# 5.1: 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.

**Introduction**

The diameter of a circle is the length of the line through the center and touching two points on the circumference of the circle.

If you had a ruler, you could easily measure the length of this line. However, if your teacher gave you a golf ball and asked you to use a ruler to measure its diameter, you would have to create your own method of measuring its diameter.

Using your ruler and the method that you have created, make two measurements of the diameter of the golf ball (to the nearest tenth of an inch). Your teacher will prepare a chart for the class to create a dot plot of all the measurements. Can you describe the shape of the plot? Do the dots seem to be clustered around one spot (value) on the chart? Do some dots seem to be far away from the clustered dots? After you have answered these questions, pick two 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.

You have probably noticed that the measurements of the diameter of the golf ball 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 1.6 inches, with a few measurements to the right of this value and a few measurements to the left of this value. The resulting shape looks like a bell and is the shape that represents the **normal distribution** of the data.

In the real world, no examples match this smooth curve perfectly, but many data plots, like the one you made, are approximately normal. For this reason, it is often said that normal distribution is ‘assumed.’ When normal distribution is assumed, the resulting bell-shaped curve is symmetric - the right side is a mirror image of the left side. If the blue line is the mirror (the line of symmetry) you can see that the green section is the mirror image of the yellow section. The line of symmetry also goes through the \begin{align*}x-\end{align*}axis.

If you took all of the measurements for the diameter of the golf ball, added them and divided the total by the number of measurements, you would know the mean (average) of the measurements. It is at the mean that the line of symmetry intersects the \begin{align*}x-\end{align*}axis. For this reason, the mean is used to describe the center of a normal distribution.

You can see that the two colors spread out from the line of symmetry and seem to flatten out the further left and right they go. This tells you that the data spreads out, in both directions, away from the mean. This spread of the data is called the standard deviation and it describes exactly how the data moves away from the mean. In a normal distribution, on either side of the line of symmetry, the curve appears to change its shape from being concave down (looking like an upside-down bowl) to being concave up (looking like a right side up bowl). Where this happens is called the inflection point of the curve. If a vertical line is drawn from the inflection point to the \begin{align*}x-\end{align*}axis, the difference between where the line of symmetry goes through the \begin{align*}x-\end{align*}axis and where this line goes through the \begin{align*}x-\end{align*}axis represents the amount of the spread of the data away from the mean.

Approximately 68% of all the data is located between these inflection points.

For now, that is all you have to know about standard deviation. It is the spread of the data away from the mean. In the next lesson, you will learn more about this topic.

Now you should be able to complete the statement that was given in the introduction.

“The typical measurement of the diameter is approximately \begin{align*}{\color{blue}\underline{1.6}}\end{align*} inches, give or take \begin{align*}{\color{blue}\underline{0.4}}\end{align*} inches.”

*Example 1:*

For each of the following graphs, complete the statement “The typical measurement is approximately ______ give or take ______.”

a)

“The typical measurement is approximately \begin{align*}{\color{blue}\underline{400 \ \text{houses built}}}\end{align*} give or take \begin{align*}{\color{blue}\underline{100}}\end{align*}.”

b)

“The typical measurement is approximately \begin{align*}{\color{blue}\underline{8 \ \text{games won}}}\end{align*} give or take \begin{align*}{\color{blue}\underline{3}}\end{align*}.”

**Lesson Summary**

In this lesson you learned what was meant by the bell curve and how data is displayed on this shape. You also learned that when data is plotted on the bell curve, you can estimate the mean of the data with a give or take statement.

**Points to Consider**

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