Standard Deviation

1
TOPIC 13

• Standard Deviation

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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
• Standard Deviation (SD) v ?x2 - ( ?x )2
• n (n)
• Or SD v ?x2 - x2
• n
• where, x individual data values
• n number of data values
• x mean

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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 v ?x2 - ( ?x)2
• n (n)
• v111643 (857)2
• 7 (7)
• v15949 122.4292
• v15949 14988.7551
• v960. 245
• 30.99

x x2
179 32041
86 7396
137 18769
140 19600
86 7396
104 10816
125 15625

?x 857 ?x2 111643
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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.

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Standard Deviation For a Frequency Distribution

• STANDARD DEVIATION FOR FREQUENCY DISTRIBUTION
• The formula used to calculate the STANDARD
  DEVIATION of
• a FREQUENCY DISTRIBUTION is
• Standard Deviation (SD) v ?fx2 - ( ?fx )2
• n (n)
• Or SD v ?fx2 - x2
• n
• where, x data values
• f frequency
• n total frequency
• x mean

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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

n 60 ?fx2 367 ?fx 125
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Standard Deviation For a Frequency Distribution

• Answer 2
• SD v ?fx2 - ( ?fx)2
• n (n)
• v367 (125)2
• 60 (60)
• v6.117 2.0832
• v6.117 4.339
• v1.778
• 1.33

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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
• Standard Deviation (SD) v ?fx2 - ( ?fx )2
• n (n)
• Or SD v ?fx2 - x2
• n
• where, x midpoint of group
• f frequency of group
• n total frequency
• x mean

9
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

n 60 ?fx2 12780 ?fx 780
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Standard Deviation For a Grouped Frequency
Distribution

• Answer 3
• SD v ?fx2 - ( ?fx)2
• n (n)
• v12780 (780)2
• 60 (60)
• v213 132
• v213 169
• v44
• 6.63
• Variance ?fx2 - ( ?fx )2 44
• n (n)