1

Chapter 2: The Normal Distribution

2.1 Density Curves and the Normal Distributions

2.2 Standard Normal Calculations

2

Histogram for Strength of Yarn Bobbins

15.60 16.10 16.60 17.10 17.60 18.10 18.60 19.10 19.60

X X X X X

X X X X X X X

X X X X X X X X

X X X X

Bobbin #1: 17.15 g/tex

Bobbin #2: 17.42 g/tex

Bobbin #3: 17.93 g/tex

3

X X X X X

X X X X X X X

X X X X X X X X

X X X X

Histogram for Strength of Yarn Bobbins

15.60 16.10 16.60 17.10 17.60 18.10 18.60 19.10 19.60

4

Density Curves

•  The smooth curve drawn over the histogram in the previous example is a density curve. –  A mathematical model

•  Here, we are looking for an overall pattern; we may ignore outliers or slight irregularities.

–  Area under the curve represents proportions of scores/observations.

•  The area under a density curve is always 1.0.

5

Practice Problem

•  Exercise 2.3, p. 83

6

Figure 2.7, p. 83

7

We can look at certain areas under the density curve and calculate proportions of observations between certain values.

15.60 16.10 16.60 17.10 17.60 18.10 18.60 19.10 19.60

8

Figure 2.4, p. 81: The area under the density curve is the proportion of observations taking values between 7 and 8.

9

Mean and Median of a Density Curve

•  Median: Equal-areas point of a density curve •  Mean: Gets pulled towards any skew or

outliers. •  See Figure 2.5 (p. 81)

•  What about the mean and median of a

symmetric density curve?

10

Figure 2.5, p. 81

11

Practice

•  Exercise 2.4, p. 84

12

Normal Distribution

•  If we draw a large sample from many populations of interest in science, scores tend to “stack up” in the middle, with a fewer number of scores towards the tails of the distribution.

•  Many, but certainly not all, distributions can be approximated by a normal density curve. –  Normal density curves describe normal

distributions.

13

Characteristics of a Normal Curve

•  Symmetric, single-peaked (uni-modal), and bell-shaped

•  The exact density curve for a particular normal distribution is described exactly by giving its mean µ and standard deviation σ. –  Notation: N(µ , σ)

•  See Figure 2.10, p. 85 •  We can “eye” µ and σ.

14

Figure 2.10, p. 85

15

15.60 16.10 16.60 17.10 17.60 18.10 18.60 19.10 19.60

σ can be found by locating the point of inflection of the density curve.

16

µ - 2σ µ - 1σ µ µ + 1σ µ + 2σ µ - 3σ µ + 3σ

1 68.3%

2 95%

3 99.7%

68-95-99.7 rule

17

Practice Problem

•  Exercise 2.8, p. 89

18

Homework

•  Reading: Section 2.1, pp. 78-90 •  Exercises:

–  2.2, p. 83 –  2.5, p. 84 –  2.7, p. 89

19

Standard Normal Curve

•  Statisticians have made one particular normal distribution the standard and computed the proportions for this distribution:

1 σ 0, µ ==

•  This mean and standard deviation define the standard normal curve.

20

Standardizing

•  Use the following transformation:

deviation standard population σand mean; population µ variable;random aon score x

score; standard z

where,σ

µ-x z

====

=

•  A z-score tells us how far a given score falls from the mean, in terms of the standard deviation.

21

Example Problems

2.19 and 2.20, p. 95

22

Normal Distribution Calculations

•  We first compute a z-score for a particular value, given its mean µ and standard deviation σ.

•  Then, because an area under a density curve is a proportion of observations in a distribution, any question about what proportion of observations lie in some range of values can be answered by finding an area under the curve.

•  Table A (front cover of your book) is a table of areas under the standard normal curve.

23

24

Practice

•  Exercises 2.21 and 2.23, p. 103

25

Homework for Rockmont Weekend

•  Read/finish reading Chapter 2 (the whole thing).

•  Chapter 2 test on Thursday (9/17).

26

Homework

•  Read through p. 103 •  Problems 2.22, 2.24, and 2.25, p. 103

27

Homework

•  Exercises 2.41-2.45, pp. 113-114

28

Assessing Normality

•  Suppose that we obtain a simple random sample from a population whose distribution is unknown. Many of the statistical tests that we perform on small data sets (sample size less than 30) require that the population from which the sample is drawn be normally distributed. –  One way we can assess whether the sample is drawn from a

normally-distributed population is to draw a histogram and observe its shape.

–  What should it look like? •  Enter data from p. 17.

•  What other ways can we assess whether we have drawn a sample from a normally-distributed population?

29

Assessing Normality, cont.

•  This method works well for large data sets, but the shape of a histogram drawn from a small sample of observations does not always accurately represent the shape of the population. For this reason, we need additional methods for assessing the normality of a random variable when we are looking at sample data. –  The normal probability plot is used most often to assess

the normality of a population from which a sample was drawn.

30

Normal Probability Plots

•  A normal probability plot shows observed data versus normal scores. –  A normal score is the expected Z-score of the data value if the

distribution of the random variable is normal. The expected Z-score of an observed value will depend upon the number of observations in the data set.

–  See Example 2.12, p. 106 for details. •  If sample data is taken from a population that is normally

distributed, a normal probability plot of the actual values versus the expected Z-scores will be approximately linear. –  In drawing the straight line, you should be influenced more

by the points near the middle of the plot than by the extreme points.

Normal Probability Plot Interpretation*

positively skewed

negatively skewed

normal

*From http://www.stat.psu.edu/~resources/ClassNotes/hrm_05/index.htm

32

Normal Probability Plot from SYSTAT

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4RESIDUAL

-2

-1

0

1

2

Exp

ecte

d Va

lue

for N

orm

al D

istri

butio

n

Residual … Observed Data

33

Problem 2.27 (all parts except c), p. 108

34

Practice Problems

•  Exercise 2.30, p. 110 •  Exercises, pp. 113-116:

–  2.38 –  2.47 –  2.49

35

More stuff on density curves … The Y axis in the normal distribution represents the " density of probability." Intuitively, it shows the chance of obtaining values near corresponding points on the X axis. In the figure below, for example, the probability of an observation with value near 40 is about half of the probability of an observation with value near 50. You should keep the following ideas in mind about the curve that describes a continuous distribution (like the normal distribution). First, the area under the curve equals 1. Second, the probability of any exact value of X is 0. Finally, the area under the curve and bounded between two given points on the X axis is the probability that a number chosen at random will fall between the two points.