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Normal Distribution

Drawing and Interpreting a Normal Distribution Curve

Concept

Normal Distribution

Data can be distributed in different ways. One type is called normal distribution. A data set that is normally distributed is symmetric about the mean, Its graph is a bell-shaped curve and shows the spread of the data in terms of the standard deviation,

In general, a normally distributed data set has the following properties.

  • The area under the curve is or
  • Approximately of the data lies within one standard deviation of the mean.
  • Approximately of the data lies within two standard deviations of the mean.
  • Approximately of the data lies within three standard deviations of the mean.

The above percents can be divided into individual segments as is shown below.

The area under a normal curve can be analyzed as probabilities or percents to interpret the data set.

Method

Drawing a Normal Curve

It's possible to sketch a normal distribution curve by hand. Consider a normally distributed data set with a mean and standard deviation The first step is to mark the mean, in this case on a horizontal axis.

normal distribution with a standard range and percentages

The next step is to find the numbers corresponding to one standard deviation above and below the mean. These are and respectively, because Mark and on the axis.

normal distribution with a standard range and percentages

Continue adding and subtracting until the horizontal axis has been labeled to three standard deviations in both directions.

normal distribution with a standard range and percentages

Lastly, draw a bell-shaped curve with its peak at the mean,

normal distribution with a standard range and percentages

Sometimes it's helpful to label the percents.

normal distribution with a standard range and percentages

Concept

Interpreting a Normal Distribution Curve

The predictability of a normal curve helps make sense of a data set. Suppose the mean and the standard deviation of a normally distributed data set are known — and Then, it can be seen that of the data points are greater than This is often shown by shading that part of the distribution.

Normal distribution with the two middle intervals selected

It is possible to also discuss the area under the curve as probabilities. Here, it can be seen that of the data points are between and Thus, it can be said that the probability that a randomly chosen data point lies between and is

Normal distribution with the two middle intervals selected
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Exercise

A data set consists of the reaction times for a particular test. It is normally distributed with a mean of ms and a standard deviation of ms. How many of the test results are greater than ms and less than ms?

Normal distribution with specified values
Show Solution
Solution

To begin, it can be helpful to find the indicated values on the normal curve. lies one standard deviation below the mean, and lies two standard deviations above the mean. Let's shade the area between these bounds.

Normal distribution with specified values and a selected interval

To determine the total area that satisfies the requirements, we can add the percentages of the shaded areas. This means that approximately of the reaction times were between and ms.

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