Basic Engineering Mathematics, Fifth Edition

302 Basic Engineering Mathematics

3. The gain of 90 similar transistors is measured
   and the results are as shown. By drawing a
   histogramofthisfrequencydistribution,deter-
   mine the mean, median and modal values of
   the distribution.
   83 .5–85. 56
   86 .5–88. 539
   89 .5–91. 527
   92 .5–94. 515
   95 .5–97. 53


4. 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

32.4 Standard deviation

32.4.1 Discrete data

Thestandarddeviationofaset ofdatagivesanindication

of the amount of dispersion, or the scatter, of members

of the set from the measure of central tendency. Its value

is the root-mean-square value of the members of the set

and for discrete data is obtained as follows.

(i) Determine the measure of central tendency, usu-

ally the mean value, (occasionally the median or

modal values are specified).

(ii) Calculate the deviation of each member of the set

from the mean, giving

(x 1 −x), (x 2 −x), (x 3 −x),...

(iii) Determine the squares of these deviations; i.e.,

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

(iv) Find the sum of the squares of the deviations, i.e.

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

(v) Divide by the number of members in the set,n,

giving

(x 1 −x)^2 +(x 2 −x)^2 +

(

x^3 −x

) 2

+···

n

(vi) Determine the square root of (v).

The standard deviation is indicated byσ(the Greek

letter small ‘sigma’) and is written mathematically 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 deviation 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 deviation

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 ,is 1+ 0 + 4 + 4 + 16 + 9 = 34

and

∑

(x−x)^2

n

=

34

6

= 5.

·

6

since there are 6 members in the set.

Hence,standard deviation,

σ=

√

√

√

√

{∑

(x−x)^2

n

}

=

√

5.

·

6 =2.380,

correct to 4 significant figures.