The table shows the distribution of
goals scored in the first half of a football season. Find the standard deviation of the
number of goals scored. Give your answer to three decimal
Looking at the table we’ve been
given, we can see that the data has been presented in a frequency table. The first row gives the number of
goals scored, and the second row gives the number of matches in which that number of
goals were scored, or in other words, the frequencies. For example, there were five
matches in which zero goals were scored. We could therefore add the labels
𝑥 sub 𝑖 to represent the data, which are the number of goals, and 𝑓 sub 𝑖 to
represent the frequencies.
We are asked to find the standard
deviation of the number of goals scored, which is a measure of how dispersed or
spread out the data is around its mean value. We can recall that for a data set
𝑋 containing the values 𝑥 sub one, 𝑥 sub two, up to 𝑥 sub 𝑛 with corresponding
frequencies 𝑓 sub one, 𝑓 sub two, up to 𝑓 sub 𝑛 and mean 𝜇, the standard
deviation, which we denote as 𝜎 𝑋, is given by 𝜎 𝑋 equals the square root of the
∑ from one to 𝑛 of 𝑥 𝑖 minus 𝜇 squared multiplied by 𝑓 i over the ∑ from one to
𝑛 of 𝑓 i.
We should also recall that the mean
of a set of data presented in a frequency table, 𝜇, is given by the ∑ from one to
𝑛 of 𝑥 𝑖 multiplied by 𝑓 𝑖 over the ∑ from one to 𝑛 of 𝑓 i.
Practically, what this means when
we calculate the standard deviation of a set of data presented in a frequency table
is we calculate the mean 𝜇 first. We then subtract this from each
𝑥-value and square. We multiplied by the frequency for
that 𝑥-value and find the ∑. We then divide this quantity by the
∑ of the frequencies or the total frequency. And finally, we take the square
We can extend the table we’ve been
given in order to work through the process. The first thing we need to find in
order to calculate the mean is the product of each 𝑥-value with its corresponding
frequency. We have zero times five, which is
zero; one times two, which is two; three times seven, which is 21; four times seven,
which is 28; and six times four, which is 24. The ∑ of these values, and that’s
the ∑ for 𝑖 from one to five because there are five 𝑥-values in the data set, is
We also need to calculate the total
frequency by summing the values in the second row of the table, five plus two plus
seven plus seven plus four, which is equal to 25.
To find the mean, we divide the ∑
of each 𝑥-value multiplied by its frequency, which is 75, by the total frequency of
25 giving 𝜇 is equal to three. So, we’ve calculated the mean, and
now we need to calculate the standard deviation. In the next row of our table, we’re
going to subtract the mean of three from each 𝑥-value. That gives negative three, negative
two, zero, one, three.
In the next row of the table, we
square each of these values, giving nine, four, zero, one, and nine. Finally, we multiply each of these
values by the frequency for that 𝑥-value. So we’re multiplying the values in
the fifth row of our table by the values in the second row, giving 45, eight, zero,
seven, and 36.
Now, returning to the formula for
the standard deviation, the numerator of the fraction underneath the square root is
the ∑ of each 𝑥 𝑖 value minus the mean squared multiplied by the frequency. Summing the values in the final row
of the table gives 96. So this will be the numerator of
the fraction. And we’ve already worked out that
the total frequency is 25.
So we have that the standard
deviation 𝜎 sub 𝑋 is equal to the square root of 96 over 25. In exact form, that simplifies to
four root six over five. But we’ve been asked to give our
answer to three decimal places. So evaluating this as a decimal
gives 1.9595 continuing. The value in the fourth decimal
place is a five, so we round up.
We found then that the standard
deviation of the number of goals scored in the first half of the football season, to
three decimal places, is 1.960.