Higher Engineering Mathematics

MEASURES OF CENTRAL TENDENCY AND DISPERSION 541

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2. The gain of 90 similar transistors is measured
   and the results are as shown.

83.5–85.5 6, 86.5–88.5 39,

89.5–91.5 27, 92.5–94.5 15,

95.5–97.5 3

By drawing a histogram of this frequency dis-

tribution, determine the mean, median and

modal values of the distribution.

[mean 89.5, median 89, mode 88.2]

3. The diameters, in centimetres, of 60 holes
   bored in engine castings are measured and
   the results are as shown. Draw a histogram
   depicting these results and hence determine
   the mean, median and modal values of the
   distribution.

2.011–2.014 7, 2.016–2.019 16,

2.021–2.024 23, 2.026–2.029 9,

2.031–2.034 5

⎡

⎣

mean 2.02158 cm,

median 2.02152 cm,

mode 2.02167 cm

⎤

⎦

55.4 Standard deviation

(a) Discrete data

The standard deviation of a set of data gives an indi-
cation of the amount of dispersion, or the scatter, of
members of the set from the measure of central ten-
dency. Its value is the root-mean-square value of the
members of the set and for discrete data is obtained
as follows:

(a) determine the measure of central tendency, usu-

ally the mean value, (occasionally the median or

modal values are specified),

(b) calculate the deviation of each member of the
set from the mean, giving
(x 1 −x), (x 2 −x), (x 3 −x),...,

(c) determine the squares of these deviations, i.e.

(x 1 −x)^2 ,(x 2 −x)^2 ,(x 3 −x)^2 ,...,

(d) find the sum of the squares of the deviations,
that is
(x 1 −x)^2 +(x 2 −x)^2 +(x 3 −x)^2 ,...,

(e) divide by the number of members in the set,n,

giving

(x 1 −x)^2 +(x 2 −x)^2 +(x 3 −x)^2 +···

n

(f) determine the square root of (e).

The standard deviation is indicated byσ(the Greek

letter small ‘sigma’) and is written mathemati-

cally as:

Standard deviation,σ=

√

√

√

√

{∑

(x−x)^2

n

}

wherexis a member of the set,xis the mean value of

the set andnis the number of members in the set. The

value of standard deviation gives an indication of the

distance of the members of a set from the mean value.

The set:{1, 4, 7, 10, 13}has a mean value of 7 and a

standard deviation of about 4.2. The set{5, 6, 7, 8, 9}

also has a mean value of 7, but the standard devi-

ation is about 1.4. This shows that the members of

the second set are mainly much closer to the mean

value than the members of the first set. The method

of determining the standard deviation for a set of

discrete data is shown in Problem 5.

Problem 5. Determine the standard devia-

tion from the mean of the set of numbers:

{5, 6, 8, 4, 10, 3}correct to 4 significant figures.

The arithmetic mean,

x=

∑

x

n

=

5 + 6 + 8 + 4 + 10 + 3

6

= 6

Standard deviation, σ=

√{∑

(x−x)^2

n

}

The (x−x)^2 values are: (5−6)^2 ,(6−6)^2 ,(8−6)^2 ,

(4−6)^2 , (10−6)^2 and (3−6)^2.

The sum of the (x−x)^2 values,

i.e.

∑

(x−x)^2 = 1 + 0 + 4 + 4 + 16 + 9 = 34

and

∑

(x−x)^2

n

=

34

6

= 5. 6 ̇

since there are 6 members in the set.