544 Higher Engineering Mathematics
Now try the following exercise
Exercise 208 Further problems on mean,
median and mode for grouped data
1. The frequency distribution given below refers
to the heights in centimetres of 100 people.
Determine the mean value of the distribution,
correct to the nearest millimetre.
150–156 5, 157–163 18,
164–170 20, 171–177 27,
178–184 22, 185–191 8
[171.7 cm]
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 distribution, 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 indication
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:
(a) determine the measure of central tendency, usually
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 by σ (the Greek
letter small ‘sigma’) and is written mathematically as:
Standard deviation, σ =
(x − x )
2
n
where x is a member of the set, x is the mean value of
the set and n is 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.
Now try the following exercise
Exercise 208 Further problems on mean,
median and mode for grouped data
1. The frequency distribution given below refers
to the heights in centimetres of 100 people.
Determine the mean value of the distribution,
correct to the nearest millimetre.
150–156 5, 157–163 18,
164–170 20, 171–177 27,
178–184 22, 185–191 8
[171.7 cm]
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 distribution, 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 indication
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:
(a) determine the measure of central tendency, usually
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 by σ (the Greek
letter small ‘sigma’) and is written mathematically as:
Standard deviation, σ =
(x − x )
2
n
where x is a member of the set, x is the mean value of
the set and n is 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.