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Normal Distribution

Drawing and Interpreting a Normal Distribution Curve

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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 1 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 10, 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 12 and 8, respectively, because
Mark 8 and 12 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, 10.

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 11. 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 8 and 11. Thus, it can be said that the probability that a randomly chosen data point lies between 8 and 11 is 0.34.

Normal distribution with the two middle intervals selected

Example

Interpret the normal distribution

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A data set consists of the reaction times for a particular test. It is normally distributed with a mean of 250 ms and a standard deviation of 50 ms. How many of the test results are greater than 200 ms and less than 350 ms?

Normal distribution with specified values
Show Solution expand_more

To begin, it can be helpful to find the indicated values on the normal curve. 200 lies one standard deviation below the mean, and 350 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 200 and 350 ms.
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