082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

29.6 Standard deviation 941
29.6 STANDARDDEVIATION
Althoughthemeanindicateswherethecentreofasetofnumberslies,itgivesnomeasure
ofthespreadofthenumbers.Forexample, −1,0,1and −10,0,10bothhaveameanof
0 but clearly the numbers in the second set are much more widely dispersed than those
inthe first. Acommonly used measure of dispersion isthestandard deviation.
Let x 1
,x 2
,...,x n
be n measurements with a meanx. Then x i
−x is the amount by
whichx i
differs from the mean. The quantityx i
−x is called the deviation ofx i
from
the mean. Some of these deviations will be positive, some negative. The mean of these
deviations is always zero (see Question 3 in Exercises 29.6) and so this is not helpful
in measuring the dispersion of the numbers. To avoid positive and negative deviations
summing to zero the squared deviation is taken, (x i
−x) 2 . The variance is the mean of
the squared deviations:
variance =
∑ (xi −x) 2
n
and
standarddeviation = √ variance
Standard deviation has the same units as thex i
.
Example29.5 Calculate the standarddeviation of
(a) −1,0,1
(b) −10, 0,10
Solution (a) x 1
= −1,x 2
= 0,x 3
= 1.Clearlyx=0.
x 1
−x=−1 x 2
−x=0 x 3
−x = 1
variance = (−1)2 +0 2 +1 2
3
= 2 3
standard deviation =
√
2
3 = 0.816
(b) x 1
= −10,x 2
= 0,x 3
= 10. Againx=0andsox i
−x=x i
,fori=1,2,3.
variance = (−10)2 +0 2 +10 2
= 200
3 3
√
200
standard deviation =
3 = 8.165
As expected, the second set has a much higher standard deviation than the first.