Descriptive Statistics
Measures of Variation
Essentials: Measures of Variation
(Variation – a must for statistical analysis.)
• Know the types of measures used to look at variation and the type data
to which they apply.
• Be able to calculate the range, standard deviation and inter-quartile
range.
• Be able to determine the distance away from the mean a given value lies
in terms of standard deviations (think z-score).
• Be able to apply the Empirical Rule and Chebychev’s Theorem to
specific situations.
Measures of Variation
•
•
•
•
Range
Variance
Standard Deviation
Interquartile Range
• (IQR; see Measures of Position)
Range
• The Range of a data set is the difference between the highest value
and the lowest value.
• Example: Given
– the following data values, identify the range of the distribution.
• Values: 2, 4, 6, 8, 10
• Range = 10 – 2 = 8
Variance
• For a sample the variance is a measure of variation equal to the sum
of the squared deviation scores divided by n-1. It is also the square
of the standard deviation.
Sample Variance:
2

(
x

x
)
s 
2
n 1
Sample Standard Deviation
• Standard deviation is a measure of the typical amount an entry
deviates (or varies) from the mean.
• The more the entries are spread out, the greater the standard
deviation.
• Sample Standard Deviation(s):
Definition Formula
s  ( x  x )
2
n 1
Calculation Formula
s
n x 2  (x) 2
n(n  1)
Interpreting Standard Deviation
• Standard deviation is a measure of the typical amount an entry
deviates from the mean.
• The more the entries are spread out, the greater the standard
deviation.
Larson/Farber 4th ed.
7
Anatomy of the Standard Deviation
The Standard Deviation is the most used measure of dispersion (how spread out the data are from one another).
The value of the Standard Deviation tells us how closely the values of observations for a data set are
clustered around the mean. A lower value of the Standard Deviation for a data set indicates that the
values of that data set are spread over a relatively smaller range around the mean. A large value of
the Standard Deviation for a data set indicates that the values of that data set are spread over a
relatively larger range around the mean.
Mean = 120
Standard Deviation = 2
n = 500
Mean = 120
Standard Deviation = 20
n = 500
60
80
70
60
40
Frequency
Frequency
50
30
20
50
40
30
20
10
10
0
0
112
117
122
127
80
NOTATION
When we refer to the Population Standard Deviation, it is denoted by 
When we refer to the Sample Standard Deviation, it is denoted by s
130
180
Interpreting Standard Deviation: Empirical Rule
(68 – 95 – 99.7 Rule)
For data with a (symmetric) bell-shaped distribution, the
standard deviation has the following characteristics:
• About 68.26% of the data lie within one standard deviation of the
mean.
• About 95.44% of the data lie within two standard deviations of the
mean.
• About 99.74% of the data lie within three standard deviations of
the mean.
Interpreting Standard Deviation: Empirical
Rule (68 – 95 – 99.7 Rule)
99.7% within 3 standard deviations
95% within 2 standard deviations
68% within 1 standard
deviation
34%
2.35%
2.35%
x  3s
34%
13.5%
x  2s
Source: Larson/Farber 4th ed.
13.5%
x s
x
xs
x  2s
x  3s
Example: Using the Empirical
Rule
Example:
In a survey conducted by the National Center for
Health Statistics, the sample mean height of
women in the United States (ages 20-29) was 64
inches, with a sample standard deviation of 2.71
inches. Estimate the percent of the women whose
heights are between 64 inches and 69.42 inches.
Source: Larson/Farber 4th ed.
Solution: Using the Empirical Rule
• Because the distribution is bell-shaped, you can use the
Empirical Rule.
34%
13.5%
55.87
x  3s
58.58
x  2s
61.29
x s
64
x
66.71
xs
69.42
x  2s
72.13
x  3s
34% + 13.5% = 47.5% of women are between 64 and 69.42 inches
tall. (64 + 2.71 = 66.71 + 2.71 = 69.42; all inches)
Source: Larson/Farber 4th ed.
ADDITIONAL TOPICS
Range Rule of Thumb
• To obtain a rough estimate of the standard deviation, s,
range
s
4
• Conversely, the “minimum” value would be approximately equal to
the mean – 2*(standard deviation). The “maximum” value would
be approximately equal to the mean + 2*(standard deviation).
Population Variance & Standard Deviation
• The population variance, 2 (sigma-squared) is a measure of
variation equal to the sum of the squared deviation scores divided by
N. It is also the square of the standard deviation.
Population Variance:
2

(
x


)
 
2
Population Standard Deviation:
N
2

(
x


)

N
Interquartile Range (IQR)
• The Interquartile Range is a
measure of variation. It is the
difference between the first quartile,
Q1(25th percentile) and the third
quartile, Q3 (75th percentile).
• The Interquartile Range enables us
to determine the existence of
outliers.
• Outliers exist in a data set if any of the
values are
– Less than
or
– Greater than
Q1  1.5(IQR )
Q3  1.5(IQR )
Chebyshev’s Theorem
• The Empirical Rule applies if the distribution of the
data is approximately bell-shaped.
• Chebyshev’s Theorem applies to distributions
regardless of shape. It states that the proportion
(fraction) of data lying within K standard deviations of
the mean is always at least 1 – 1/K2, where K is any
possible number > 1.
– When K = 2: At least 3/4 (75%) of
all values lie within 2 standard
deviations of the mean.
– When K = 3: At least 8/9 (89%) of
all values lie within 3 standard
deviations of the mean.
1
1 3
 or 75%
2
2
4
1
1 8
 or 88.9%
32 9
Example: Using Chebychev’s Theorem
The age distribution for Florida is shown in the
histogram. Apply Chebychev’s Theorem to the
data using k = 2. What can you conclude?
Source: Larson/Farber 4th ed.
Solution: Using Chebychev’s Theorem
Given k = 2:
•Two S.D. below the mean = μ – 2σ = 39.2 – 2(24.8) = -10.4
(use 0 since age can’t be negative)
•Two S.D. above the mean = μ + 2σ = 39.2 + 2(24.8) = 88.8
At least 75% of the population of Florida is between 0 and
88.8 years old.
Source: Larson/Farber 4th ed.
End of Slides