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Variance, standard deviation ... Range and Inter-quartile range are relatively easy to compute. ... Definition: The Sample Standard Deviation is defined by:.
Typology: Lecture notes
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0.02 0 0.040.
0.080.
0.120.
0 5 10 15 20 25
Variability
Definition Let min = the smallest observation Let max = the largest observation Then Range =max - min
0.02 0 0.040.
0.080.
0.120.
0 5 10 15 20 25
Range
Definition Let Q 1 = the first quartile, Q 3 = the third quartile Then the Inter-Quartile Range = IQR = Q 3 - Q 1
0
0 5 Q 1 10 Q 315 20 25
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
min = 80 Q 1 = 89 Q 2 = 96 Q^3 = 105 max = 119
The numbers
are called deviations from the the mean
d (^) 1 = x 1 − x d (^) 2 = x 2 − x d (^) 3 = x 3 − x M d (^) n = xn − x
The sum
is called the sum of squares of deviations from the the mean. Writing it out in full:
or
= =
n i i
n i
di x x 1
2 1
2
2 2 3 2 2 2 d 1 (^) + d + d +L+ dn
Is defined as the quantity:
and is denoted by the symbol
1
2 1
2
−
= = n
x x n
d
n i i
n i i
s^2
Example Let x 1 , x 2 , x 3 , x 3 , x 4 , x 5 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 = x 1 + x 2 + x 3 + x 4 + x 5 = 10 + 15 + 21 + 7 + 13 = 66 and
5 i 1 i
x
n
x x x x x n
x x n n
n
The deviations from the mean d 1 , d 2 , d 3 , d 4 , d 5 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.
0
0 5 10 15 20 25
s
Inflection point
Mode
s
2/
s
2s
A researcher collected data on 1500 males aged 60-65. The variable measured was cholesterol and blood pressure.
The sum of squares of deviations from the the mean can also be computed using the following identity:
n
x x x x
n i n i i i
n i i
2
1 1
2 1
2
= = =
To use this identity we need to compute:
1 2 and 1 n
n i
=
12 22 2 1
n (^2) n i
=
Then:
n
x x x x
n i n i i i
n i i
2
1 1
2 1
2
= = =
and
2
1 1
2 1
2 2 −
= = = n
n
x x n
x x s
n i n i i i
n i i
and 2
1 1
2 1
2
−
= = = n
n
x x n
x x s
n i n i i i
n i i
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
=
n i
xi 1
=
n i
xi 1
2
A quick (rough) calculation of s
The reason for this is that approximately all (95%) of the observations are between and Thus
s ≈^ Range
x − 2 s x + 2 s. max ≈ x + 2 s and min≈ x − 2 s.
= 4 s 4 Hence s ≈Range
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.
s ≈^119 -^80 = =
The Pseudo Standard Deviation (PSD)
Definition: The Pseudo Standard Deviation (PSD) is defined by:
InterQuartile Range
Properties
Example Verbal IQ on n = 23 students Inter-Quartile Range = IQR = Q 3 - Q 1 = 105 – 89 = 16 Pseudo standard deviation
This compares with the standard deviation
=PSD = 1 IQR. 35 = 116. 35 = 11. 85
s = 10. 782
Inner fences Outer fence
Mild outliers
Box-Whisker plot Extreme outlier representing the data that are not outliers
0.020.04 0 0.060.080.
0.120.140.
0 5 10 15 20 25 0.02^0 0.040.060.
0.120.140.
0.02 (^00 5 10 15 20 ) 0.040.060.
0.120.140. 0 5 10 15 20 25
0.020.04 0 0.060.080.
0.120. -3 -2 -1 00 1 2 3 0 5 10 15 20 25 -3 -2 -1 00 1 2 3
=
n i
xi x 1
3
=
n i
xi x 1
4
