Mean, median, mode and range

Two skills, and most preparation only covers the first. Skill 3 is the arithmetic. Skill 4 asks what the number means, and skill 5 asks whether choosing it was honest.

The four definitions

Two points on the mechanics. Order the data before you take the median, every time, and with an even number of values the median is the average of the middle two. A set can have more than one mode or none at all.

Why the mean and the median disagree

The mean uses every value, so a single extreme value drags it. The median only cares about position, so it barely moves.

Take the set 1, 2, 3, 4 and 100. The mean is 22. The median is 3. Neither is wrong and they describe different things, and an item can ask which better represents a typical value. The answer there is the median, because 22 is larger than four of the five values in the set.

Dispersion, and what the rule includes

Skill 4 names range and standard deviation as the measures of dispersion. Range is in skill 3 as something to calculate; standard deviation is not, and it appears only in the interpretation skill.

Read that carefully, because it tells you what to expect. You are asked to interpret what a standard deviation means in context, not to compute one. A larger standard deviation means more spread. That is the level being tested.

Choosing a statistic dishonestly

Skill 5 is about exactly this: how the selection of a statistic can lead to different or inappropriate interpretations. Somebody reporting the mean salary at a company with one very high earner is choosing a number that flatters.

• Use the median when the data is skewed or has outliers
• Use the mean when the data is roughly symmetric and every value should count
• Use the mode for categories, where averaging makes no sense

A question asking which measure best represents a data set is asking you to spot the skew. Look for one value far from the rest, and if it is there, the median is almost always the answer.

What a missing value question wants

A common item gives you a mean and all but one value, and asks for the missing one. Work backwards: multiply the mean by the number of values to get the total, then subtract what you have.

It is two operations and it feels like more, which is why it is a good item. Write the total down before you subtract, because doing it in your head under a clock is where the arithmetic goes wrong even with a calculator on the screen.

Calculate each measure from an ordered data set

The mean is the sum of all values divided by the number of values. For 2, 4, 4, 8, add to get 18 and divide by 4; the mean is 4.5. The median is the middle value after ordering: with four observations it is the average of the two middle values, (4 + 4)/2 = 4. The mode is the most frequent value, here 4. The range is maximum minus minimum: 8 − 2 = 6.

For an odd number of values, the median is the single middle observation after sorting. For an even number, average the two central observations. Always order the data first; the middle value in the original list may not be the median. A data set can have no mode, one mode, or multiple modes depending on the frequency pattern and the convention used.

Choose a measure that fits the question

The mean uses every value and is sensitive to extreme observations. The median is resistant to outliers and often better describes the center of skewed data such as incomes or home prices. The mode identifies the most common value and can be used for categories that cannot meaningfully be averaged. The range describes total spread but depends only on the smallest and largest observations.

If a very large value is added to a data set, the mean usually rises, while the median may change little or not at all. This difference is why “average” should be specified: it could mean mean, median, or mode. In a word problem, identify which measure is requested rather than assuming mean.

Use units and check the result

The mean and median use the same units as the observations. Range also uses those units, while a variance would use squared units. A quick check is to confirm that the mean lies between the minimum and maximum values. If it does not, arithmetic or transcription is wrong. For positive values, the mean also cannot be below zero.

A frequency table can calculate the mean by multiplying each value by its frequency, adding the products, and dividing by the total frequency. For grouped intervals, using class midpoints produces an estimate, not an exact mean, because the precise observations are unknown. State that limitation when interpreting a grouped-data estimate.

• Sort values before finding the median.
• Compute mean as sum divided by count.
• Use frequencies to identify mode and calculate a weighted mean.
• Consider outliers when choosing mean versus median.
• Check that the center and spread measures have sensible units and bounds.