The Chemistry Maths Book, Second Edition

600 Chapter 21Probability and statistics

Qualitative measures of spread are the range, the difference between the extreme

data values, and the interquartile range, containing the middle 50% of values, with

upper quartileand lower quartileranges on either side, each containing 25%. These

measures have little use in physical applications. Quantitative measures of spread

are obtained from consideration of the deviationsx

i

1 −1Eof data values from the mean.

The mean deviation (summing over all data elements) is necessarily zero because of

the definition of the mean:

that is, the positive and negative deviations from the mean cancel. The spread of a

distribution is instead nearly always measured in terms of the mean of the squaresof

the deviations,

(21.4)

or, in terms of the kdistinct values (or classes),

(21.5)

The quantityV(x)is called the varianceof the distribution. The square root of the

variance is called the standard deviation;V 1 = 1 s

2

and

(21.6)

(but see Section 21.11 for corrections to Vand swhen Nis not large). The standard

deviation has the same units and dimensions as the data elements.

We note that if then

i

in (21.3) and (21.5) are interpreted as masses at positionsx

i

on a straight line, thenEis the position of the centre of mass,NV(x)is the moment

of inertia with respect to the centre of mass, and sis the radius of gyration.

EXAMPLES 21.2Standard deviation

The mean value of the data in Table 21.2 is 4.98 and the variance is

The standard deviation is therefore , and 70% of the data lies within s

of the mean.

0 Exercise 5

sV==. 195

Vnx

i

ii

=−.=.

=

∑

1

50

498 380

1

11

2

()

sVx

N

xx

i

N

i

== −

=





























∑

() ( )

1

2

1

1

2

Vx

N

nx x

i

k

ii

()=−( )

=

∑

1

1

2

Vx

N

xx

i

N

i

()=−( )

=

∑

1

1

2

111

10

111

N

xx

N

xx

N

xx

i

N

i

i

N

i

i

N

===

∑∑∑

()− = − =−=