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Normal Distribution of Data: Examples, Definition & Characteristics

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  • 0:01 Data and Distribution
  • 0:33 What Is Normal Distribution?
  • 1:02 Characteristics of…
  • 1:52 Percentiles
  • 2:55 Example
  • 4:11 Lesson Summary
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Lesson Transcript
Instructor
Yolanda Williams

Yolanda has taught college Psychology and Ethics, and has a doctorate of philosophy in counselor education and supervision.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

In this lesson, we'll explore the normal distribution of data. Learn about the characteristics of normal distribution, how to plot histograms, the empirical rule, and more.

Data and Distribution

Imagine that you are a professor teaching an intro to psychology course. At the end of the semester, you have all 100 of your students complete a final exam consisting of 100 multiple-choice questions. The scores that your students received are as follows:

You can tell from looking at the data that the highest score a student received was 100% and the lowest score was 60%. What you might not have been able to tell just by glancing at the table is that the data is normally distributed.

What Is Normal Distribution?

A normal distribution is a bell-shaped frequency distribution curve. Most of the data values in a normal distribution tend to cluster around the mean. The further a data point is from the mean, the less likely it is to occur. There are many things, such as intelligence, height, and blood pressure, that naturally follow a normal distribution. For example, if you took the height of one hundred 22-year-old women and created a histogram by plotting height on the x-axis, and the frequency at which each of the heights occurred on the y-axis, you would get a normal distribution.

Characteristics of Normal Distribution

Here, we see the four characteristics of a normal distribution. Normal distributions are symmetric, unimodal, and asymptotic, and the mean, median, and mode are all equal.

A normal distribution is perfectly symmetrical around its center. That is, the right side of the center is a mirror image of the left side. There is also only one mode, or peak, in a normal distribution. Normal distributions are continuous and have tails that are asymptotic, which means that they approach but never touch the x-axis. The center of a normal distribution is located at its peak, and 50% of the data lies above the mean, while 50% lies below. It follows that the mean, median, and mode are all equal in a normal distribution.

Percentiles

Now, look at the line that says standard deviations (SD). You can see that 34.13% of the data lies between 0 SD and 1 SD. Since a normal distribution is perfectly symmetric, it follows that 34.13% of the data lies between -1 SD and 0 SD. If you continue to add the percentages together, you will see that on either side:

  • Approximately 68% of the data lies within 1 SD of the mean
  • Approximately 95% of the data lies within 2 SD of the mean
  • Approximately 99.7% of the data lies within 3 SD of the mean

This is known as the empirical rule.

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Additional Activities

Determining if Data is Normally Distributed

In the following activity, students will graph histograms for three different sets of data. After examining the data and histograms, students will determine whether the data appears to follow a normal distribution or not - and defend their viewpoint.

Activity

Materials needed:

  • paper
  • writing utensils (just pencil is fine, but anything can be used)
  • ruler

Directions:

For the following three sets of data, create a histogram displaying the data. Use a separate sheet of paper for each histogram. Then, determine whether each data set appears to follow a normal distribution - and include supporting details.

Data Set 1: 50 people were given an IQ test, and the results are below:

Frequency IQ
2 80
5 85
6 90
7 95
10 100
8 105
6 110
5 115
1 120

Data Set 2: 50 people were surveyed about their favorite color out of the choices red, blue, green, or yellow

Frequency Color
14 red
12 blue
13 green
11 yellow

Data Set 3: 50 people were surveyed about the number of pets that their household has

Frequency Number of Pets
12 0
20 1
10 2
4 3
2 4
1 5
1 6


Solutions

The histograms for the three data sets should appear similar to the histograms below:

Of the three data sets, the one that most closely resembles a normal distribution is the "IQ test results". Reasons to support this include having the highest peak occur in the middle of the data (so the mean, median, and mode are all approximately equal) and the roughly bell-shaped curve. The ends are lower than the rest of the histogram. The "favorite color" histogram is not normally distributed, as all colors are liked fairly equally. The "number of pets" histogram is not normally distributed because most of the data falls on one side of the distribution.

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