### Standard Deviation of a Normal Distribution

Let's go back to the basketball example presented in the introduction to this set of Concepts. When you made the observations regarding the measurements of the diameter of the basketball, you must have noticed that they were not all the same. In spite of the different measurements, you should have seen that the majority of the measurements clustered around the value of 9.4 inches. This value represents the approximate diameter of a basketball. Also, you should have noticed that a few measurements were to the right of this value, and a few measurements were to the left of this value. The resulting shape looks like a bell, and this is the shape that represents a **normal distribution** of data.

In the real world, no examples match this smooth curve perfectly. However, many data plots, like the one you made, will approximate this smooth curve. For this reason, you will notice that the term *assume* is often used when referring to data that deals with normal distributions. When a normal distribution is assumed, the resulting bell-shaped curve is symmetric. That is, the right side is a mirror image of the left side. In the figure below, if the blue line is the mirror (the line of symmetry), you can see that the pink section to the left of the line of symmetry is the mirror image of the yellow section to the right of the line of symmetry. The line of symmetry also goes through the \begin{align*}x\end{align*}-axis.

If you knew all of the measurements that were plotted for the diameter of the basketball, you could calculate the mean (average) diameter by adding the measurements and dividing the sum by the total number of values. It is at this value that the line of symmetry intersects the \begin{align*}x\end{align*}-axis. In other words, the mean of a normal distribution is the center, or balance point, of the distribution.

You can see that the 2 colors form a peak at the top of the line of symmetry and then spread out to the left and to the right from the line of symmetry. The shape of the bell flattens out the further it moves away from the line of symmetry. In other words, the data spreads out in both directions away from the mean. This spread of the data is measured by the **standard deviation**, and it describes exactly how the data moves away from the mean. You will learn more about standard deviation in the next concept. For now, that is all you have to know about standard deviation\begin{align*}-\end{align*}it is a measure of the spread of the data away from the mean.

Now you should be able to complete the following statement regarding the measurements of the diameter of the basketball:

“The typical measurement of the diameter is approximately 9.4 inches, give or take 0.4 inches.”

This statement assumes that the mean of the measurements was 9.4 inches and the standard deviation of the measurements was 0.4 inches. It also assumes that the standard deviation is the difference between the mean and the first tick mark to the left of the mean.

In each of the following examples, complete the statement. Fill in the first blank in each statement with the mean and the second blank in each statement with the standard deviation. Assume that the standard deviation is the difference between the mean and the first tick mark to the left of the mean.

#### Fill in the Blank According to the Directions Above

1. “The typical measurement is approximately ______ in the bank, give or take ______.”

“The typical measurement is approximately $500 in the bank, give or take $50.”

2. “The typical measurement is approximately ______ minutes played, give or take ______ minutes.”

“The typical measurement is approximately 64 minutes played, give or take 6 minutes.”

3. “The typical measurement is approximately ______ ounces, give or take ______ ounces.”

“The typical measurement is approximately 20.0 ounces, give or take 0.1 ounces.”

**Points to Consider**

- Is there a way to determine the actual values for a give or take statement?
- Can a give or take statement go beyond a single give or take?
- Can all the actual values be represented on a bell curve?

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### Examples

Suppose that the mean number of calories consumed by an adult in a particular country per day is 3,200, with a standard deviation of 150 calories. Determine each of the following intervals and shade the area under a normal distribution curve for each interval.

**Example 1**

The values within 1 standard deviation of the mean

The values within 1 standard deviation of the mean are from \begin{align*}3,200-150\end{align*} calories to \begin{align*}3,200+150\end{align*} calories, or from 3,050 calories to 3,350 calories.

#### Example 2

The values within 2 standard deviations of the mean

The values within 2 standard deviations of the mean are from \begin{align*}3,200-(2 \times 150)\end{align*} calories to \begin{align*}3,200+(2 \times 150)\end{align*} calories, or from 2,900 calories to 3,500 calories.

#### Example 3

The values within 3 standard deviations of the mean

The values within 3 standard deviations of the mean are from \begin{align*}3,200-(3 \times 150)\end{align*} calories to \begin{align*}3,200+(3 \times 150)\end{align*} calories, or from 2,750 calories to 3,650 calories.

### Review

For each of the following, complete the statement. Fill in the first blank in each statement with the mean and the second blank in each statement with the standard deviation. Assume that the standard deviation is the difference between the mean and the first tick mark to the left of the mean.

- “The typical measurement is approximately ______ kilograms, give or take ______ kilograms.”
- “The typical measurement is approximately ______ miles per hour, give or take ______ miles per hour.”
- “The typical measurement is approximately ______ feet, give or take ______ feet.”

Suppose that the mean age of the students in a high school is 16.8 years, with a standard deviation of 0.7 years. Determine each of the following intervals.

- The values within 1 standard deviation of the mean
- The values within 2 standard deviations of the mean
- The values within 3 standard deviations of the mean

Suppose that the mean amount of toothpaste in a tube is 170.0 grams, with a standard deviation of 0.4 grams. Shade the area under a normal distribution curve for each of the following intervals.

- The values within 1 standard deviation of the mean
- The values within 2 standard deviations of the mean
- The values within 3 standard deviations of the mean

Closely examine your graphs for the previous 3 questions.

- For each of the graphs, estimate the percentage of the space under the graph that is shaded.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 6.1.