Percentiles For Grouped Data

The mathematics marks of \(\text{100}\) grade \(\text{10}\) learners at a school have been collected. The data are presented in the following table:

Percentage mark	Number of learners
\(0 \le x < 20\)	2
\(20 \le x < 30\)	5
\(30 \le x < 40\)	18
\(40 \le x < 50\)	22
\(50 \le x < 60\)	18
\(60 \le x < 70\)	13
\(70 \le x < 80\)	12
\(80 \le x < 100\)	10

1. Calculate the mean of this grouped data set.

2. In which intervals are the quartiles of the data set?

3. In which interval is the \(30^{\text{th}}\) percentile of the data set?

Calculate the mean

Since we are given grouped data rather than the original ungrouped data, the best we can do is approximate the mean as if all the learners in each interval were located at the central value of the interval.

\begin{align*} \text{Mean } & = \cfrac{2(10) + 5(25) + 18(35) + 22(45) + 18(55) + 13(65) + 12(75) + 10(90)}{100} \\ & = \text{54}\% \end{align*}

Find the quartiles

Since the data have been grouped, they have also already been sorted. Using the percentile formula and the fact that there are \(\text{100}\) learners, we can find the rank of the \(25^{\text{th}}\), \(50^{\text{th}}\) and \(75^{\text{th}}\) percentiles as

\begin{align*} {r}_{25} & =\cfrac{25}{100}(100 - 1) + 1 \\ & = \text{24.75} \\ {r}_{50}& = \cfrac{50}{100}(100 - 1) + 1 \\ & = \text{50.5} \\ {r}_{75} & = \cfrac{75}{100}(100 - 1) + 1 \\ & = \text{75.25} \end{align*}

Now we need to find in which ranges each of these ranks lie.

• For the lower quartile, we have that there are \(2 + 5 = 7\) learners in the first two ranges combined and \(2 + 5 + 18 = 25\) learners in the first three ranges combined. Since \(7 < {r}_{25} < 25\), this means the lower quartile lies somewhere in the third range: \(30 \le x < 40\).

• For the second quartile (the median), we have that there are \(2 + 5 + 18 + 22 = 47\) learners in the first four ranges combined. Since \(47 < {r}_{50} < 65\), this means that the median lies somewhere in the fifth range: \(50 \le x < 60\).

• For the upper quartile, we have that there are \(\text{65}\) learners in the first five ranges combined and \(65 + 13 = 78\) learners in the first six ranges combined. Since \(65 < {r}_{75} < 78\), this means that the upper quartile lies somewhere in the sixth range: \(60 \le x < 70\).

Find the \(30^{\text{th}}\) percentile

Using the same method as for the quartiles, we first find the rank of the \(30^{\text{th}}\) percentile.

\begin{align*} r & = \cfrac{30}{100}(100 - 1) + 1 \\ & = \text{30.7} \end{align*}

Now we have to find the range in which this rank lies. Since there are \(\text{25}\) learners in the first \(\text{3}\) ranges combined and \(\text{47}\) learners in the first \(\text{4}\) ranges combined, the \(30^{\text{th}}\) percentile lies in the fourth range: \(40 \le x < 50\)