Lesson 1 What 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.

,

a box plot with a point at 40 and a median at 75. The Interquartile range is 15. 404040505050606060707070808080909090100100100
B.

,

a histogram with a mean of 73.21. The range is between 60 and 100 606060707070808080909090100100100222444666888
C.

,

a dot plot where the mean is 84.7 and the range is between 70 and 100 555555606060656565707070757575808080858585909090959595100100100222444666
D.

,

a histogram with a mean of 80.27. The range is between 50 and 100 606060707070808080909090100100100555101010

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:

a histogram where the distribution is normal –4–4–4–2–2–20002224440.10.10.10.20.20.20.30.30.30.40.40.4

This is not normal:

a histogram where the distribution is not normal 160160160180180180200200200220220220240240240202020404040606060808080100100100120120120

What differences do you see between these distributions?

2.

This is normal:

a distribution curve where the distribution is normal. the peak of the distribution is the mean, median, and mode. Normal Distribution CenterMeanMedianMode

This is not normal:

a distribution curve where the distribution is not normal. the peak of the distribution is not mean, median, and mode. ModeMedianMean

What differences do you see between these distributions?

3.

This is approximately normal:

a histogram where the distribution is normal. the peak of the distribution is the mean, median, and mode. This is called a bell curve. 808080100100100120120120140140140160160160180180180200200200220220220240240240000101010202020303030404040505050“Bell Curve”

This is not normal:

a histogram where the distribution is not normal and is more skewed to the left 116116116118118118120120120122122122124124124126126126128128128130130130Frequency000111222333444555666777888Height of StudentsHeight (cm)

What differences do you see between these distributions?

4.

This is normal:

a distribution curve where the distribution is normal. the points of inflection are equidistant from the peak Concave downPoints of inflectionConcave upConcave up

This is not normal:

a histogram where the distribution is not normal 222333444555666777888999101010111111121212131313141414Frequency000101010202020303030404040505050

What differences do you see between these distributions?

5.

This is approximately normal:

a histogram where the distribution is normal. the peak of the distribution is about where the mean, median, and mode are 606060656565707070757575808080858585909090959595Density0000.010.010.010.020.020.020.030.030.030.040.040.040.050.050.050.060.060.06Test Scores

This is not normal:

a histogram where the distribution is not normal 120120120140140140160160160180180180200200200220220220# of Individuals202020404040606060808080100100100120120120Body Mass (g)

What differences do you see between these distributions?

6.

This is approximately normal:

a histogram where the distribution is normal. the peak of the distribution is about where the mean, median, and mode are

This is not normal:

a histogram where the distribution is not normal. the peak of the distribution is not where the mean, median, and mode are

What differences do you see between these distributions?

7.

This is normal:

a curve where the distribution is normal. the peak of the distribution is about where the mean, median, and mode are 2.15%13.59%34.13%34.13%13.59%68.26%95.44%99.74%0.13%0.13%2.15%

This is not normal:

a curve where the distribution is not normal. the peak of the distribution is not where the mean, median, and mode are MeanMedianMode

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 =

a bell curve where the mean is 3 and the standard deviation is .5 0.50.50.51111.51.51.52222.52.52.53333.53.53.54444.54.54.55555.55.55.56660.50.50.5111000

Mean = , Standard Deviation =

a bell curve where the mean is 3 and the standard deviation is 1 0.50.50.51111.51.51.52222.52.52.53333.53.53.54444.54.54.55555.55.55.56660.50.50.5111000

Mean = , Standard Deviation =

a bell curve where the mean is 3 and the standard deviation is .25 0.50.50.51111.51.51.52222.52.52.53333.53.53.54444.54.54.55555.55.55.56660.50.50.51111.51.51.5000

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 =

a bell curve where the mean is 1 and the standard deviation is .25 0.50.50.51111.51.51.52222.52.52.53333.53.53.54444.54.54.55555.55.55.56660.50.50.51111.51.51.5000

Mean = , Standard Deviation =

a bell curve where the mean is 2 and the standard deviation is .25 0.50.50.51111.51.51.52222.52.52.53333.53.53.54444.54.54.55555.55.55.56660.50.50.51111.51.51.5000

Mean = , Standard Deviation =

a bell curve where the mean is 3 and the standard deviation is .25 0.50.50.51111.51.51.52222.52.52.53333.53.53.54444.54.54.55555.55.55.56660.50.50.51111.51.51.5000

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.

Is this statement true or false? Explain your answer.

b.

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

Is this statement true or false? Explain your answer.

c.

A normal distribution is bimodal.

Is this statement true or false? Explain your answer.

d.

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

Is this statement true or false? Explain your answer.

Ready for More?

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:

Adding Notation, Vocabulary, and Conventions

Point of inflection:

a curve that resembles a sine function where the point of reflection is (3,0) x222444666y–1–1–1111000

On a normal distribution:

a bell curve where the points of inflection are 1 unit away from the origin –2–2–2222000

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 .