Averages and Reading Data

Introduction

An average summarises a list of numbers with one typical value. But there are several averages, and choosing the wrong one can mislead. Learning to read data critically is one of the most useful maths skills for everyday life.

The three averages

Take the list: 3, 5, 5, 7, 10.

• Mean: add everything and divide by the count. (3 + 5 + 5 + 7 + 10) / 5 = 30 / 5 = 6.
• Median: the middle value when sorted. The sorted list is 3, 5, 5, 7, 10, so the median is 5. With an even count, take the mean of the two middle values.
• Mode: the most common value. Here 5.
• Range: largest minus smallest. 10 - 3 = 7. It measures spread.

Which average to use

• The mean uses every value and is common, but outliers (very large or small values) pull it.
• The median is better when data is skewed, such as incomes or house prices.
• The mode suits categories, such as the most common shoe size.

Example: five salaries of 30, 32, 35, 38 and 200 (in thousands). The mean is 67, but only one person earns more than that. The median is 35, which describes a typical person better.

Reading charts

• Bar charts compare categories. Line charts show change over time. Pie charts show parts of a whole and work best with few slices.
• Check the title, labels and units.
• Look at the axes. A bar chart whose axis starts at a high number exaggerates differences. A bar chart should normally start at zero.
• Watch the scale and intervals.
• Ask how much data lies behind a number: 3 people or 3,000?

Misleading statistics

• Cherry-picking: showing only data that supports a claim.
• Relative versus absolute: "risk doubled" could mean from 1 in 100,000 to 2 in 100,000.
• Sample bias: a poll of people at one event does not represent everyone.
• Percent of what? A 50 percent rise from a very small base may be tiny.
• Averages hiding spread. Two groups can have the same mean and very different distributions.
• Missing context: compared with what, over what period?

Ask: Who collected this data, how, and what is left out?

Try it yourself

Find the mean, median, mode and range of 12, 15, 15, 18, 20, 20, 20, 95. Which average best describes a typical value? Find a chart in the news and list two questions you would ask about it.

Summary

• Mean, median and mode are different averages, and the range shows spread.
• Use the median when outliers or skew distort the mean.
• Read charts carefully for axes, scale, sample size and missing context.