation directly underneath it . . .

-411 = +107 + 496 + 122 - 349 + LE

LE = -411 - 107 - 496 - 122 + 349

Note: Notice that in the calculation, we aren't making any assumptions about the sign of the lattice enthalpy (despite the fact that it is obviously negative because the arrow is pointing downwards). In the first line of the calculation, I have just written "+ LE", and have left it to the calculation to work out that it is a negative answer.

How would this be different if you had drawn a lattice dissociation enthalpy in your diagram? (Perhaps because that is what your syllabus wants.)

Your diagram would now look like this:

The only difference in the diagram is the direction the lattice enthalpy arrow is pointing. It does, of course, mean that you have to find two new routes. You can't use the original one, because that would go against the flow of the lattice enthalpy arrow.

This time both routes would start from the elements in their standard states, and finish at the gaseous ions.

-411 + LE = +107 + 496 + 122 - 349

LE = +107 + 496 + 122 - 349 + 411

LE = +787 kJ mol -1

Once again, the cycle sorts out the sign of the lattice enthalpy for you.

Note: You will find more examples of calculations involving Born-Haber cycles in my chemistry calculations book . This includes rather more complicated cycles involving, for example, oxides.

If you compare the figures in the book with the figures for NaCl above, you will find slight differences - the main culprit being the electron affinity of chlorine, although there are other small differences as well. Don't worry about this - the values in the book come from an older data source. In an exam, you will just use the values you are given, so it isn't a problem.

Theoretical values for lattice energy

Let's assume that a compound is fully ionic. Let's also assume that the ions are point charges - in other words that the charge is concentrated at the centre of the ion. By doing physics-style calculations, it is possible to calculate a theoretical value for what you would expect the lattice energy to be.

And no - I am not being careless about this! Calculations of this sort end up with values of lattice energy , and not lattice enthalpy . If you know how to do it, you can then fairly easily convert between the two.

There are several different equations, of various degrees of complication, for calculating lattice energy in this way. You won't be expected to be able to do these calculations at this level, but you might be expected to comment on the results of them.

There are two possibilities:

There is reasonable agreement between the experimental value (calculated from a Born-Haber cycle) and the theoretical value.

Sodium chloride is a case like this - the theoretical and experimental values agree to within a few percent. That means that for sodium chloride, the assumptions about the solid being ionic are fairly good.

The experimental and theoretical values don't agree.

A commonly quoted example of this is silver chloride, AgCl. Depending on where you get your data from, the theoretical value for lattice enthalpy for AgCl is anywhere from about 50 to 150 kJ mol -1 less than the value that comes from a Born-Haber cycle.

In other words, treating the AgCl as 100% ionic underestimates its lattice enthalpy by quite a lot.

The explanation is that silver chloride actually has a significant amount of covalent bonding between the silver and the chlorine, because there isn't enough electronegativity difference between the two to allow for complete transfer of an electron from the silver to the chlorine.

Comparing experimental (Born-Haber cycle) and theoretical values for lattice enthalpy is a good way of judging how purely ionic a crystal is.

Note: If you have forgotten about electronegativity it might pay you to revise it now by following this link.

Why is magnesium chloride MgCl 2 ?

This section may well go beyond what your syllabus requires. Before you spend time on it, check your syllabus (and past exam papers as well if possible) to make sure.

The question arises as to why, from an energetics point of view, magnesium chloride is MgCl 2 rather than MgCl or MgCl 3 (or any other formula you might like to choose).

It turns out that MgCl 2 is the formula of the compound which has the most negative enthalpy change of formation - in other words, it is the most stable one relative to the elements magnesium and chlorine.

Let's look at this in terms of Born-Haber cycles.

In the cycles this time, we are interested in working out what the enthalpy change of formation would be for the imaginary compounds MgCl and MgCl 3 .

That means that we will have to use theoretical values of their lattice enthalpies. We can't use experimental ones, because these compounds obviously don't exist!

I'm taking theoretical values for lattice enthalpies for these compounds that I found on the web. I can't confirm these, but all the other values used by that source were accurate. The exact values don't matter too much anyway, because the results are so dramatically clear-cut.

We will start with the compound MgCl, because that cycle is just like the NaCl one we have already looked at.

The Born-Haber cycle for MgCl

Find two routes around this without going against the flow of any arrows. That's easy:

ΔH f = +148 + 738 + 122 - 349 - 753

ΔH f = -94 kJ mol -1

So the compound MgCl is definitely energetically more stable than its elements.

I have drawn this cycle very roughly to scale, but that is going to become more and more difficult as we look at the other two possible formulae. So I am going to rewrite it as a table.

You can see from the diagram that the enthalpy change of formation can be found just by adding up all the other numbers in the cycle, and we can do this just as well in a table.

The Born-Haber cycle for MgCl 2

The equation for the enthalpy change of formation this time is

So how does that change the numbers in the Born-Haber cycle?

You need to add in the second ionisation energy of magnesium, because you are making a 2+ ion.

You need to multiply the atomisation enthalpy of chlorine by 2, because you need 2 moles of gaseous chlorine atoms.

You need to multiply the electron affinity of chlorine by 2, because you are making 2 moles of chloride ions.

You obviously need a different value for lattice enthalpy.

You can see that much more energy is released when you make MgCl 2 than when you make MgCl. Why is that?

You need to put in more energy to ionise the magnesium to give a 2+ ion, but a lot more energy is released as lattice enthalpy. That is because there are stronger ionic attractions between 1- ions and 2+ ions than between the 1- and 1+ ions in MgCl.

So what about MgCl 3 ? The lattice energy here would be even greater.

The Born-Haber cycle for MgCl 3

So how does that change the numbers in the Born-Haber cycle this time?

You need to add in the third ionisation energy of magnesium, because you are making a 3+ ion.

You need to multiply the atomisation enthalpy of chlorine by 3, because you need 3 moles of gaseous chlorine atoms.

You need to multiply the electron affinity of chlorine by 3, because you are making 3 moles of chloride ions.

You again need a different value for lattice enthalpy.

This time, the compound is hugely energetically unstable, both with respect to its elements, and also to other compounds that could be formed. You would need to supply nearly 4000 kJ to get 1 mole of MgCl 3 to form!

Look carefully at the reason for this. The lattice enthalpy is the highest for all these possible compounds, but it isn't high enough to make up for the very large third ionisation energy of magnesium.

Why is the third ionisation energy so big? The first two electrons to be removed from magnesium come from the 3s level. The third one comes from the 2p. That is closer to the nucleus, and lacks a layer of screening as well - and so much more energy is needed to remove it.

The 3s electrons are screened from the nucleus by the 1 level and 2 level electrons. The 2p electrons are only screened by the 1 level (plus a bit of help from the 2s electrons).

Magnesium chloride is MgCl 2 because this is the combination of magnesium and chlorine which produces the most energetically stable compound - the one with the most negative enthalpy change of formation.

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Experimental vs Observational Studies: Differences & Examples

Understanding the differences between experimental vs observational studies is crucial for interpreting findings and drawing valid conclusions. Both methodologies are used extensively in various fields, including medicine, social sciences, and environmental studies.

Researchers often use observational and experimental studies to gather comprehensive data and draw robust conclusions about their investigating phenomena.

This blog post will explore what makes these two types of studies unique, their fundamental differences, and examples to illustrate their applications.

What is an Experimental Study?

An experimental study is a research design in which the investigator actively manipulates one or more variables to observe their effect on another variable. This type of study often takes place in a controlled environment, which allows researchers to establish cause-and-effect relationships.

Key Characteristics of Experimental Studies:

Manipulation: Researchers manipulate the independent variable(s).
Control: Other variables are kept constant to isolate the effect of the independent variable.
Randomization: Subjects are randomly assigned to different groups to minimize bias.
Replication: The study can be replicated to verify results.

Types of Experimental Study

Laboratory Experiments: Conducted in a controlled environment where variables can be precisely controlled.
Field Research : These are conducted in a natural setting but still involve manipulation and control of variables.
Clinical Trials: Used in medical research and the healthcare industry to test the efficacy of new treatments or drugs.

Example of an Experimental Study:

Imagine a study to test the effectiveness of a new drug for reducing blood pressure. Researchers would:

Randomly assign participants to two groups: receiving the drug and receiving a placebo.
Ensure that participants do not know their group (double-blind procedure).
Measure blood pressure before and after the intervention.
Compare the changes in blood pressure between the two groups to determine the drug’s effectiveness.

What is an Observational Study?

An observational study is a research design in which the investigator observes subjects and measures variables without intervening or manipulating the study environment. This type of study is often used when manipulating impractical or unethical variables.

Key Characteristics of Observational Studies:

No Manipulation: Researchers do not manipulate the independent variable.
Natural Setting: Observations are made in a natural environment.
Causation Limitations: It is difficult to establish cause-and-effect relationships due to the need for more control over variables.
Descriptive: Often used to describe characteristics or outcomes.

Types of Observational Studies:

Cohort Studies : Follow a control group of people over time to observe the development of outcomes.
Case-Control Studies: Compare individuals with a specific outcome (cases) to those without (controls) to identify factors that might contribute to the outcome.
Cross-Sectional Studies : Collect data from a population at a single point to analyze the prevalence of an outcome or characteristic.

Example of an Observational Study:

Consider a study examining the relationship between smoking and lung cancer. Researchers would:

Identify a cohort of smokers and non-smokers.
Follow both groups over time to record incidences of lung cancer.
Analyze the data to observe any differences in cancer rates between smokers and non-smokers.

Difference Between Experimental vs Observational Studies

Topic	Experimental Studies	Observational Studies
Manipulation	Yes	No
Control	High control over variables	Little to no control over variables
Randomization	Yes, often, random assignment of subjects	No random assignment
Environment	Controlled or laboratory settings	Natural or real-world settings
Causation	Can establish causation	Can identify correlations, not causation
Ethics and Practicality	May involve ethical concerns and be impractical	More ethical and practical in many cases
Cost and Time	Often more expensive and time-consuming	Generally less costly and faster

Choosing Between Experimental and Observational Studies

The researchers relied on statistical analysis to interpret the results of randomized controlled trials, building upon the foundations established by prior research.

Use Experimental Studies When:

Causality is Important: If determining a cause-and-effect relationship is crucial, experimental studies are the way to go.
Variables Can Be Controlled: When you can manipulate and control the variables in a lab or controlled setting, experimental studies are suitable.
Randomization is Possible: When random assignment of subjects is feasible and ethical, experimental designs are appropriate.

Use Observational Studies When:

Ethical Concerns Exist: If manipulating variables is unethical, such as exposing individuals to harmful substances, observational studies are necessary.
Practical Constraints Apply: When experimental studies are impractical due to cost or logistics, observational studies can be a viable alternative.
Natural Settings Are Required: If studying phenomena in their natural environment is essential, observational studies are the right choice.

Strengths and Limitations

Experimental studies.

Establish Causality: Experimental studies can establish causal relationships between variables by controlling and using randomization.
Control Over Confounding Variables: The controlled environment allows researchers to minimize the influence of external variables that might skew results.
Repeatability: Experiments can often be repeated to verify results and ensure consistency.

Limitations:

Ethical Concerns: Manipulating variables may be unethical in certain situations, such as exposing individuals to harmful conditions.
Artificial Environment: The controlled setting may not reflect real-world conditions, potentially affecting the generalizability of results.
Cost and Complexity: Experimental studies can be costly and logistically complex, especially with large sample sizes.

Observational Studies

Real-World Insights: Observational studies provide valuable insights into how variables interact in natural settings.
Ethical and Practical: These studies avoid ethical concerns associated with manipulation and can be more practical regarding cost and time.
Diverse Applications: Observational studies can be used in various fields and situations where experiments are not feasible.
Lack of Causality: It’s easier to establish causation with manipulation, and results are limited to identifying correlations.
Potential for Confounding: Uncontrolled external variables may influence the results, leading to biased conclusions.
Observer Bias: Researchers may unintentionally influence outcomes through their expectations or interpretations of data.

Examples in Various Fields

Experimental Study: Clinical trials testing the effectiveness of a new drug against a placebo to determine its impact on patient recovery.
Observational Study: Studying the dietary habits of different populations to identify potential links between nutrition and disease prevalence.
Experimental Study: Conducting a lab experiment to test the effect of sleep deprivation on cognitive performance by controlling sleep hours and measuring test scores.
Observational Study: Observing social interactions in a public setting to explore natural communication patterns without intervention.

Environmental Science

Experimental Study: Testing the impact of a specific pollutant on plant growth in a controlled greenhouse setting.
Observational Study: Monitoring wildlife populations in a natural habitat to assess the effects of climate change on species distribution.

How QuestionPro Research Can Help in Experimental vs Observational Studies

Choosing between experimental and observational studies is a critical decision that can significantly impact the outcomes and interpretations of a study. QuestionPro Research offers powerful tools and features that can enhance both types of studies, giving researchers the flexibility and capability to gather, analyze, and interpret data effectively.

Enhancing Experimental Studies with QuestionPro

Experimental studies require a high degree of control over variables, randomization, and, often, repeated trials to establish causal relationships. QuestionPro excels in facilitating these requirements through several key features:

Survey Design and Distribution: With QuestionPro, researchers can design intricate surveys tailored to their experimental needs. The platform supports random assignment of participants to different groups, ensuring unbiased distribution and enhancing the study’s validity.
Data Collection and Management: Real-time data collection and management tools allow researchers to monitor responses as they come in. This is crucial for experimental studies where data collection timing and sequence can impact the results.
Advanced Analytics: QuestionPro offers robust analytical tools that can handle complex data sets, enabling researchers to conduct in-depth statistical analyses to determine the effects of the experimental interventions.

Supporting Observational Studies with QuestionPro

Observational studies involve gathering data without manipulating variables, focusing on natural settings and real-world scenarios. QuestionPro’s capabilities are well-suited for these studies as well:

Customizable Surveys: Researchers can create detailed surveys to capture a wide range of observational data. QuestionPro’s customizable templates and question types allow for flexibility in capturing nuanced information.
Mobile Data Collection: For field research, QuestionPro’s mobile app enables data collection on the go, making it easier to conduct studies in diverse settings without internet connectivity.
Longitudinal Data Tracking: Observational studies often require data collection over extended periods. QuestionPro’s platform supports longitudinal studies, allowing researchers to track changes and trends.

Experimental and observational studies are essential tools in the researcher’s toolkit. Each serves a unique purpose and offers distinct advantages and limitations. By understanding their differences, researchers can choose the most appropriate study design for their specific objectives, ensuring their findings are valid and applicable to real-world situations.

Whether establishing causality through experimental studies or exploring correlations with observational research designs, the insights gained from these methodologies continue to shape our understanding of the world around us.

Whether conducting experimental or observational studies, QuestionPro Research provides a comprehensive suite of tools that enhance research efficiency, accuracy, and depth. By leveraging its advanced features, researchers can ensure that their studies are well-designed, their data is robustly analyzed, and their conclusions are reliable and impactful.

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COMMENTS

Theoretical vs. Experimental Probability: How do they differ?
Theoretical probability is calculated using mathematical formulas, while experimental probability is based on results from experiments or surveys. In order words, theoretical probability represents how likely an event is to happen. On the other hand, experimental probability illustrates how frequently an event occurs in an experiment.
Theoretical vs. Experimental Probability
Theoretical probability is based on reasoning and mathematics. Experimental probability is based on the results of several trials or experiments. Theoretical probability is calculated by taking ...
Empirical vs Theoretical Probability
1) Empirical (experimental) probability is the probability observed in the chart above. The 8 was rolled 8 times out of 50 rolls. The empirical probability = 8/50 = 16%. 2) Theoretical probability is based upon what is expected when rolling two dice, as seen in the "sum" table at the right. The theoretical probability of rolling an 8 is 5 times ...
Theoretical Probability: Definition + Examples
The experimental probability for the dice landing on "2" can be calculated as: P(land on 2) = (lands on 2 three times) / (rolled the dice 11 times) = 3/11. How to Remember the Difference. You can remember the difference between theoretical probability and experimental probability using the following trick:
Khan Academy
When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to.
Comparing Theoretical and Experimental Probability: Key Insights
We can simply continue the experimental by flipping the coin for many more times —say, 20,000 times. When more trials are performed, the difference between experimental probability and theoretical probability will diminish. The experimental probability will gradually get closer to the value of the theoretical probability.
Theoretical Probability versus Experimental Probability
The theoretical probability is 8.3% and the experimental probability is 4%. Although the experimental probability is slightly lower, this is not a significant difference. In most experiments, the theoretical probability and experimental probability will not be equal; however, they should be relatively close. If the calculations are not close ...
Theoretical vs. Experimental Probability
Theoretical probability is what we expect to happen, where experimental probability is what actually happens when we try it out. The probability is still calculated the same way, using the number of possible ways an outcome can occur divided by the total number of outcomes. As more trials are conducted, the experimental probability generally ...
Experimental Probability vs. Theoretical Probability
Experimental probability relies on observed data, making it subjective and dependent on the specific experiments or observations conducted. On the other hand, theoretical probability is objective and based on mathematical calculations, providing a more systematic and consistent approach. Another difference lies in the precision of the ...
Theoretical and experimental probability
Experimental probability describes how frequently an event actually occurred in an experiment. So if you tossed a coin 20 times and got heads 8 times, the experimental probability of getting heads would be 8/20, which is the same as 2/5, or 0.4, or 40%. The theoretical probability of an event will always be the same, but the experimental ...
4.2: Three Types of Probability
Figure 4-4 shows a graph of experimental probabilities as n gets larger and larger. The dashed yellow line is the theoretical probability of rolling a four of 1/6 $\neq$ 0.1667. Note the x-axis is in a log scale. Note that the more times you roll the die, the closer the experimental probability gets to the theoretical probability. Figure 4-4
Using Theoretical and Experimental Probability to Make Predictions
Experimental probability is the ratio of the number of times an event occurs to the total number of trials. In other words, theoretical probability is a ratio that describes what should happen, but experimental probability is a ratio that describes what actually happened. Differences between Theoretical and Experimental Probabilities
Difference Between Experimental and Theoretical Probability
1.Experimental probability measures the ratio of outcome vs. attempt while theoretical probability measures the favorable or wanted outcome vs. the possible outcome. 2.Experimental probability is widely used in experimental research while theoretical probability is widely used in businesses. Author. Recent Posts.
Experimental and Theoretical Probability
Experimental probability is that the results of the quantity of. occurrences of a happening / the whole number of trials. Theoretical probability is that the results of the quantity. of favorable outcomes / the entire number of possible outcomes. Example: A coin is tossed 10 times.
Experimental vs Theoretical Probability
The experimental probability it's based on data and facts. In contrast, the theoretical probability it's based on assumption. The formula of experimental probability is the ratio of possible outcomes of a specific event to the total number of trials. In comparison, the formula of theoretical probability is the ratio of the number of ...
Khan Academy
Learn how to compare theoretical and experimental probability with coin flips and die rolls. Practice with interactive exercises and quizzes.
What is the Difference Between Theoretical and Experimental Probability?
School Learning. Number System - MAQ. Theoretical probability is based on mathematical calculations and predicts the likelihood of an event occurring in ideal conditions, while experimental probability is based on actual observations and results from experiments or real-world data. Learn more about probability at GeeksforGeeks.
Experimental vs Theoretical Probability
Solution: 1) Experimental probability is the probability observed during an experiment of rolling two dice. The results are in the chart above. The 8 was rolled 8 times out of 50 rolls. The experimental probability = 8/50 = 16% 2) Theoretical probability is based upon what is expected when rolling two dice, as seen in the "sum" table at the right. This table shows all of the possible sums when ...
Percent Error / Percent Difference: Definition, Examples
What is Percent Difference? Percent difference is practically the same as percent error, only instead of one "true" value and one "experimental" value, you compare two experimental values. The formula is: Where: E 1 is the first experimental measurement. E 2 is the second experimental measurement.
AQA A Level Chemistry Revision Notes 2017
This has been calculated for a number of ionic solids and allows a comparison between theoretical lattice enthalpies and experimental lattice enthalpies obtained from Born-Haber cycles. Table comparing theoretical and experimental lattice enthalpies. You can see from the table that there is quite close agreement between the two values for the ...
5: Experimental Design
Experimental design is a discipline within statistics concerned with the analysis and design of experiments. Design is intended to help research create experiments such that cause and effect can be established from tests of the hypothesis. We introduced elements of experimental design in Chapter 2.4. Here, we expand our discussion of ...
What is the difference between Accepted Value vs. Experimental Value
#"% error" = "|experimental value - accepted value|"/"experimental value" × 100 %# For example, suppose that you did an experiment to determine the boiling point of water and got a value of 99.3 °C.
LATTICE ENTHALPY (LATTICE ENERGY)
There is reasonable agreement between the experimental value (calculated from a Born-Haber cycle) and the theoretical value. Sodium chloride is a case like this - the theoretical and experimental values agree to within a few percent. That means that for sodium chloride, the assumptions about the solid being ionic are fairly good.
Experimental vs Observational Studies: Differences & Examples
Choosing between experimental and observational studies is a critical decision that can significantly impact the outcomes and interpretations of a study. QuestionPro Research offers powerful tools and features that can enhance both types of studies, giving researchers the flexibility and capability to gather, analyze, and interpret data ...

Theoretical vs. Experimental Probability: How do they differ?

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How Is Theoretical Probability Used in Real Life?

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How Is Experimental Probability Used in Real Life?

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