Mean, Median, Mode Fundamentals

Mean, median, and mode are three different ways to describe a "typical" value in a data set, and they can give noticeably different answers for the same data. Knowing how each one is computed, and how each one reacts to an unusually large or small value, makes it possible to choose the right measure for a given question.

What you will learn

• Compute the mean, median, and mode of a data set
• Explain how each measure responds to an outlier
• Choose the appropriate measure for a given question

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

Three measures, three definitions

The mean is the sum of all the values divided by how many values there are — the familiar "average." The median is the middle value once the data is sorted in order; with an odd number of values there is a single middle value, and with an even number of values the median is the average of the two middle values. The mode is the value (or values) that appears most often in the data set; a data set can have one mode, more than one mode if there is a tie, or no mode at all if every value appears the same number of times.

Computing the median always requires sorting the data first — reading off the middle of an unsorted list gives a meaningless answer. Computing the mode requires counting how many times each distinct value appears, not just looking for a value that seems common.

How each measure reacts to an outlier

An outlier — a value far larger or smaller than the rest of the data — pulls the mean noticeably toward itself, since every value contributes to the sum. In the data set $2,3,4,5,50$, the mean is $\frac{2+3+4+5+50}{5}=12.8$, far above every value except the outlier itself, because the single value 50 dominates the sum.

The median, by contrast, only depends on which value sits in the middle position once the data is sorted, so a single extreme value has no effect on it unless it changes which value occupies that middle spot. For the same data set, the median is just 4 — the middle of $2,3,4,5,50$ — completely unaffected by how large the outlier happens to be. The mode is similarly unaffected by a single outlier, unless that outlier happens to repeat often enough to become the most frequent value itself.

Choosing the right measure

Which measure best describes a "typical" value depends on what the data represents and what question is being asked. When a data set has an outlier or is heavily skewed — such as home prices in a neighborhood with one unusually expensive mansion — the median usually gives a more representative sense of a typical value, since it resists being dragged toward the extreme.

The mode is the right choice when the data represents categories rather than numerical quantities, such as favorite colors or shoe sizes sold, where "most common" is the natural notion of typical. The mean remains the standard choice whenever every value should contribute proportionally to the result, such as computing a total that will later be redistributed evenly.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Finding the median of an unsorted list instead of sorting the data first.
• Forgetting to average the two middle values when a data set has an even number of entries.
• Assuming the mode must be a value near the middle of the data, rather than checking which value actually repeats most often.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• The mean is the sum of the values divided by the count, the median is the middle of the sorted data, and the mode is the most frequent value.
• An outlier pulls the mean toward itself, but leaves the median — and usually the mode — unaffected.
• Choose the median for skewed data with outliers, the mode for categorical "most common" questions, and the mean when every value should count proportionally.