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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, $μ.$ Its graph is a bell-shaped curve and shows the spread of the data in terms of the standard deviation, $σ.$

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

- The area under the curve is 1 or $100%.$
- Approximately $68%$ of the data lies within one standard deviation of the mean.
- Approximately $95%$ of the data lies within two standard deviations of the mean.
- Approximately $100%$ 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.It's possible to sketch a normal distribution curve by hand. Consider a normally distributed data set with a mean $μ=10$ and standard deviation $σ=2.$ The first step is to mark the mean, in this case 10, on a horizontal axis.

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

$10+2=12and10−2=8.$

Mark 8 and 12 on the axis. Continue adding and subtracting $σ$ until the horizontal axis has been labeled to three standard deviations in both directions.

Lastly, draw a bell-shaped curve with its peak at the mean, 10.

Sometimes it's helpful to label the percents.

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 — $μ=8$ and $σ=3.$ Then, it can be seen that $16%$ of the data points are greater than 11. This is often shown by shading that part of the distribution.

It is possible to also discuss the area under the curve as probabilities. Here, it can be seen that $34%$ 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.

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?

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.

To determine the total area that satisfies the requirements, we can add the percentages of the shaded areas.

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