Chapter 2:
The Normal Distributions
Distribution of
data can be
approximated by
a smooth density
curve
Red shaded region
represents an
approximation of the
fraction of scores
between 6 and 8
6>scores>8 same as
6<scores>8
Area A, represents
the proportion
observations falling
between values a
and b
Median
Symmetric density
curve.
Area
0.25
Area Area
0.25 0.25
Q1
Q2
Area
0.25
Density curve for uniform distribution
1/5 or 0.2
because total
area must
equal 1
If d = 7 and c
=2, what would
be the height of
the curve?
Area = l x w = (7-2) x 1/5 = 1
Normal Distributions
All Normal distributions
have this general shape.
m indicates the mean of a
density curve.
s indicates the standard
deviation of a density curve.
Three normal distributions
Differing m results in center of graph
at different location on the x axis
Differing s results in varying
spread
Inflection points
-1s
+1s
Mean
(aka – The Empirical rule)
Normal distributions are
abbreviated as ; N(m,s)
The Normal distribution
with mean of 0 and
standard deviation of 1 is
called the standard
Normal curve; N(0,1)
Example
What is the z score
for Iowa test score of
3.74?
z = (x-m)/s
N(6.84,1.55)
z = (3.74 – 6.84)/1.55
z = -2
This says that a score of
3.74 is 2 standard deviation
below the mean
Using the 68 – 95 – 99.7 rule to solve problems
Example
What percentage of scores are greater than
5.29?
Finding Normal Percentiles by
• Table A is the standard Normal table. We have to
convert our data to z-scores before using the table.
• The figure shows us how to find the area to the left
when we have a z-score of 1.80:
Using Table A to find the area under the standard normal
curve that lies (a) to the left of a specified z-score, (b) to
the right of a specified z-score, and (c) between two
specified z-scores
•
•
Say a toy car goes an average of 3,000 yards
between recharges, with a standard deviation of
50 yards (i.e., µ = 3,000 and s = 50)
What is the probability that the car will go more
than 3,100 yards without recharging?
3100  3000 

P ( x  3100 )  P  z 

50


P ( z  2 . 00 )  1  P ( z  2 . 00 ) 
1  P ( z  2 . 00 ) 
1  . 9772 
. 0228
Determine the percentage of people having IQs between
115 and 140
P[115< x < 140]
P[(115-100)/16 < z < (140-100)/16]
P[0.94< z < 2.50]
= 0.9938 – 0.8264
= 0.1674 = 16.74%
From Percentiles to Scores: z in Reverse
• Sometimes we start with areas and need to find the
corresponding z-score or even the original data value.
• Example: What z-score represents the first quartile in a
Normal model?
From Percentiles to Scores: z in Reverse
• Look in Table A for an area of 0.2500.
• The exact area is not there, but 0.2514 is pretty close.
• This figure is associated with z = –0.67, so the first
quartile is 0.67 standard deviations below the mean.
• To unstandardize; solve x = m + zs
Example
What score is the 90th
percentile for N(504,22)?
X = zs + m
= 1.28(22) + 504
= 532.16
m
z
Methods for Assessing Normality
•
If the data are normal
– A histogram or stem-and-leaf display will look like the normal
curve
– The mean ± s, 2s and 3s will approximate the empirical rule
percentages. (68%,95%,99.7%)
– The ratio of the interquartile range to the standard deviation
will be about 1.3
– A normal probability plot , a scatterplot with the ranked data on
one axis and the expected z-scores from a standard normal
distribution on the other axis, will produce close to a straight
line
Errors per MLB team in 2003
–
– Mean: 106
Standard Deviation: 17
– IQR: 22
Frequency
Histogram
10
9
8
7
6
5
4
3
2
1
0

IQR

s
22

 1 . 29
17
x  s  106  17
89  123

22 out of 30: 73%
x  2 s  106  34
Frequency
77
89.8 102.6 115.4 128.2 More
Errors per team, 2003
72  140
28 out of 30: 93%
x  3 s  106  51
55  157
30 out of 30: 100%

3
Normal Quantile
2
1
0
-1
-2
-3
60
80
100
Errors
120
140
160
A normal probability
plot is a scatterplot with
the ranked data on one
axis and the expected zscores from a standard
normal distribution on
the other axis
• A skewed distribution might have a histogram and
Normal probability plot like this:
Khan Academy Videos
http://www.khanacademy.org/math/statistics/v/introduction-to-the-normaldistribution
http://www.khanacademy.org/math/statistics/v/ck12-org-normaldistribution-problems--qualitative-sense-of-normal-distributions
http://www.khanacademy.org/math/statistics/v/ck12-org-normaldistribution-problems--z-score
http://www.khanacademy.org/math/statistics/v/ck12-org-normaldistribution-problems--empirical-rule
http://www.khanacademy.org/math/statistics/v/ck12-org-exercise-standard-normal-distribution-and-the-empirical-rule
http://www.khanacademy.org/math/statistics/v/ck12-org--more-empiricalrule-and-z-score-practice