Mind Expanding Exercises
695
9
559504
62572
2
9
1
9
1
2
=
=
⎟
⎠
⎞
⎜
⎝
⎛
=
∑
∑
=
=
n
x
x
i
i
i
i
11
.
7
61
.
50
61
.
50
1
9
9
559504
2572
6
1
2
9
1
9
1
2
2
=
=
=
−
−
=
−
⎟
⎠
⎞
⎜
⎝
⎛
−
=
∑
∑
=
=
s
n
n
x
x
s
i
i
i
i
Subtract 30 and multiply by 10
9
22848400
2579200
2
9
1
9
1
2
=
=
⎟
⎠
⎞
⎜
⎝
⎛
=
∑
∑
=
=
n
x
x
i
i
i
i
658
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.
71
1
.
5061
1
.
5061
1
9
9
22848400
579200
2
1
2
9
1
9
1
2
2
=
=
=
−
−
=
−
⎟
⎠
⎞
⎜
⎝
⎛
−
=
∑
∑
=
=
s
n
n
x
x
s
i
i
i
i
Yes, the rescaling is by a factor of 10.
Therefore, s
2
and s would be rescaled by multiplying s
2
by 10
2
(resulting in 5061.1) and s by 10 (71.14). Subtracting 30 from each value has no affect on the variance or
standard deviation.
This is because
.
)
(
)
(
2
X
V
a
b
aX
V
=
+
696
2
1
2
1
2
)
(
)
(
)
(
a
x
n
x
x
a
x
n
i
i
n
i
i
−
+
−
=
−
∑
∑
=
=
;
The sum written in this form shows that the quantity is
minimized when a =
x .
697
Of the two quantities
()
∑
=
−
n
i
i
x
x
1
2
and
, the
quantity
(
∑
=
−
n
i
i
x
1
2
μ
)
(
∑
=
−
n
i
i
x
x
1
2
)
will be smaller
given that
≠
x
.
This is because
x
is based on the values of the
’s.
The value of
μ
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 Spring '14
 Standard Deviation, Mean, 10%

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