Calculate the mean, median and mode, including from frequency tables, and find the range as a measure of spread

What this dot point is asking

SEAB wants you to calculate the three averages, the mean, median and mode, from a list and from a frequency table, to find the range as a simple measure of spread, and to comment on which average best represents a data set. Averages summarise data in a single representative value.

The answer

The mean

The mean is the total of all the values divided by how many there are:

It uses every value, so it is affected by extreme values (outliers).

The median

The median is the middle value when the data is arranged in order. With an odd number of values it is the single middle one; with an even number it is the mean of the two middle values. The median is not distorted by outliers.

The mode

The mode is the value that occurs most often. A data set can have more than one mode, or none if all values are different. The mode is the only average that works for non-numerical categories.

Averages from a frequency table

For a frequency table, the mean is the sum of (value times frequency) divided by the total frequency. The mode is the value with the highest frequency, and the median is the value at the middle position, found by counting through the frequencies.

The range

The range measures spread as the difference between the largest and smallest values:

A larger range means the data is more spread out.

Finding the median position from a frequency table

For a frequency table, the median is the value at the middle position, located by counting through the cumulative frequencies. With $n$ values, the median sits at position $\tfrac{n+1}{2}$ (for odd $n$) or between positions $\tfrac{n}{2}$ and $\tfrac{n}{2}+1$ (for even $n$). Build a running total of the frequencies and find which value contains that position. For $20$ matches the median is between the $10$th and $11$th values, so you count through the frequencies until the running total reaches them. Locating the median by its position in the cumulative count, rather than by eye, is the dependable method for tabulated data.

Working backwards from a known mean

A common twist gives the mean and all but one value, and asks for the missing one. Because the mean times the count equals the total, you can recover the total and subtract the known values. If five numbers have a mean of $8$, their total is $5 \times 8 = 40$; if four of them are $6, 7, 9, 10$ (summing to $32$), the fifth is $40 - 32 = 8$. The same idea handles a missing frequency in a table, using the formula mean $=$ (sum of value times frequency) over total frequency. Turning the mean back into a total is the key step in these reverse problems.

Examples in context

Example 1. House prices. A few very expensive houses pull the mean price up, so the median is often quoted as a fairer typical price. This is why property reports favour the median over the mean.

Example 2. Most popular size. A shop restocking shirts cares about the mode, the size that sells most, rather than the mean size, which might not even be a real size sold. The mode guides practical decisions about stock.

Try this

Q1. Find the mean of $4, 8, 10, 14$. [1 mark]

• Cue. $\dfrac{4 + 8 + 10 + 14}{4} = \dfrac{36}{4} = 9$.

Q2. Find the median of $2, 5, 9, 11, 20$. [1 mark]

• Cue. Middle of five ordered values is the $3$rd, which is $9$.

Q3. Find the mode and range of $3, 3, 5, 7, 7, 7, 12$. [2 marks]

• Cue. Mode is $7$ (most frequent); range is $12 - 3 = 9$.