Higher Engineering Mathematics, Sixth Edition

Measures of central tendency and dispersion 545

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.

Hence,standard deviation,

σ=

√{∑

(x−x)^2

n

}

=

√

5. 6

=2.380,correct to 4 significant figures.

(b) Grouped data

Forgrouped data, standard deviation

σ=

√√

√

√

{∑

{f(x−x)^2 }

∑

f

}

wherefis the class frequency value,xis the class mid-
point value andxis the mean value of the grouped data.
The method of determining the standard deviation for a
set of grouped data is shown in Problem 6.

Problem 6. The frequency distribution for the

values of resistance in ohms of 48 resistors is as

shown. Calculate the standard deviation from the

mean of the resistors, correct to 3 significant figures.

20.5–20.9 3, 21.0–21.4 10,

21.5–21.9 11, 22.0–22.4 13,

22.5–22.9 9, 23.0–23.4 2

The standard deviation for grouped data is given by:

σ=

√{∑

{f(x−x)^2 }

∑

f

}

From Problem 3, the distribution mean value,

x= 21 .92, correct to 4 significant figures.

The ‘x-values’ are the class mid-point values, i.e. 20.7,

21.2, 21. 7 ,...

Thus the (x−x)^2 values are ( 20. 7 − 21. 92 )^2 ,

( 21. 2 − 21. 92 )^2 ,( 21. 7 − 21. 92 )^2 ,...

and the f(x−x)^2 values are 3( 20. 7 − 21. 92 )^2 ,

10 ( 21. 2 − 21. 92 )^2 ,11( 21. 7 − 21. 92 )^2 ,...

The

∑

f(x−x)^2 values are

4. 4652 + 5. 1840 + 0. 5324 + 1. 0192 + 5. 4756

+ 3. 2768 = 19. 9532

∑{

f(x−x)^2

}

∑

f

=

19. 9532

48

= 0. 41569

andstandard deviation,

σ=

√

√

√

√

{∑{

f(x−x)^2

}

∑

f

}

=

√

0. 41569

=0.645,correct to 3 significant figures.

Now try the following exercise

Exercise 209 Further problemson

standard deviation

1. Determine the standard deviation from the
   mean of the set of numbers:

{ 35 , 22 , 25 , 23 , 28 , 33 , 30 }

correct to 3 significant figures. [4.60]

2. The values of capacitances, in microfarads, of
   ten capacitors selected at random from a large
   batch of similar capacitors are:

34.3, 25.0, 30.4, 34.6, 29.6, 28.7, 33.4,

32.7, 29.0 and 31.3

Determine the standard deviation from the

mean for these capacitors, correct to 3 signifi-

cant figures. [2.83μF]