Post on 12-Jan-2016
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Measures of Variability
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0 5 10 15 20 25
Variability
Measure of Variability (Dispersion, Spread)
• Variance, standard deviation
• Range
• Inter-Quartile Range
• Pseudo-standard deviation
Range
Range
Definition
Let min = the smallest observation
Let max = the largest observation
Then Range =max - min
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Range
Inter-Quartile Range (IQR)
Inter-Quartile Range (IQR)
Definition
Let Q1 = the first quartile,
Q3 = the third quartile
Then the
Inter-Quartile Range
= IQR = Q3 - Q1
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0 5 10 15 20 25Q1 Q3
25% 25%
50%
Inter-Quartile Range
Example
The data Verbal IQ on n = 23 students arranged in increasing order is:
80 82 84 86 86 89 90 94
94 95 95 96 99 99 102 102
104 105 105 109 111 118 119
Example
The data Verbal IQ on n = 23 students arranged in increasing order is:
80 82 84 86 86 89 90 94 94 95 95 96 99 99 102 102 104 105 105 109 111 118 119
Q2 = 96Q1 = 89 Q3 = 105min = 80 max = 119
Range
Range = max – min = 119 – 80 = 39
Inter-Quartile Range
= IQR = Q3 - Q1 = 105 – 89 = 16
Some Comments
• Range and Inter-quartile range are relatively easy to compute.
• Range slightly easier to compute than the Inter-quartile range.
• Range is very sensitive to outliers (extreme observations)
Varianceand
Standard deviation
Sample Variance
Let x1, x2, x3, … xn denote a set of n numbers.
Recall the mean of the n numbers is defined as:
n
xxxxx
n
xx nn
n
ii
13211
The numbers
are called deviations from the the mean
xxd 11
xxd 22
xxd 33
xxd nn
The sum
is called the sum of squares of deviations from the the mean.
Writing it out in full:
or
n
ii
n
ii xxd
1
2
1
2
223
22
21 ndddd
222
21 xxxxxx n
The Sample Variance
Is defined as the quantity:
and is denoted by the symbol
111
2
1
2
n
xx
n
dn
ii
n
ii
2s
Example
Let x1, x2, x3, x3 , x4, x5 denote a set of 5 denote the set of numbers in the following table.
i 1 2 3 4 5
xi 10 15 21 7 13
Then
= x1 + x2 + x3 + x4 + x5
= 10 + 15 + 21 + 7 + 13
= 66
and
5
1iix
n
xxxxx
n
xx nn
n
ii
13211
2.135
66
The deviations from the mean d1, d2, d3, d4, d5 are given in the following table.
i 1 2 3 4 5
xi 10 15 21 7 13
di -3.2 1.8 7.8 -6.2 -0.2
The sum
and
n
ii
n
ii xxd
1
2
1
2
22222 2.02.68.78.12.3
80.112
04.044.3884.6024.324.10
2.28
4
8.112
11
2
2
n
xxs
n
ii
The Sample Standard Deviation s
Definition: The Sample Standard Deviation is defined by:
Hence the Sample Standard Deviation, s, is the square root of the sample variance.
111
2
1
2
n
xx
n
ds
n
ii
n
ii
In the last example
31.52.28
4
8.112
11
2
2
n
xxss
n
ii
Interpretations of s
• In Normal distributions– Approximately 2/3 of the observations will lie
within one standard deviation of the mean– Approximately 95% of the observations lie
within two standard deviations of the mean– In a histogram of the Normal distribution, the
standard deviation is approximately the distance from the mode to the inflection point
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s
Inflection point
Mode
s
2/3
s
2s
Example
A researcher collected data on 1500 males aged 60-65.
The variable measured was cholesterol and blood pressure.
– The mean blood pressure was 155 with a standard deviation of 12.
– The mean cholesterol level was 230 with a standard deviation of 15
– In both cases the data was normally distributed
Interpretation of these numbers
• Blood pressure levels vary about the value 155 in males aged 60-65.
• Cholesterol levels vary about the value 230 in males aged 60-65.
• 2/3 of males aged 60-65 have blood pressure within 12 of 155. Ii.e. between 155-12 =143 and 155+12 = 167.
• 2/3 of males aged 60-65 have Cholesterol within 15 of 230. i.e. between 230-15 =215 and 230+15 = 245.
• 95% of males aged 60-65 have blood pressure within 2(12) = 24 of 155. Ii.e. between 155-24 =131 and 155+24 = 179.
• 95% of males aged 60-65 have Cholesterol within 2(15) = 30 of 230. i.e. between 230-30 =200 and 230+30 = 260.
Measures of Variability
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 5 10 15 20 25
Variability
Measure of Variability (Dispersion, Spread)
• Variance, standard deviation
• Range
• Inter-Quartile Range
• Pseudo-standard deviation
Range
Range =max – min
Interquartile range (IQR)
IQR = Q3 – Q1
The Sample Variance
111
2
1
2
2
n
xx
n
ds
n
ii
n
ii
2s
The Sample standard deviation
111
2
1
2
n
xx
n
ds
n
ii
n
ii
2s
A Computing formula for:
Sum of squares of deviations from the the mean :
The difficulty with this formula is that will have many decimals.
The result will be that each term in the above sum will also have many decimals.
n
ii xx
1
2
x
The sum of squares of deviations from the the mean can also be computed using the following identity:
n
x
xxx
n
iin
ii
n
ii
2
1
1
2
1
2
To use this identity we need to compute:
and 211
n
n
ii xxxx
222
21
1
2n
n
ii xxxx
Then:
n
x
xxx
n
iin
ii
n
ii
2
1
1
2
1
2
11 and
2
1
1
2
1
2
2
nn
x
x
n
xxs
n
iin
ii
n
ii
11
and
2
1
1
2
1
2
nn
x
x
n
xxs
n
iin
ii
n
ii
Example
The data Verbal IQ on n = 23 students arranged in increasing order is:
80 82 84 86 86 89 90 94
94 95 95 96 99 99 102 102
104 105 105 109 111 118 119
= 80 + 82 + 84 + 86 + 86 + 89
+ 90 + 94 + 94 + 95 + 95 + 96 + 99 + 99 + 102 + 102 + 104
+ 105 + 105 + 109 + 111 + 118 + 119 = 2244
= 802 + 822 + 842 + 862 + 862 + 892
+ 902 + 942 + 942 + 952 + 952 + 962 + 992 + 992 + 1022 + 1022 + 1042
+ 1052 + 1052 + 1092 + 1112
+ 1182 + 1192 = 221494
n
iix
1
n
iix
1
2
Then:
n
x
xxx
n
iin
ii
n
ii
2
1
1
2
1
2
652.2557
23
2244221494
2
11 and
2
1
1
2
1
2
2
nn
x
x
n
xxs
n
iin
ii
n
ii
26.116
22
652.2557
2223
2244221494
2
11 Also
2
1
1
2
1
2
nn
x
x
n
xxs
n
iin
ii
n
ii
26.116
22
652.2557
2223
2244221494
2
782.10
A quick (rough) calculation of s
The reason for this is that approximately all (95%) of the observations are between
and
Thus
4
Ranges
sx 2.2sx
sx 2max .2min and sx .22minmax and sxsxRange
s4
4
Range Hence s
Example
Verbal IQ on n = 23 students min = 80 and max = 119
This compares with the exact value of s which is 10.782.The rough method is useful for checking your calculation of s.
75.94
39
4
80-119s
The Pseudo Standard Deviation (PSD)
The Pseudo Standard Deviation (PSD)
Definition: The Pseudo Standard Deviation (PSD) is defined by:
35.1
Range ileInterQuart
35.1
IQRPSD
Properties
• For Normal distributions the magnitude of the pseudo standard deviation (PSD) and the standard deviation (s) will be approximately the same value
• For leptokurtic distributions the standard deviation (s) will be larger than the pseudo standard deviation (PSD)
• For platykurtic distributions the standard deviation (s) will be smaller than the pseudo standard deviation (PSD)
Example
Verbal IQ on n = 23 students Inter-Quartile Range
= IQR = Q3 - Q1 = 105 – 89 = 16
Pseudo standard deviation
This compares with the standard deviation
85.1135.1
16
35.1
IQRPSD
782.10s
Summary