Higher Engineering Mathematics

542 STATISTICS AND PROBABILITY

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

}

wheref is the class frequency value,xis the class

mid-point value andxis the mean value of the

grouped data. The method of determining the stan-

dard 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 210 Further problems on 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 sig-

nificant figures. [2.83μF]

3. The tensile strength in megapascals for 15
   samples of tin were determined and found
   to be:
   34.61, 34.57, 34.40, 34.63, 34.63,

34.51, 34.49, 34.61, 34.52, 34.55,

34.58, 34.53, 34.44, 34.48 and 34.40

Calculate the mean and standard deviation

from the mean for these 15 values, correct

to 4 significant figures.

[

mean 34.53 MPa, standard

deviation 0.07474 MPa

]

4. Determine the standard deviation from the
   mean, correct to 4 significant figures, for the