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TOPIC 13
Standard Deviation
Standard Deviation
The STANDARD DEVIATION is a measure of dispersion and
it allows us to assess how spread out a set of data is:
STANDARD DEVIATION FOR A SET OF NUMBERS
The formula used to calculate the STANDARD DEVIATION of
a SET OF NUMBERS is:
1.
Standard Deviation (SD) = √ ∑x2 - ( ∑x )2
n
(n)
Or
SD = √ ∑x2 - x2
n
where, x = individual data values
n = number of data values
x = mean
Standard Deviation For a Set of Numbers
Example 1
Calculate the standard deviation of this
set of numbers:
179, 86, 137, 140, 86, 104, 125
Answer 1
SD = √ ∑x2 - ( ∑x)2
n
(n)
= √111643 – (857)2
7
(7)
= √15949 – 122.4292
= √15949 – 14988.7551
= √960. 245
= 30.99
x
x2
179
32041
86
7396
137
18769
140
19600
86
7396
104
10816
125
15625
∑x = 857
∑x2 = 111643
Standard Deviation For a Set of Numbers
Another important measure in statistics is the VARIANCE.
VARIANCE = (STANDARD DEVIATION)2
Therefore, for a SET OF NUMBERS:
Variance = ∑x2 - ( ∑x )2
n
(n)
So for Example 1, variance = 960.245
Note: Adding the same number to (or subtracting the same
number from) all data values has no effect on the SD.
Multiplying (or dividing) all the data values by the same
number means the SD is also multiplied (or divided) by
this number.
Standard Deviation For a Frequency Distribution
STANDARD DEVIATION FOR FREQUENCY
DISTRIBUTION
The formula used to calculate the STANDARD DEVIATION of
a FREQUENCY DISTRIBUTION is:
2.
Standard Deviation (SD) = √ ∑fx2 - ( ∑fx )2
n
(n)
Or
SD = √ ∑fx2 - x2
n
where, x = data values
f = frequency
n = total frequency
x = mean
Standard Deviation For a Frequency Distribution
Example 2
Find the standard deviation of the following distribution of the number
of children per family.
Answer 2
Children (x) Frequency (f)
x2
fx2
fx
0
5
0
0
0
1
16
1
16
16
2
22
4
88
44
3
8
9
72
24
4
5
16
80
20
5
3
25
75
15
6
1
36
36
6
∑fx2 = 367
∑fx = 125
n = 60
Standard Deviation For a Frequency Distribution
Answer 2
SD = √ ∑fx2 - ( ∑fx)2
n
(n)
= √367 – (125)2
60
(60)
= √6.117 – 2.0832
= √6.117 – 4.339
= √1.778
= 1.33
Standard Deviation For a Grouped Frequency Distribution
STANDARD DEVIATION FOR GROUPED FREQUENCY
DISTRIBUTION
The formula used to ESTIMATE the STANDARD DEVIATION of a
GROUPED FREQUENCY DISTRIBUTION is also:
3.
Standard Deviation (SD) = √ ∑fx2 - ( ∑fx )2
n
(n)
Or
SD = √ ∑fx2 - x2
n
where, x = midpoint of group
f = frequency of group
n = total frequency
x = mean
Standard Deviation For a Grouped Frequency Distribution
Example 3
Find an estimate for the standard deviation of the following distribution.
Answer 3
Age (years) Frequency (f) Midpoint (x)
x2
fx2
fx
0-4
8
2
4
32
16
5-9
11
7
49
539
77
10-14
13
12
144
1872
156
15-19
19
17
289
5491
323
20-24
7
22
484
3388
154
25-29
2
27
729
1458
54
∑fx2 =
12780
∑fx =
780
n = 60
Standard Deviation For a Grouped Frequency Distribution
Answer 3
SD = √ ∑fx2 - ( ∑fx)2
n
(n)
= √12780 – (780)2
60
(60)
= √213 – 132
= √213 – 169
= √44
= 6.63
Variance = ∑fx2 - ( ∑fx )2 = 44
n
(n)