How to find the mean, median, mode and range

The mean is what most people call the average. Add up every value, then divide by how many values there are.

The mean doesn't have to be one of the values, or even a whole number. The mean of $3, 4, 4, 6$ is $\dfrac{17}{4} = 4.25$.

The median is the middle value once the data is in order. Forgetting to order the data first is the most common mistake.

• Odd number of values: there's one middle value. For $n$ values, it's value number $\dfrac{n + 1}{2}$.
• Even number of values: there are two middle values. The median is halfway between them.

The mode is the value that appears most often. Ordering the data first makes repeats easy to spot.

• $2, 4, 5, 6, 8, 8, 9$: the mode is $8$.
• $3, 5, 8, 10$: every value appears once, so there's no mode.
• $2, 4, 4, 6, 7, 7$: $4$ and $7$ both appear twice, so there are two modes, $4$ and $7$ (the data is bimodal).

The mode is the only one of the three that also works for categories, like the most popular sport in a class.

The range measures spread, not the centre. For $2, 4, 5, 6, 8, 8, 9$ the range is $9 - 2 = 7$.

The range is always a single number. Writing "2 to 9" is not the range.

A frequency table tells you how many times ($f$) each value ($x$) appears. Add an $f \times x$ column to find the total.

A common mistake is dividing by the number of rows (5) instead of the total frequency (20).

Quartiles split ordered data into four equal parts. $Q_1$ (the lower quartile) has a quarter of the data below it, the median ($Q_2$) has half, and $Q_3$ (the upper quartile) has three quarters.

The interquartile range is the spread of the middle half of the data:

The method most Australian textbooks use:

1. Put the data in order and find the median.
2. Split the data into a lower half and an upper half. If there's an odd number of values, leave the median out of both halves.
3. $Q_1$ is the median of the lower half, and $Q_3$ is the median of the upper half.

Methods for quartiles vary slightly between textbooks and calculators (some include the median in each half, some use a formula for the position). They can give slightly different answers for the same data, so follow the method your teacher uses.

Outliers (Year 10). A value is usually called an outlier if it's below $Q_1 - 1.5 \times \text{IQR}$ or above $Q_3 + 1.5 \times \text{IQR}$. The five-number summary (minimum, $Q_1$, median, $Q_3$, maximum) is what you draw a box plot from.

An outlier (a value far from the rest) pulls the mean towards it, but barely moves the median.

Five people earn $500, $550, $600, $650 and $5000 a week. The mean is $\dfrac{7300}{5} = \text{\textdollar}1460$, which is more than four of the five people earn. The median, $600, is a better summary of a typical wage.

• Use the mean when the data has no outliers.
• Use the median when there are outliers or the data is skewed.
• Use the mode for categories, or when you want the most common value.

The same goes for spread: the IQR isn't affected by outliers, but the range is.