Variation Coefficient
Calculate the sample coefficient of variation (%) for the values: 10, 1, 0, 5, 7
Expand Hint
The sample coefficient of variation:
$$$CV=\frac{s}{\mu}$$$
where
$$s$$
is the sample standard deviation, and
$$\mu$$
is the arithmetic mean.
Hint 2
Sample standard deviation:
$$$s =\sqrt{\frac{1}{n-1}\sum_{i=1}^{n} (X_{1}-\mu )^{2}}$$$
where
$$n$$
is the number of items or observations,
$$X$$
is the value from the set, and
$$\mu$$
is the mean.
First, let's find the mean of all the values:
$$$\mu=\frac{sum\:of\:terms}{number\:of\:terms}=\frac{10+1+0+5+7}{5}=\frac{23}{5}=4.6$$$
Next, let’s find the sample standard deviation:
$$$s =\sqrt{\frac{1}{n-1}\sum_{i=1}^{n} (X_{1}-\mu )^{2}}$$$
where
$$n$$
is the number of items or observations,
$$X$$
is the value from the set, and
$$\mu$$
is the mean. (Note, do not confuse this with a population standard deviation.)
$$$s =\sqrt{\frac{(10-4.6)^{2}+(1-4.6)^{2}+(0-4.6)^{2}+(5-4.6)^{2}+(7-4.6)^{2}}{(5-1)}}$$$
$$$=\sqrt{\frac{(5.4)^{2}+(-3.6)^{2}+(-4.6)^{2}+(0.4)^{2}+(2.4)^{2}}{4}}$$$
$$$=\sqrt{\frac{29.16+12.96+21.16+0.16+5.76}{4}}=\sqrt{\frac{69.2}{4}}=4.159$$$
Finally, the sample coefficient of variation:
$$$CV=\frac{s}{\mu}$$$
where
$$s$$
is the sample standard deviation, and
$$\mu$$
is the arithmetic mean.
$$$CV=\frac{4.159}{4.6}=0.90$$$
The sample coefficient of variation is
$$0.90 \times 100=90\%$$
.
90%
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