When we have many numbers, such as the incomes in a town or the waiting times at a clinic, we need ways to summarize them. The two big questions are: what is typical, and how much do the values vary?

Three kinds of average

• The mean is the sum of the values divided by how many there are. It is what most people call “the average.”
• The median is the middle value when the data are put in order. Half the values lie below it and half above. With an even number of values, it is the mean of the two middle ones.
• The mode is the value that appears most often.

Take the five numbers 2, 3, 3, 5, 7. Their sum is 20, so the mean is 20 divided by 5, which is 4. The middle value in order is 3, so the median is 3. The mode is also 3, since it appears twice.

When the mean misleads

The mean is pulled by extreme values; the median is not. If nine people in a room earn between $30,000 and $65,000 and a billionaire walks in, the mean income soars while the median barely moves. Neither number is false, but only the median describes a typical person in the room. That is why the median is often a better summary when data are skewed or contain extreme outliers, such as incomes or house prices. Government reports on household income usually feature the median for this reason.

The mode is most useful for categories, where the other averages make no sense. You cannot calculate a mean favorite color, but you can say which color was chosen most often. The mode suits categories such as favorite color, shoe size or the most common reason for a complaint.

Spread

Two sets of data can share an average yet look completely different. Two towns might both have a mean temperature of 15°C, one with mild weather all year and the other with hot summers and freezing winters. Measures of spread capture this difference.

• The range is the difference between the largest and smallest values. It is simple but depends entirely on the two most extreme values.
• The interquartile range covers the middle half of the data, ignoring the top and bottom quarters, so it is less affected by outliers.
• The standard deviation measures roughly how far values typically sit from the mean. A small standard deviation means values cluster tightly; a large one means they are scattered.

Distributions

A distribution describes how often each value occurs, often drawn as a histogram. Many measurements, such as adult heights or errors in repeated measurements, form a symmetrical, bell-shaped curve called the normal distribution. In a normal distribution, about 68 percent of values lie within one standard deviation of the mean and about 95 percent within two.

Many other distributions are skewed. Incomes have a long tail to the right: most people earn moderate amounts and a few earn enormous sums. In a right-skewed distribution the mean is larger than the median, another reason to look at both.

Outliers

An outlier is a value far from the rest. It might be an error, such as a person's age typed as 250, or a genuine and important case, such as a hospital with unusually many deaths. The right response is to check whether an outlier is an error or a real value before deciding what to do. Deleting inconvenient values without explanation is a classic way to mislead, and so is letting one strange value drive a conclusion.

Worked example: the typical worker

A small company has nine employees with these annual salaries: $30,000, $35,000, $40,000, $40,000, $45,000, $50,000, $55,000, $60,000 and $65,000.

Mean. The total is $420,000. Divided by 9, the mean is about $46,700.

Median. With nine values in order, the middle one is the fifth: $45,000.

Mode. $40,000 appears twice, so it is the mode.

The three averages are close, because the salaries are fairly evenly spread. Now add the owner, who pays herself $1,000,000.

New mean. The total becomes $1,420,000. Divided by 10, the mean is $142,000, more than anyone except the owner earns.

New median. With ten values, the median is the mean of the fifth and sixth: $45,000 and $50,000 give $47,500.

A recruitment ad saying “average salary $142,000” would be technically true and deeply misleading. The median, $47,500, describes the typical worker far better.

Spread. Now consider two bus routes, each with a mean wait of 10 minutes. On Route 1, waits range from 9 to 11 minutes. On Route 2, they range from 0 to 30 minutes. The averages are identical, but the experience is not: on Route 2 you might miss an appointment. The range (2 minutes against 30) tells you what the mean hides.

Whenever you see an average, ask two things: which average is it, and how spread out are the values around it?