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{{ printedBook.courseTrack.name }} {{ printedBook.name }} 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

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. $34%+34%+13.5%=81.5%$ This means that approximately $82%$ of the reaction times were between $200$ and $350$ ms.

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