# 6.1: Estimating the Mean and Standard Deviation of a Normal Distribution

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

**Learning Objectives**

- Understand the meaning of normal distribution and bell-shape.
- Estimate the mean and the standard deviation of a normal distribution.

Now that you have created your plot on a large sheet of grid paper, can you describe the shape of the plot? Do the dots seem to be clustered around 1 spot (value) on the chart? Do some dots seem to be far away from the clustered dots? After you have made all the necessary observations to answer these questions, pick 2 numbers from the chart to complete this statement:

“The typical measurement of the diameter is approximately______inches, give or take______inches.”

We will complete this statement later in the lesson.

**Normal Distribution**

The shape below should be similar to the shape that has been created with the dot plot.

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 lesson. 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 statement that was presented earlier in this lesson.

“The typical measurement of the diameter is approximately 9.4 inches, give or take 0.2 inches.”

This statement assumes that the mean of the measurements was 9.4 inches and the standard deviation of the measurements was 0.2 inches.

*Example 1*

For each of the following graphs, 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.

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

*Solution:*

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

b) “The typical measurement is approximately ______ goals scored, give or take ______ goals.”

*Solution:*

“The typical measurement is approximately 64 goals scored, give or take 6 goals.”

**Lesson Summary**

In this lesson, you learned what was meant by normal distribution. You also learned about the smooth bell curve that is used to represent a data set that is normally distributed. In addition, you learned that when data is plotted on a bell curve, you can estimate the mean by using the value where the line of symmetry crosses the \begin{align*}x\end{align*}-axis. Finally, the spread of data in a normal distribution was represented by using a give or take statement.

**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?