Normal distribution

Histogram for Female Ability to Match Stick Angle

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Classes of number of sticks correctly matched

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An example from class

HEIGHTS OF MOTHERS

CLASS LIMITS(in.) FREQUENCY52-53 0.553-54 1.554-55 155-56 256-57 6.557-58 1858-59 34.559-60 79.560-61 135.561-62 16362-63 18363-64 16364-65 114.565-66 78.566-67 4167-68 1668-69 7.569-70 4.570-71 2TOTAL 1052

Example Of a Normal Variable

Histogram Of Heights Of Mothers (in.)

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Height

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Normal distribution

40 50 60 70 80 90 100

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Bell-shaped curve

Mean = 70 SD = 5

Mean = 70 SD = 10

Characteristics of normal distribution

• Symmetric, bell-shaped curve.• Shape of curve depends on population mean ()

and standard deviation ().• Center of distribution is mean () and mode and

median.• Spread is determined by standard deviation().• Most values fall around the mean, but some values

are smaller and some are larger.

Examples of normalrandom variables

• testosterone level of male students• head circumference of adult females• length of middle finger of Stat/Soc students• Height• Weight• IQ scores• Body temperature• Repeated measurement of same quantity

Probability between 65 and 70?

55 60 65 70 75 80 85

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P(65 < X < 70)

Probability above 75?

55 60 65 70 75 80 85

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Probability student scores higher than 75?

P(X > 75)

Probability below 65?

55 65 75 85

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P(X < 65)

Normal Percents

The 68-95-99.7 Rule

Example: Young Women’s Height

• The heights of young women are approximately normal with mean = 64.5 inches and std.dev. = 2.5 inches.

Example: Young Women’s Height

• % of young women between 62 and 67?• % of young women lower than 62 or taller than 67?• % between 59.5 and 62?• % taller than 68.25?

Example: Young Women’s Height

• The heights of young women are approximately normal with mean = 64.5 inches and std.dev. = 2.5 inches.

Example: Young Women’s Height

• The heights of young women are approximately normal with mean = 64.5 inches and std.dev. = 2.5 inches.

Example: Young Women’s Height

• % of young women between 62 and 67?• % of young women lower than 62 or taller than 67?• % between 59.5 and 62?• % taller than 68.25?

Working With the General NormalEXAMPLE: IQ Scores

|100

s.d. = 16

IQ Scores have a normal distribution with a mean of 100 and a standard deviation of 16. What is the 99% percentile of IQ Scores?

The Standard Normal Table: Table A

• Table A is a table of areas under the standard normal density curve. The table entry for each value z is the area under the curve to the left of z.