The Variance -- A common measurement of the variability between numbers in a data set.
Two Formulas
a.
Population Variance (
b.
Sample Variance (s
2
)
4.
The Standard Deviation – Probably the most common and widely used measure of variability.
Two Formulas
a.
Population Standard Deviation
b.
Sample Standard Deviation
Note: The standard deviation is simply the square root of the variance.

Module 3: Measures of Variability – Ungrouped Data
5.
The Coefficient of Variation (CV) – Another measure of variability that is less common than the variance or standard
deviation.
The coefficient of variation is always expressed as a percent.
a.
Population CV
x 100
=
b.
Sample CV =
x 100
=
•

Module 3: Measures of Variability – Ungrouped Data
Example 1: For the following small data set, calculate the following: (5, 9, 16, 17, 18)
a.
The Population Variance
b.
The Population Standard Deviation
c.
The Population Coefficient of Variation
d.
The Sample Variance
e.
The Sample Standard Deviation
f.
The Sample Coefficient of Variation

Module 3: Measures of Variability – Ungrouped Data
Solution for Example 1: For the following small data set, calculate the following: (5, 9, 16, 17, 18)
a.
2 =
= 26
b.
=
= 5.0990
c. Population CV = (100) = 39.2231%
d. s
2
=
= 32.5
e. s = = 5.7009
f. Sample CV = (100) = 43.8531%
Note: The sample statistics are larger then the population parameters, reflecting the added uncertainty associated with
sample vs. population data.
•
x
x
2
5
25
9
81
16
256
17
289
18
324
∑x = 65
∑x
2
= 975
mean = 65/5 = 13

Module 3: Measures of Variability – Ungrouped Data
Solution for Example 1 using Excel
Note: The data was in rows 12 through 16 in Column A.
There is no CV function in Excel.
You must divide the standard deviation by the mean and express as a %.
x
5
9
16
17
18
5
= Sample Size
=COUNT(A12:A16)
13
= Mean
=AVERAGE(A12:A16)
26
=
2
=VARP(A12:A16)
5.0990
=
=STDEVP(A12:A16)
39.2232%
= Population CV
=(A21/A19)
32.5
= s
2
=VAR(A12:A16)
5.7009
= s
=STDEV(A12:A16)
43.85%
= Sample CV
=(A24/A19)

Module 3: Measures of Variability – Ungrouped Data
Example 2: Assume the following data is a sample randomly obtained from a population. Calculate:
a.
Sample Size (n)
b.
The Sample Mean (
)
c.
The Sample Variance (s
2
)
d.
The Sample Standard Deviation (s)
e.
The Sample Coefficient of Variation
66
89
79
62
85
72
88
73
72
83
79
86
82
82
60
81
88
80
82
69
65
69
85
70
51
51
83
82
64
59
77

Module 3: Measures of Variability – Ungrouped Data
Example 2 Solution
51
a.
n =
31
51
b.
=
74.6452
59
c.
s
2
=
116.5032
60
d.
s =
10.79367
62
e.
Sample CV =
14.4600%
64
65
66
69
69
70
72
72
73
77
79
79
80
81
82
82

Module 3: Measures of Variability – Ungrouped Data
Example 3: Assume the following data is the population of bags of cheese curds, measured in grams, a bags of cheese
curds at Middlefield Cheese on a particular day. Calculate:
a.
The Population Size (N)
b.
The Population Mean (
)
c.
The Population Variance (
2
)
d.
The Population Standard Deviation(
)
e.
The Population Coefficient of Variation

Module 3: Measures of Variability – Ungrouped Data
Example 3 Solution:
150
a.
N =
24
89
b.
=
135.0417
112
c.
2
=
542.1233
156
d.

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- Standard Deviation, Ungrouped Data