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The Mean Deviation and the Standard Deviation
Vartanian: SW 540
n
Mean Deviation = ∑
i
| xi − x|
n
The value for the mean deviation gives the average distance from the mean.
Example: Income in thousands of dollars.
Person 1: 10
Person 2: 15
Person 3: 20
Person 4: 25
Person 5: 30
X = 20
MD =
|10 − 20|+ |15 − 20|+ |20 − 20|+ |25 − 20|+ |30 − 20| 30
=
=6
5
5
Standard Deviation for a sample:
The standard deviation for the population is the same as above except the denominator is simply
n, not –1.
Thus, the standard deviation is:
S=/(10-20)2+(15-20)2+(20-20)2+(25-20)2+(30-20)2/4
S=/(100+25+0+25+100)/4
S=/250/4=
S=/62.5
S=7.906
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A computationally easier method for computing the standard deviation for the population is
In the case above,
s=
s=
s=
s=
s=
1
5(100 + 225 + 400 + 625 + 900) − (100) 2 =
5
1
5 * 2250 − 10000 =
5
1
11250 − 10000 =
5
1
1250 =
5
1
(35.3553) = 7.071067812
5
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The Mean for Grouped Data
Vartanian: SW 540
Example:
$
0-100
101-200
201-300
301-400
Cases
10
50
70
20
Midpoint
50
150
250
350
To determine the mean, use the midpoints and multiply by the number of cases in each category.
Then divide by the total number of cases.
10 * 50 + 50 *150 + 70 * 250 + 20 * 350
=
150
500 + 7500 + 17500 + 7000
X=
=
150
32500
X=
= 216.667
150
X=
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