Chapter 2: The Normal Distribution
2.1
Density Curves and the Normal
Distributions
2.2
Standard Normal Calculations
1
Histogram for Strength of Yarn Bobbins
Bobbin #1: 17.15 g/tex
Bobbin #2: 17.42 g/tex
Bobbin #3: 17.93 g/tex
X
X
X
X
15.60
16.10
16.60
17.10
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
17.60
X
X
X
X
18.10
X
18.60
19.10
19.60
2
Histogram for Strength of Yarn Bobbins
X
15.60
16.10
16.60
X
X
X
17.10
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
17.60
X
X
X
X
18.10
18.60
X
19.10
19.60
3
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.
4
Practice Problem
• Exercise 2.3, p. 83
5
Figure 2.7, p. 83
6
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
7
Figure 2.4, p. 81: The area under the density curve is the
proportion of observations taking values between 7 and 8.
8
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?
9
Figure 2.5, p. 81
10
Practice
• Exercise 2.4, p. 84
11
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.
12
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.
13
Figure 2.10, p. 85
14
σ can be found by locating the point of
inflection of the density curve.
15.60
16.10
16.60
17.10
17.60
18.10
18.60
19.10
19.60
15
68-95-99.7 rule
  1
68.3%
  2
95%
  3
99.7%
µ - 3σ
µ - 2σ
µ - 1σ
µ
µ + 1σ
µ + 2σ µ + 3σ
16
Practice Problem
• Exercise 2.8, p. 89
17
Homework
• Reading: Section 2.1, pp. 78-90
• Exercises:
– 2.2, p. 83
– 2.5, p. 84
– 2.7, p. 89
18
Standard Normal Curve
• Statisticians have made one particular normal
distribution the standard and computed the
proportions for this distribution:
μ  0, σ  1
• This mean and standard deviation define the
standard normal curve.
19
Standardizing
• Use the following transformation:
x -μ
z
, where
σ
z  standard score;
x  score on a random variable;
μ  population mean; and
σ  population standard deviation
• A z-score tells us how far a given score falls from the mean, in
terms of the standard deviation.
20
Example Problems
2.19 and 2.20, p. 95
21
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.
22
23
Practice
• Exercises 2.21 and 2.23, p. 103
24
Homework for Rockmont Weekend
• Read/finish reading Chapter 2 (the whole
thing).
• Chapter 2 test on Thursday (9/17).
25
Homework
• Read through p. 103
• Problems 2.22, 2.24, and 2.25, p. 103
26
Homework
• Exercises 2.41-2.45, pp. 113-114
27
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?
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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.
29
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.
30
Normal Probability Plot Interpretation*
negatively skewed
normal
positively skewed
*From http://www.stat.psu.edu/~resources/ClassNotes/hrm_05/index.htm
Expected Value for Normal Distribution
Normal Probability Plot from SYSTAT
2
1
0
-1
-2
-0.3 -0.2 -0.1 0.0 0.1 0.2
RESIDUAL
Residual … Observed Data
0.3
0.4
32
Problem 2.27 (all parts except c), p. 108
33
Practice Problems
• Exercise 2.30, p. 110
• Exercises, pp. 113-116:
– 2.38
– 2.47
– 2.49
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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.
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