The Normal Distribution

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The Normal Distribution

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Consider a large population. Data is collected on a sample. That information is placed into a histogram.

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More data is collected and placed in the histogram…and more…

When ALL the population is placed into a histogram, you have a

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When ALL the population is placed into a histogram, you have a

NORMAL DISTRIBUTION

More data is collected and placed in the histogram…and more…

Also called the Gaussian distribution

(after Carl Gauss)

Also called the

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It is symmetric.

The mean, median, and mode are equal.

The mean, median, and mode are located at

the center of the distribution.

The distribution can be used to make

predictions about the entire population given

only data from a sample.

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What about the standard deviation?

Mean, median, and mode are equal and located

at the center of the distribution.

mean median mode mean median mode mean median mode

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7.

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mean median

mode

The 68-95-99.7 Rule

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0

1 -1

68%

mean median

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mean median

mode

0

1 -1

2 -2

68%

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mean median

mode

0

1 -1

2 -2

3 -3

68%

95%

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mean median mode

0

1 -1

2 -2

3 -3

68%

95%

99.7%

50%

34%

13.5%

Think:

95% - 68% = ?

Think:

95% - 68% =

27%

2.35%

Think:

99.7% - 95% = ?

Think:

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70”

Example

The heights of young men are normally distributed. The mean height is 70 inches.

The standard deviation of heights is 2.5 inches.

72.5”

75”

77.5” 67.5”

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70”

Example

72.5” 75” 77.5” 67.5” 65” 62.5”

What height is 3 standard deviations above the mean?

77.5 inches

What height is 2 standard deviations below the mean?

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70”

Example

72.5” 75” 77.5” 67.5” 65” 62.5”

What percent of young men are between 65 and 75 inches tall?

95%

What percent of young men are under 67.2 inches tall?

16%

✓

✓ ✓ ✓

Think:

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70”

Example

72.5” 75” 77.5” 67.5” 65” 62.5”

What percent of young men are over 75 inches tall?

2.5%

What percent of young men are between 62.5 and 72.5 inches tall?

83.85%

✓ ✓

✓ ✓ ✓

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A

z

-score is the number of standard deviations

a data item is above or below the mean.

How is the location of

a particular score found?

•

Data items above the mean have positive

z

-scores.

•

Data items below the mean have negative

z

-scores.

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Example

The weights of newborn infants are normally distributed.

The mean weight is 7 pounds; the standard deviation is 0.8 pound.

Find the z-score for a weight of 9.4 pounds.

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Example

Find the z-score for a weight of 5 pounds.

7 6.2

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Example

Intelligence quotients (IQs) are normally distributed with a mean of 100 and a standard deviation of 15.

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To compare scores on two different tests, in

relation to the mean on each test,

z

-scores can be

used.

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Example

A student scored 70 on a math test and 66 on an English test. The scores for both tests are normally distributed.

The math test had a mean of 60 and a standard deviation of 20. The English test had a mean of 60 and a standard deviation of 2.

On which test did the student have the better score?

Math Test:

English Test:

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Example

A student scored 72 on a science test and 255 on a language test. The scores for both tests are normally distributed.

The science test had a mean of 84 and standard deviation of 10. The language test had a mean of 300 and standard deviation of 30.

On which test did the student have the better score?

Science Test:

Language Test:

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If

n

% of the items in a distribution are less than a particular data

item, we say that the data item is in the

n

th percentile of the

distribution.

What is a percentile?

A percentile is NOT an actual score.

A percentile is NOT the percent of answers that are correct.

What a percentile is not…

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70”

PREVIOUS Example

72.5” 75” 77.5” 67.2” 65” 62.5”

What percent of young men are under 67.2 inches tall?

16%

✓

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PREVIOUS Example . . . here’s the problem

Find the z-score for a weight of 5 pounds.

7 6.2 5.4 4.6 5 2.35% NOT Half of 2.35%

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Z

-Scores and Percentiles

Cow book page 672

Find the

z-score of -2.5 in the table.

So, infants of 5 pounds are in 0.62 percentile,

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Example

According to the Department of Health and Education, cholesterol levels are normally distributed. For men between 18 and 24 years, the mean is 178.1 (mg/mL) and the standard deviation is 40.7.

What percentage of men in this age range have a cholesterol level less than 239.15?

First, find the

z

-score.

Next, find the

z

-score in the table of percentiles.

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Example

Lengths of pregnancy of women are normally distributed with a mean of 266 days and a standard deviation of 16 days.

What percentage of children are born from pregnancies lasting more than 274 days?

First, find the

z

-score.

Next, find the

z

-score in the table of percentiles.

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Example

The National Federation of State High School Associations states that the amount of time that high school students work at jobs each week is

normally distributed with mean of 10.7 hours and standard deviation of 11.2 hours.

What percentage of high school students work between 5.1 and 38.7 hours each week?

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Example

For 5.1 hours:

30.85% are below

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Example

For 38.7 hours:

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Example

30.85% are below 5.1 hours 99.38% are below 38.7 hours 99.38% are below 38.7 hours 30.85% are below 5.1 hours