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Normal distribution
Histogram for Female Ability to Match Stick Angle
0
1
2
3
4
5
6
7
8
1 3 5 7 9
11
13
15
17
19
Classes of number of sticks correctly matched
Fre
qu
en
cy
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.)
0
50
100
150
200
Height
Fre
qu
ency
Normal distribution
40 50 60 70 80 90 100
0.00
0.01
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0.08
Grades
Den
sity
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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0.01
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0.08
Grades
Den
sity
P(65 < X < 70)
Probability above 75?
55 60 65 70 75 80 85
0.00
0.01
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0.05
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0.08
Grades
Den
sity
Probability student scores higher than 75?
P(X > 75)
Probability below 65?
55 65 75 85
0.00
0.01
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0.05
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0.08
Grades
Den
sity
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.