The Chemistry Maths Book, Second Edition

21.3 Frequency and probability 601

The variance (21.5) can be written in simpler form as ‘the mean of the squares minus

the square of the mean’:

Therefore, writing for the mean of the squares,

(21.7)

Mean absolute deviation

An alternative measure of the spread of a distribution is the mean of the absolute

deviations ,

(21.8)

Although this is seldom used, it can be useful for broad distributions with significant

numbers of points remote from the centre.

Skewness and kurtosis

The asymmetry of a set of data is measured by the mean cube deviation, with skewness

(or skew) defined by

(21.9)

This quantity is zero for a symmetrical distribution, positive if a tail extends further

to the right than to the left, and negative if a tail extends further to the left than to

the right. It is an important statistic of a distribution, but seldom used in the physical

sciences. Even less frequently used is the kurtosis, defined in terms of the mean

fourth-power deviation.

0 Exercise 6

21.3 Frequency and probability

The simplest nontrivial statistical experiment is one that has only two possible

outcomes: true or false, on or off, success or failure. Table 21.4 shows some results for

the tossing of a coin (an unbiased coin); Nis the number of tosses,n(H)is the number,

or frequency, of heads, andf(H) 1 = 1 n(H) 2 Nis the fraction, or relative frequency, of heads.

γ=

−

















=−+

( )

=

∑

11

32

1

3

3

323

N

xx

s

s

xxxx

i

N

i

1

1

N

xx

i

N

i

=

∑

−

xx

i

−

Vx x x()=−

22

x

N

x

i

N

i

2

1

2

1

=

=

∑

=−+

=

∑

1

2

1

22

N

xxxx

i

N

i

=−+=−

===

∑∑∑

1

2

1

2

1

1

22

1

2

1

N

xxxx

N

xx

N

x

i

N

ii

i

N

i

i

N

()

ii

i

N

x

N

−

=

∑

2

1

1

1

Vx

N

xx

i

N

i

()=−( )

=

∑

1

1

2