Chapter 2: The Normal Distributions

CHAPTER 2: THE NORMAL DISTRIBUTIONS

SECTION 2.1: DENSITY CURVES AND THE NORMAL DISTRIBUTIONS

2

Chapter 1 gave a strategy for exploring data on a single quantitative variable.Make a graph.

Usually a histogram or stemplotDescribe the distribution.

Shape, center, spread, and any striking deviations.

Calculate numerical summaries to briefly describe the center and spread.Mean and standard deviation for symmetric distributions

Five-number summary for skewed distributions

DENSITY CURVES Chapter 2 tells us the next step.

If the overall pattern of a large number of observations is very regular, describe it with a smooth curve.

This curve is a mathematical model for the distribution. Gives a compact picture of the overall pattern. Known as a density curve.

3

DENSITY CURVES

4

A density curve describes the overall pattern of a distribution. Is always on or above the horizontal axis. The area under the curve represents a proportion.

Has an area of exactly 1 underneath it.

The median of a density curve is the equal-areas point. Point that divides the area under the curve in half. The quartiles divide the area into quarters

¼ of the area is to the left of Q1 ¾ of the area is to the left of Q3

The mean of a density curve is the balance point. Point that the curve would balance at if made of solid

material.

MATHEMATICAL MODEL A density curve is an idealized description of the

distribution of data. It gives a quick picture of the overall pattern

ignoring minor irregularities as well as outliers Since a density curve is an idealized description of

the data (not the actual data), we need to differentiate between the mean and standard deviation of the curve and the mean and standard deviation of the actual observations.

5

  Population Sample

Mean  Greek letter mu ”x-bar” 

Standard Deviation  Greek letter sigma  

xs

NORMAL DISTRIBUTIONS: Normal curves

Curves that are symmetric, single-peaked, and bell-shaped. They are used to describe normal distributions.

The mean is at the center of the curve.The standard deviation controls the

spread of the curve.The bigger the St Dev, the wider the curve.

There are roughly 6 widths of standard deviation in a normal curve, 3 on one side of center and 3 on the other side. 6

NORMAL CURVE

7

1 2 3 1 2 3

HERE ARE 3 REASONS WHY NORMAL CURVES ARE IMPORTANT IN STATISTICS.

Normal distributions are good descriptions for some distributions of real data.

Normal distributions are good approximations to the results of many kinds of chance outcomes.

Most important is that many statistical inference procedures based on normal distributions work well for other roughly symmetric distributions.

8

THE 68-95-99.7 RULE OR EMPIRICAL RULE:

9

68% of the observations fall within one standard deviation of the mean.

95% of the observations fall within two standard deviation of the mean.

99.7% of the observations fall within three standard deviation of the mean.

10

69 71.5 74 76.566.56461.5

11

95%

2.5%

69 71.5 74 76.566.56461.5

2.5%

12

95%

69 71.5 74 76.566.56461.5

2.5%64 to 74 in

13

69 71.5 74 76.566.56461.5

2.5%64 to 74 in

68%

16%

16%

14

69 71.5 74 76.566.56461.5

2.5%64 to 74 in

68%

16%

34%50%84%

15

NORMAL DISTRIBUTION NOTATION Since normal distributions are so

common, a short notation is helpful

Abbreviate the normal distribution with mean and standard deviation as:

The distribution of men’s heights would be

16

( , )N

(69,2.5)N

Find the proportion of observations within the given interval

P(0 < X < 2) P(.25 < X < .5) P(.25 < X < .75) P(1.25 < X <

1.75) P(.5 < X < 1.5) P(1.75 < X < 2)

17

0 .25 .5 .75 1.0 1.25 1.5 1.75 2.00

.25

.5

.75

1.0

= 1.0= .125= .25= .25= .46875= .15625

SECTION 2.1 COMPLETE

Homework: p.83-91 #’s 2-4, 9, 12 & 14

18