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Distributions

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Distributions

After we get the results of our study, it is essential to examine the data set to see what kind of distribution it has. Distributions will tell us a lot about our data.

The distribution is a visible representation of the data we have collected. Based on the shape of the distribution, we can interpret whether our data is usually distributed.

It is vital to have a normal distribution. This is because we can run the most powerful inferential (parametric) tests for data that is typically distributed. If the data is not normally distributed (skewed distribution), we can run non-parametric tests, but they are less meaningful. In this article, we will look at both normal and skewed distributions.

What is a normal distribution?

The normal distribution has a bell-curve shape. It is also known as the Gaussian distribution, named after the German mathematician Carl Gauss, who first described this distribution. The normal distribution is a probability distribution. This means that it tells us about the probability of an outcome in the real world.

Distributions Normal distribution StudySmarter

Normal distribution example, Dan Kernler, Wikimedia Commons

It is a normal distribution because we would get this form of results if we were to measure the population in terms of some trait, e.g. IQ. Most people would score about average and be in the middle of the curve. The further we get from the bell curve, the fewer people reach extreme values.

In the middle of the graph, ‘μ’ (population mean) is the mean, median and mode. Both sides of the curve are symmetrical. 50% of the values are on the left side and 50% on the right side.

From the graph, we can see that it is a normal distribution:

  • 68% of the results would deviate from the mean by one standard deviation. We have already mentioned that the normal distribution is a probability distribution. If we pick a value at random, it has a 68% probability of being between -1 and +1 standard deviation from the mean. This is usually the average.
  • 95% of the results are within two deviations of the mean. If we randomly select a value, it is 95% likely between -2 and +2 standard deviations from the mean.
  • 99.7% of the values are within three standard deviations of the mean. If we select a value at random, it is between -3 and +3 standard deviations from the mean with 99.7% probability.

What are skewed distributions?

In a skewed distribution, the data are not normally distributed. There is a large cluster at one end. The data may be positively or negatively skewed. The mode (the value that occurs most often) is still at the highest point of the curve, but the mean and median are not. The mode is still at the highest point because extreme values are not affected.

Positive skew

There is a large group on the left side with positive skewness, while the values drop off to the right.

Distributions Positive skew StudySmarter

Example of positive skew, Erika Hae, StudySmarter Originals

A positive skew would result if most participants received low scores. The mean is pulled to the right because the mean is affected by extreme scores.

There was a psychology test for your class, and most people found it very difficult. Most people scored low (mode), with higher scores dropping to the right.

You can remember what positive skew looks like when you imagine the positive, higher scores dropping off to the right.

Negative skew

There is a big cluster on the right in a negative skew, while the values decrease towards the left.

Distributions Negative skew StudySmarter

Example of negative skew, Erika Hae, StudySmarter Originals

A negative skew would be observed if most participants scored high. The mean is pulled to the left. You can recall what a negative skew looks like by imagining the negative, lower scores dropping off to the back.

You may have come across the term binomial distribution before. This is essentially a distribution where there are only two possible outcomes in a data set (think heads or tails when you flip a coin), and the probability of those two outcomes occurring is constant.

Examples of distributions scores

Suppose in a psychology test these were the results of the class:

Mean
Median
Mode
25
25.6
25.3

This would be a normal distribution because the mean, median, and mode are all approximately the same and would fall on the same point on the curve.

Let us take a look at these results:

Mean
Median
Mode
28
23
18

This would be a positive skew as most people scored low. The mode and median are below the mean.

What about these results?

Mean
Median
Mode
17
26
30

This would be a negative skew since most people scored high. The mode and median are higher here.

Usually, data is presented with the mean, median, and mode for a group of items or people in an exam. It is then your task to identify the distribution pattern. You must indicate what type of distribution it is, positive, negative, or normally distributed, and what that means.

Do the results we have given indicate anything?

If we look at our example above, the positively skewed shape indicates that the participants scored lower on average, so perhaps the test was too hard. Similarly, the negatively skewed graph shows higher scores, so maybe the test was too easy. The mode is the highest point that illustrates the distribution.

We can then suggest making the test harder or easier, depending on the distributions above.


Distributions - Key takeaways

  • A distribution is a visible representation of the data collected. By looking at the shape of the distribution, we can interpret whether our data is usually distributed.
  • The data must have a normal distribution because we can run the most powerful inferential tests (parametric) on the data.
  • The normal distribution has the shape of a bell curve. The mode, median, and mean are all at the same point in the highest part of the curve. In a population, most people would get around the mean. The further we get from the bell curve, the fewer people reach extreme values.
  • With a skewed distribution, the mode (the value that occurs most often) is still at the highest point of the curve, but the mean and median are not.
  • There is a large group on the left with positive skewness, while the values drop to the right. This would be the case if most participants scored low. There is a large group on the right with negative skewness, with scores dropping off to the left. This would be the case if most participants scored high.

Frequently Asked Questions about Distributions

The normal distribution has the shape of a bell curve. The mode, median, and mean are all at the same point in the highest part of the curve. Both sides of the curve are symmetrical: 50% of scores are on the left side and 50% on the right side. As we go out from the bell curve, fewer and fewer people get extreme scores.

This means that the data is spread evenly. Most people would get around the average score, and fewer people get extreme scores as the bell curve goes out.

Frequency distribution measures how often a value occurs in a data set. Frequency distributions are usually presented in tables.

If data is positively skewed, the test/experiment could be easier as a positive skew indicates that many people found it challenging. Conversely, the test/experiment could be harder if the data are negatively skewed. These changes should affect the results and give a normal distribution. 

Final Distributions Quiz

Question

What is a distribution?

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Answer

A distribution is a visible representation of the collected data.

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Question

Why is it important for data to be normally distributed?

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Answer

The most potent inferential tests can only be used on normally distributed data.

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Question

What shape does a normal distribution make?

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Answer

A bell curve shape.

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Question

What does it mean if we have a normal distribution?

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Answer

A normal distribution shows us the data is representative of the population. If we were to measure the population on some quality, for example, IQ, we would get this bell-curved shape of results.

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Question

Why is the mode still at the highest point of the curve in a skewed distribution?

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Answer

The mode is not affected by extreme scores.

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Question

What does a positive skew look like?

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Answer

There is a big cluster on the left, with scores tailing off to the right.

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Question

Would a negative skew be seen if most participants got low or high scores?

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Answer

High scores.

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Question

For these results, would the distribution be normal, positive skewed, or negative skewed: mean 25, median 25.6, mode 25.3?

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Answer

Normal distribution.

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Question

Would the distribution be normal, positive skewed or negative skewed for these results: mean 28, median 23, mode 18?

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Answer

Positive skew.

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Question

Would the distribution be normal, positive skewed, or negative skewed for these results: mean 17, median 26, mode 30?

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Answer

Negative skew.

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