2. DESCRIPTIVE STATISTICS
21
Sample Variance (grouped)
X
(xi x)2fi
s2 = P
fi 1
Example (Protein). x = 73.77
Class intervals
30  x < 50
50  x < 70
70  x < 90
90  x < 110
110  x < 130
130  x < 150
150  x < 170
xi
40
60
80
100
120
140
160
s2 =
xi x
-33.77
-13.77
6.23
26.23
46.23
66.23
86.23
(xi x)2
1140.4129
189.6129
38.8129
688.0129
2137.2129
4386.4129
7435.6129
fi
5
26
20
6
2
1
1
—
61
(xi x)2fi
5702.0645
4929.9354
776.2580
4128.0774
4274.4258
4386.4129
7435.6129
——
31632.7869
31632.7869
= 527.213115
60
Notice howPthe variance changes P
with the grouping. We divide by n 1 instead
of n and
fi 1 instead of
fi in order to use the sample variance in
inference procedures discussed later. This is because dividing by n 1 better
approximates (is an unbiased estimator) the population variance. Also, we say
we have n 1 degrees of freedom, i.e., once we have made n 1 choices, the
last choice is determined.
Population Variance
2
=
N
X
(xi
µ)2
i=1
N
Problem – the variance units are the square of the data units.