Exercise solution 6, page 66 - Mathematics Higher Student Book

⭐ What you need to know:

A) A stem-and-leaf diagram is a way of organising data while keeping every original value. The stem shows the tens digit and the leaf shows the units digit. For example, if the key says that \(5 \mid 3\) means \(53\), then \(2 \mid 7\) means \(27\). To read a value, combine the stem and the leaf.

B) To find how many times a particular number appears, look for the correct stem and then check whether the matching leaf is listed. For example, to find how many times \(24\) appears, look at stem \(2\) and count how many leaves \(4\) there are. If the leaves were \(1,4,4,6,9\), then \(24\) would appear 2 times.

C) The range tells you how spread out the data is. It is found by subtracting the smallest value from the largest value:

\[ \text{range} = \text{largest value} - \text{smallest value} \]

For example, if the smallest value is \(12\) and the largest value is \(38\), then

\[\begin{aligned} \text{range} &= 38 - 12 \\ &= 26 \end{aligned}\]

D) The mode is the value that appears most often. In a stem-and-leaf diagram, this means finding the number that is repeated the greatest number of times. For example, if the data are \(11, 14, 14, 18, 22, 22, 22, 25\), the mode is \(22\) because it appears 3 times.

E) The median is the middle value when the data are written in order. A stem-and-leaf diagram already shows the data in order, so this makes finding the median easier.

If there is an odd number of values, the median is the single middle value. For example, in

\[ 7,\ 9,\ 12,\ 15,\ 18 \]

the median is \(12\).

If there is an even number of values, the median is the mean of the two middle values. For example, in

\[ 6,\ 10,\ 13,\ 17,\ 21,\ 24 \]

the two middle values are \(13\) and \(17\), so

\[\begin{aligned} \text{median} &= \frac{13+17}{2} \\ &= \frac{30}{2} \\ &= 15 \end{aligned}\]

F) When finding the median, it is often helpful to count how many data values there are altogether first. If there are \(n\) values:

If \(n\) is odd, the median is the \(\frac{n+1}{2}\)th value.

If \(n\) is even, the median is halfway between the \(\frac{n}{2}\)th and the \(\frac{n}{2}+1\)th values.

For example, if there are \(9\) values, the median is the

\[\begin{aligned} \frac{9+1}{2} &= \frac{10}{2} \\ &= 5 \end{aligned}\]

so it is the 5th value.