Basic Engineering Mathematics, Fifth Edition

Mean, median, mode and standard deviation 303

32.4.2 Grouped data

For grouped 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. 93

21 .0–21. 410

21 .5–21. 911

22 .0–22. 413

22 .5–22. 99

23 .0–23. 42

The standard deviation for grouped data is given by

σ=

√√

√

√

{∑

{f(x−x)^2 }

∑

f

}

From Problem 3, the distribution mean value is
x= 21 .92, correct to 2 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

and standard deviation,

σ=

√

√

√

√

{∑

{f(x−x)^2 }

∑

f

}

=

√

0. 41569

=0.645,correct to 3 significant figures.

Now try the following Practice Exercise

PracticeExercise 127 Standard deviation

(answers on page 354)

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.


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 signif-
   icant figures.


3. The tensile strength in megapascals for 15
   samplesoftinweredeterminedandfoundtobe
   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.


4. Calculate the standard deviation from the
   mean for the mass of the 50 bricks given in
   problem 1 of Practice Exercise 126, page 301,
   correct to 3 significant figures.


5. Determine the standard deviation from the
   mean, correct to 4 significant figures, for the
   heights of the 100 people given in problem 2
   of Practice Exercise 126, page 301.


6. Calculate the standard deviation from the
   mean for the data given in problem 4 of
   Practice Exercise 126, page 302, correct to 3
   decimal places.

32.5 Quartiles, deciles and percentiles

Other measures of dispersion which are sometimes

used are the quartile, decile and percentile values. The

quartile valuesof a set of discrete data are obtained by

selecting the values of members which divide the set