9. INTERPRETING DENSITY CURVES USING THE EMPIRICAL

NORMAL Distribution The Empirical Rule

Normal Distribution? These density curves are symmetric, single-peaked, and bell-shaped. We capitalize Normal to remind you that these curves are special.

Normal distribution is described by giving its mean μ and its standard deviation σ

Shape of the Normal curve The standard deviation σ controls the spread of a Normal curve

The Empirical Rule 68-95-99.7 rule In the Normal distribution with mean μ and standard deviation σ: • Approximately 68% of the observations fall within σ of the mean μ. • Approximately 95% of the observations fall within 2σ of μ. • Approximately 99.7% of the observations fall within 3σ of μ.

The Normal Distribution and Empirical Rule

Example: YOUNG WOMEN’s HEIGHT The distribution of heights of young women aged 18 to 24 is approximately Normal with mean μ = 64.5 inches and standard deviation σ = 2.5 inches.

Importance of Normal Curve • scores on tests taken by many even though many people (such as SAT exams and sets of data follow a many psychological tests), Normal distribution,

• repeated careful measurements of the same quantity, and • characteristics of biological populations (such as yields of corn and lengths of animal pregnancies).

many do not.

Most income distributions, for example, are skewed to the right and so are not Normal

Standard Normal distribution Standard Normal Distribution

The standard Normal distribution is the Normal distribution N(0, 1) with mean 0 and standard deviation

Standard Normal Calculation The Standard Normal Table

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

How to use the table of values

Area to the

LEFT

Using the standard Normal table

Problem: Find the proportion of observations from the standard Normal distribution that are less than 2.22.

illustrates the relationship between the value z = 2.22 and the area 0.9868.

illustrates the relationship between the value z = 2.22 and the area 0.9868.

Example Area to the

RIGHT

Using the standard Normal table Problem: Find the proportion of observations from the standard Normal distribution that are greater than −2.15

z = −2.15 Area = 0.0158 Area = 1-0.0158 Area = .9842

Practice (a) z < 2.85

(a) 0.9978.

(b) z > 2.85

(b)0.0022.

(c) z > −1.66

(c) 0.9515.

(d) −1.66 < z < 2.85

(d) 0.9493.

CODY’S quiz score relative to his classmates 79 81 80 77 73 83 74 93 78 80 75 67 73 77 83 86 90 79 85 89 84 77 72 83 82 x

z = 0.99 Area = .8389

Cody’s actual score relative to the other students who took the same test is 84%