The median and the mode

Statistics and probability, grade 10 (4 units) Lesson 7 of 20Learning

The median and the mode

The median

The median is the middle datum of a list ordered from least to greatest: half the data are less than it or equal to it, and half are greater than it or equal to it.

\displaystyle \frac{n+1}{2}

This is the median’s place in the ordered list, where $\displaystyle n$ is the number of data.

An odd number of data – the median is the datum in the middle place. The waiting times of seven patients at a clinic, in minutes: 3, 5, 5, 6, 8, 12, 30. The place is $\displaystyle {\frac{7+1}{2}}={4}$ , and the median is 6 minutes.
An even number of data – the place comes out as a half, and the median is the mean of the two middle data. Rami’s marks, in order: 60, 62, 65, 70, 77, 87. The place is $\displaystyle {\frac{6+1}{2}}={3.5}$ , and the median is $\displaystyle {\frac{65+70}{2}}={67.5}$ .

The 30-minute wait does not change the median: had it been 13 minutes, the median would still be 6. The mean, $\displaystyle \frac{69}{7}\approx9.86$ , is pushed up by it.

The median from a frequency table

In a frequency table where each column is one value, the data are already in order. Find the median’s place, and count up to it with the cumulative frequency.

The fouls of the 40 jumpers:

Number of fouls	0	1	2	3	4
Frequency	4	16	9	6	5
Cumulative frequency	4	20	29	35	40

The place: $\displaystyle {\frac{40+1}{2}}={20.5}$ – between the 20th datum and the 21st.
Data 5–20 are 1, and data 21–29 are 2.
The median: $\displaystyle {\frac{1+2}{2}}={1.5}$ .

In a bar chart the count goes the same way: add up the bars’ heights, from left to right, until the median’s place is reached.

The mode

The mode is the value with the highest frequency – the value that appears most often.

In the jumpers’ fouls the mode is 1: 16 jumpers fouled once.
The mode needs no arithmetic, so a qualitative variable has a mode too: the most common way of getting to school is “on foot”. It is the only measure of centre that suits a qualitative variable.
There can be two modes: in the data 2, 3, 3, 5, 7, 7 the modes are 3 and 7.
When every value appears the same number of times, there is no mode: Rami’s marks have no mode.

The strengths and weaknesses of the measures of centre

Each measure of centre sums up the data in one number in a different way, and each one has situations it suits.

The mean
- worked out from all the data.
- strongly affected by an extreme datum.
- a qualitative variable has no mean.
The median
- not affected by extreme data.
- depends only on the data in the middle; all the others can grow or shrink without changing it.
- a qualitative variable has no median: its values cannot be put in order.
The mode
- easy to find, and suits every kind of variable.
- not affected by extreme data.
- meaningful only with many data and one value standing out; sometimes there is no mode, or there are several.

Choosing the right measure of centre

When there is an extreme datum, the median describes most of the data better than the mean.

Tamar’s marks, in order: 10, 85, 87, 92, 93, 100.

The mean: $\displaystyle \frac{467}{6}\approx77.83$ – lower than five of the six marks, because of the 10.
The median: $\displaystyle {\frac{87+92}{2}}={89.5}$ – close to most of the marks.
The mode: none, as each mark appears once.

The median represents Tamar’s marks better. When there is no extreme datum, as with Rami, the mean suits, because it uses all the data.

Exercise 1

Naama’s English marks during the year:
57, 65, 88, 90, 70, 75.

Find the median of the marks.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power