# 6.3: Connecting the Standard Deviation and Normal Distribution

**At Grade**Created by: CK-12

**Learning Objectives**

- Represent the standard deviation of a normal distribution on a bell curve.
- Use the percentages associated with normal distributions to solve problems.

In the problem presented in the previous lesson regarding your bowling average, your teacher told you that the bowling averages were normally distributed. In the previous lesson, you calculated the standard deviation of the averages by using the TI-83 calculator. Later in this lesson, you will be able to represent the value of the standard deviation in a normal distribution.

You have already learned that 68% of the data lies within 1 standard deviation of the mean, 95% of the data lies within 2 standard deviations of the mean, and 99.7% of the data lies within 3 standard deviations of the mean. To accommodate these percentages, there are defined values in each of the regions to the left and to the right of the mean.

These percentages are used to answer real-world problems when both the mean and the standard deviation of a data set are known. Also keep in mind that since 99.7% of the data in a normal distribution is within 3 standard deviations of the mean, \begin{align*}1-99.7\%=0.3\%\end{align*} of the data does not fall within 3 standard deviations of the mean. This means that \begin{align*}\frac{0.3\%}{2}=0.15\%\end{align*} of the data is beyond 3 standard deviations on either end of the normal distribution curve. This is not shown in the figure above.

*Example 7*

The lifetimes of a certain type of light bulb are normally distributed. The mean life is 400 hours, and the standard deviation is 75 hours. For a group of 5,000 light bulbs, how many are expected to last each of the following times?

a) between 325 hours and 475 hours

b) more than 250 hours

c) less than 250 hours

*Solution:*

a) 68% of the light bulbs are expected to last between 325 hours and 475 hours. This means that \begin{align*}5,000 \times 0.68=3,400\end{align*} light bulbs are expected to last between 325 and 475 hours.

b) \begin{align*}95\% + 2.35\% + 0.15\% = 97.5\%\end{align*} of the light bulbs are expected to last more than 250 hours. This means that \begin{align*}5,000 \times 0.975 = 4,875\end{align*} of the light bulbs are expected to last more than 250 hours.

c) Only \begin{align*}2.35\% + 0.15\% = 2.5\%\end{align*} of the light bulbs are expected to last less than 250 hours. This means that \begin{align*}5,000 \times 0.025 = 125\end{align*} of the light bulbs are expected to last less than 250 hours.

*Example 8*

A bag of chips has a mean mass of 70 g, with a standard deviation of 3 g. Assuming a normal distribution, create a normal curve, including all necessary values.

a) If 1,250 bags of chips are processed each day, how many bags will have a mass between 67 g and 73 g?

b) What percentage of the bags of chips will have a mass greater than 64 g?

*Solution:*

a) Between 67 g and 73 g lies 68% of the data. If 1,250 bags of chips are processed, \begin{align*}1,250 \times 0.68 = 850\end{align*} bags will have a mass between 67 g and 73 g.

b) \begin{align*}95\% + 2.35\% + 0.15\% = 97.5\%\end{align*} of the bags of chips will have a mass greater than 64 grams.

Now you can represent the data that your teacher gave to you for the bowling averages of the players in your league on a normal distribution curve. The mean bowling score was 63.7, and the standard deviation was 14.1.

From the normal distribution curve, you can see that your average bowling score of 70 is within 1 standard deviation of the mean. You can also see that 68% of all the data is within 1 standard deviation of the mean, so you did very well bowling this semester. You should definitely return to the league next semester.

**Lesson Summary**

In this lesson, you have learned what is meant by a set of data being normally distributed and the significance of standard deviation. You are now able to represent data on a bell-curve and to interpret a given normal distribution curve. In addition, you can calculate the standard deviation of a given data set both manually and by using technology. All of this knowledge can be applied to real-world problems, which you are now able to answer.

**Points to Consider**

- Is the normal distribution curve the only way to represent data?
- The normal distribution curve shows the spread of the data, but it does not show the actual data values. Do other representations of data show the actual data values?

**Vocabulary**

- Inflection point
- A point on a normal curve where it goes from being concave down to being concave up. On a normal curve, inflection points occur at 1 standard deviation from the mean.

- Normal distribution
- A symmetric bell-shaped curve with tails that extend infinitely in both directions from the mean of a data set.

- 68-95-99.7 Rule
- The rule that includes the percentages of data that are within 1, 2, and 3 standard deviations of the mean of a set of data.

- Standard deviation
- A measure of spread of a data set equal to the square root of the sum of the squared variances divided by the number of data values.

- Variance
- A measure of spread of a data set equal to the mean of the squared variations of each data value from the mean of the data set.

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