Lesson 1What Is Normal?Develop Understanding

Jump Start

Which One Doesn’t Belong?

Look at each representation of data and determine which one you believe does not belong. Be prepared to justify your choice.

A.

,

B.

,

C.

,

D.

,

Learning Focus

Understand features of a normal distribution.

Identify the effect of changing the mean or standard deviation in a normal distribution.

What is a “normal” distribution? How are normal distributions related to bell curves?

Open Up the Math: Launch, Explore, Discuss

One very important type of data distribution is called a “normal distribution.” In this case, the word “normal” has a special meaning for statistical distributions. In this task, you will be given a pair of data distributions represented with histograms and distribution curves. In each pair, one distribution is normal, and one is not. Your job is to compare each of the distributions given and come up with a list of features for normal distributions.

1.

This is approximately normal:

This is not normal:

What differences do you see between these distributions?

2.

This is normal:

This is not normal:

What differences do you see between these distributions?

3.

This is approximately normal:

This is not normal:

What differences do you see between these distributions?

4.

This is normal:

This is not normal:

What differences do you see between these distributions?

5.

This is approximately normal:

This is not normal:

What differences do you see between these distributions?

6.

This is approximately normal:

This is not normal:

What differences do you see between these distributions?

7.

This is normal:

This is not normal:

What differences do you see between these distributions?

8.

Based upon the examples you have seen in problems 1-7, what are the features of a normal distribution?

Pause and Reflect

9.

Mean = , Standard Deviation =

Mean = , Standard Deviation =

Mean = , Standard Deviation =

a.

What does the standard deviation tell us about a distribution?

b.

Use the three normal distributions provided to answer the questions, “How does changing the standard deviation affect a normal curve? Why does it have this effect?”

10.

Mean = , Standard Deviation =

Mean = , Standard Deviation =

Mean = , Standard Deviation =

a.

What does the mean tell us about a distribution?

b.

Use the three normal distributions provided to answer the questions, “How does changing the mean affect a normal curve? Why does it have this effect?”

11.

Now that you have figured out some of the features of a normal distribution, determine if the following statements are true or false. In each case, explain your answer.

a.

A normal distribution depends on the mean and the standard deviation.

b.

The mean, median, and mode are equal in a normal distribution.

c.

A normal distribution is bimodal.

d.

In a normal distribution, exactly of the population is within one standard deviation of the mean.

An automatic radar camera is used to measure the speed of cars on a freeway. The speeds are normally distributed with a mean of and a standard deviation of . What is the probability that a car picked at random is traveling at more than ?

Takeaways

Features of a normal distribution:

Point of inflection:

On a normal distribution:

Lesson Summary

In this lesson, we learned about features of a normal distribution. We learned a normal distribution is defined by the mean, which is the center of the distribution and the standard deviation, which determines the spread of the distribution. Normal distributions are represented with the rules, which describe the percent of the distribution within one standard deviation, two standard deviations, and three standard deviations, respectively, from the mean.

Retrieval

1.

DeAndre scored a on his math test. The class average was with a standard deviation of . How many standard deviations above the mean did DeAndre score?

2.

DeAndre’s score was very high. If his score was removed from the data set, would the standard deviation increase or decrease?

3.

Find the inverse of .