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Statistics

An average is one number standing in for a whole list, and there are three of them because lists differ in what needs standing in for. The mean shares the total out evenly, the median is the value in the middle once the list is in order, and the mode is whichever value turns up most often. The range is not an average at all — it measures how spread out a list is, which is often the more useful thing to know.

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The median of an even list

Find the median of 8, 3, 9, 4.

  1. Put the list in order: 3, 4, 8, 9.
  2. There are four values, so no single one sits in the middle.
  3. Take the two middle values, 4 and 8, and find the number halfway between them.
  4. 4 + 8 is 12, and half of 12 is 6. The median is 6.

Worth knowing. With an odd number of values the median is simply the middle one, and only an even-length list needs the halfway step. Not ordering the list first is where most wrong medians come from.

The mean of a list

Find the mean of 6, 10, 4, 12, 8.

  1. Add the whole list: 6 + 10 + 4 + 12 + 8 is 40.
  2. Count how many values there are: 5.
  3. Divide the total by the count: 40 divided by 5.
  4. The mean is 8.

Worth knowing. The mean is what every value would be if the total were shared out evenly. It is the only one of the three that uses every value, which is why a single very large or very small number pulls it about.

Where it usually goes wrong

  • Finding the median without ordering the list first, so whatever sits in the middle of the list as written gets reported instead.
  • Dividing by the wrong count when finding a mean — usually one short, because the last value was added and then not counted.
  • Giving how many times a value appeared instead of the value itself when asked for the mode.

The Professor writes the list out in order before he answers anything. The median is then sitting in the middle, the range is the two ends subtracted, and repeated values have gathered themselves together for the mode.

Practise it

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Questions teachers and parents ask

What is the difference between the mean, the median and the mode?

The mean shares the total out evenly, the median is the middle value once the list is in order, and the mode is the value that occurs most often. They answer different questions, and on a list with one unusually large value they can be a long way apart.

Is the range an average?

No. The range is the largest value minus the smallest, and it describes how spread out a list is rather than where its centre lies. Two lists can share a mean and have completely different ranges.

Can a list have more than one mode?

In general yes — a list where two values tie for most common has two modes. Every mode question here is built to have exactly one, so the answer is never ambiguous.

When are averages taught?

Summarising a data set is grade 6 (6.SP.B.5), and giving measures of centre and spread is 6.SP.B.5.C. Most children meet the mode and the range a year or two before the mean, because both can be read off a list without any dividing.

Where this shows up

Standards

  • 6.SP · Statistics and probability
  • CCSS.MATH.CONTENT.6.SP.B.5 — Summarise numerical data sets in relation to their context, including the number of observations and what was measured.
  • CCSS.MATH.CONTENT.6.SP.B.5.C — Give quantitative measures of centre and variability for a data set, such as the mean, the median and the range.

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