FRAME 37.
For example:
(X-u ) 2
X
X-u
15
- 15
(-15)(-15) =
225
Note that a minus times
20
- 10
(-10)(-10) =
100
a minus always equals
25
-5
(-5)(-5) =
25
a plus.
30
0
(0)(0)
=
0
35
5
(5)(5)
=
25
40
10
(10)(10) =
100
45
15
(15)(15) =
225
Σ(X-u) =0,
Σ(X-u) 2
700
By squaring the differences, we have eliminated the problem of a total being equal to zero.
But this total of the squares does not consider the number of observations that contributed to the
result is a value called variance. For our example:
variance = Σ(X-u)2 = 700 = 100
N
7
FRAME 38.
Squaring the individual differences increased their magnitude. We must adjust or correct for this
increased magnitude or our measure of dispersion will be far too large and thus not
representative. The method for correcting this is to perform the opposite operation. Since we
squared the individual differences, the opposite operation would be to take the square root of the
variance. The square root of the variance is called standard deviation. Standard deviation is a
measure of dispersion Just like the centimeter is a measure of distance. Just as 5 centimeters
represents a greater distance than 2 centimeters, a standard deviation of 4 represents a greater
dispersion than a standard deviation of 2. The more variability or scatter in a group of
observations, the larger the standard deviation will be. To find the standard deviations when the
variance is 100, the following equation is used.:
standard deviation =
variance =
10
= 10
53
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