Normal Distributions

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Chapter 13

• Normal Distributions

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Thought Question 1
Birth weights of babies born in the United States
follow, at least approximately, a bell-shaped
curve. What does that mean?
Thought Question 2 What does it mean if
a persons SAT score falls at the 20th percentile
for all people who took the test?
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Thought Question 3
A study in 1976-80 found that males (ages 18-24)
have a mean height of 70 inches and a standard
deviation of 2.8 in., while females (ages 18-24)
have a mean height of 65 in. and a standard
deviation of 2.5 in. A standardized score is
the number of standard deviations an individual
falls above or below the mean for the whole group
(is positive for values above the mean, and
negative for those below the mean). Thus, a man
who is 72.8 inches tall has a standardized score
of 1. What is the standardized score for your
height?
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Thought Question 4
Many measurements in nature tend to follow a
similar pattern. The pattern is that most of the
individual measurements take on values that are
near the average, with fewer and fewer
measurements taking on values that are farther
from the average in either direction. Describe
what shape the distribution of such measurements
would have.
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Bell-Shaped CurveThe Normal Distributionof
Population Values
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Asymmetric Distributionsof the Population Values
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The Normal Distribution
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With the Mean and Standard Deviation of the
Normal Distribution We Can Determine

• What proportion of individuals fall into any
  range of values
• At what percentile a given individual falls, if
  you know their value
• What value corresponds to a given percentile

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Empirical Rule forAny Normal Curve

• 68 of the values fall within one standard
  deviation of the mean
• 95 of the values fall within two standard
  deviations of the mean
• 99.7 of the values fall within three standard
  deviations of the mean
• 68-95-99.7 Rule

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Empirical Rule forAny Normal Curve
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Health and Nutrition Examination Study of
1976-1980(HANES)

• Heights of adults, ages 18-24
• women
• mean 65.0 inches
• standard deviation 2.5 inches
• men
• mean 70.0 inches
• standard deviation 2.8 inches

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Health and Nutrition Examination Study of
1976-1980(HANES)

• Empirical Rule
• women
• 68 are between 62.5 and 67.5 inches
• mean ? 1 std dev 65.0 ? 2.5
• 95 are between 60.0 and 70.0 inches
• 99.7 are between 57.5 and 72.5 inches
• men
• 68 are between 67.2 and 72.8 inches
• 95 are between 64.4 and 75.6 inches
• 99.7 are between 61.6 and 78.4 inches

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Health and Nutrition Examination Study of
1976-1980(HANES)

• What proportion of men are less than 72.8 inches
  tall?

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Health and Nutrition Examination Study of
1976-1980(HANES)

• What proportion of men are less than 68 inches
  tall?

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Standardized Scores

• How many standard deviations is 68 from 70?
• standardized score
• (observed value minus mean) / (std dev)
• (68 - 70) / 2.8 -0.71
• The value 68 is 0.71 standard deviations below
  the mean 70.

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Standardized Scores

• standardized score
• (observed value minus mean) / (std dev)
• z is the standardized score
• x is the observed value
• m is the population mean
• s is the population standard deviation

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Health and Nutrition Examination Study of
1976-1980(HANES)

• What proportion of men are less than 68 inches
  tall?

-0.71 0 (standardized values)
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Table B Percentiles of the Standardized Normal
Distribution

• See Table B (the Standard Normal Table) in back
  of the text (or back of the supplement).
• Look up the closest standardized score in the
  table.
• Find the percentile corresponding to the
  standardized score (this is the percent of values
  below the corresponding standardized score or
  z-value).

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Table B
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Table B Percentiles of the Standardized Normal
Distribution
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Health and Nutrition Examination Study of
1976-1980(HANES)

• What proportion of men are less than 68 inches
  tall?

24.20
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Health and Nutrition Examination Study of
1976-1980(HANES)

• What height value is the 10th percentile for men
  ages 18 to 24?

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Table B Percentiles of the Standardized Normal
Distribution

• See Table B (the Standard Normal Table) in back
  of the text (or back of the supplement).
• Look up the closest percentile in the table.
• Find the corresponding standardized score.
• The value you seek is that many standard
  deviations from the mean.

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Table B Percentiles of the Standardized Normal
Distribution
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Health and Nutrition Examination Study of
1976-1980(HANES)

• What height value is the 10th percentile for men
  ages 18 to 24?

-1.3 0 (standardized values)
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Observed Value for a Standardized Score

• What height value is the 10th percentile for men
  ages 18 to 24?
• observed value
• mean plus (standardized score) ? (std dev)
• 70 (-1.3 ) ? (2.8)
• 70 (?3.64) 66.36
• The value 66.36 is approximately the 10th
  percentile of the population.

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Observed Value for a Standardized Score

• observed value
• mean plus (standardized score) ? (std dev)
• x is the observed value
• m is the population mean
• z is the standardized score
• s is the population standard deviation

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Health and Nutrition Examination Study of
1976-1980(HANES)

• RECALL What proportion of men are less than 68
  inches tall?

NOW what proportion of men are greater than 68
inches tall?
?
100?24.2 75.8
24.20
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Health and Nutrition Examination Study of
1976-1980(HANES)

• The average height of males ages 1824 years old
  was 70.0 inches with a standard deviation of 2.8
  inches.
• It is also known that this distribution of
  heights follows a normal or bell-shaped curve.
• What proportion of men are between 68 inches
  tall and 74 inches tall?

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Health and Nutrition Examination Study of
1976-1980(HANES)

• First, draw and label a normal curve.

? 70 in. ? 2.8 in.
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Health and Nutrition Examination Study of
1976-1980(HANES)

• Shade on the graph the range of heights between
  68 and 74 inches.

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Standardized Scores

• How many standard deviations is 68 from 70?
• standardized score
• (observed value minus mean) / (std dev)
• (68 - 70) / 2.8 0.71
• The value 68 is 0.71 standard deviations below
  the mean 70.

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Health and Nutrition Examination Study of
1976-1980(HANES)

• What proportion of men are less than 68 inches
  tall?

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Table B Percentiles of the Standardized Normal
Distribution
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Health and Nutrition Examination Study of
1976-1980(HANES)

• What proportion of men are less than 68 inches
  tall?

24.20
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Health and Nutrition Examination Study of
1976-1980(HANES)

• What proportion of men are less than 74 inches
  tall?

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Standardized Scores

• How many standard deviations is 74 from 70?
• standardized score
• (observed value minus mean) / (std dev)
• (74 - 70) / 2.8 1.43
• The value 74 is 1.43 standard deviations above
  the mean 70.

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Health and Nutrition Examination Study of
1976-1980(HANES)

• What proportion of men are less than 74 inches
  tall?

24.20
(standardized values) -0.71 0
1.43
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Table B Percentiles of the Standardized Normal
Distribution
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Health and Nutrition Examination Study of
1976-1980(HANES)

• What proportion of men are less than 74 inches
  tall?

91.92
24.20
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Health and Nutrition Examination Study of
1976-1980(HANES)

• What proportion of men are between 68 inches
  tall and 74 inches tall?

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Key Concepts

• Population values are distributed with differing
  shapes, some normal, some non-normal.
• Empirical Rule (68-95-99.7 Rule)
• Standardized Score
• Percentile
• Standard Normal Table