Mean, Median, Mode and Range

The mean, median and mode are three kinds of average. An average is one number that describes a typical value in a set of data. The range is different: it measures how spread out the data is, so it is not an average.

• Mode: the value that appears most often.
• Median: the middle value when the data is in order.
• Mean: add up all the values, then divide by how many values there are.
• Range: the largest value subtract the smallest value.

In the exam you are usually given a short list of numbers and asked to write down the mode, or work out the median, the mean or the range, for 1 or 2 marks each. Questions worth 3 or 4 marks give you a mean and ask for a missing value, or ask for the mean of two groups put together.

The mode is the value with the highest frequency. Frequency means how many times something appears. Give the value itself, not how many times it appears.

A list can have two modes, if two values tie for appearing most often. If every value appears only once, there is no mode.

The range is one number: largest value - smallest value. A small range means the values are close together, or consistent. A large range means they are spread out.

Take care with negative numbers. For temperatures of -3°C, 1°C and 5°C, the range is 5 - (-3) = 8°C, not 2°C.

To find the median, first write the values in order, from smallest to largest. Keep every repeated value: if 6 appears twice, write it twice.

With an odd number of values there is one middle value, and that is the median. With an even number of values there are two middle values. The median is halfway between them: add them and divide by 2.

To find the middle quickly, cross off the smallest and largest values in pairs, working in from both ends, until one value or two values are left.

The mean is what each value would be if the total were shared out equally.

\text{mean} = \text{total of the values} \div \text{number of values}

Count every value, including any zeros. A zero adds nothing to the total, but it is still one of the values you divide by.

Check your answer: the mean is always between the smallest and the largest value. It does not have to be a whole number.

If you know the mean, you can find the total. Turn the mean round:

\text{total} = \text{mean} \times \text{number of values}

To find a missing value, work out the total that all the values must make, then subtract the values you know. If a value is taken away, find the old total, subtract that value, then divide by the new number of values.

To find the mean of two groups put together, find each group's total first. Never just add the two means and divide by 2, unless the groups are the same size.

To compare two sets of data, compare an average and the range. Write each comparison as a sentence about the context, and give the figures. Just writing down the two values, or the difference between them, is not a comparison.

For example, Class A's test marks have median 64 and range 30. Class B's have median 58 and range 45. On average Class A did better, because its median is higher. Class A's marks were more consistent, because its range is smaller.

Think about what a higher value means. For marks, higher is better; for times or prices, lower is usually better.

One value far from the rest pulls the mean towards it. For delivery times of 12, 13, 14, 15 and 46 minutes, the mean is 100 \div 5 = 20 minutes, more than four of the five times. The median, 14 minutes, describes a typical delivery better.

• Finding the median without ordering the list. Write the values in order first, and keep every repeated value.
• Giving both middle values, or their difference. With two middle values, the median is halfway between them: add them and divide by 2.
• Mixing up the averages. Read which one the question asks for: the mode is the most common value, the median the middle value.
• Writing the range as "12 to 40". The range is one number: 40 - 12 = 28.
• Dividing by the wrong number for the mean. Divide the total by how many values there are, including any zeros.
• Adding two means and halving. For two groups, find each group's total, add them, then divide by the total number of values.