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Mean vs Median

Both are averages, and on symmetric data they agree. On skewed data they can be wildly different, and choosing the wrong one is how a technically true statistic ends up misleading.

How each is calculated

  • Mean: add every value and divide by how many there are.
  • Median: sort the values and take the middle one.
  • With an even count, the median is the average of the two middle values.

The key difference

The mean uses every value, so one extreme value drags it. The median only cares about position, so an extreme value moves it by at most one place. That resistance is called robustness, and it is the whole reason both measures exist.

Which one to use

  • Symmetric data with no outliers: the mean, since it uses all the information.
  • Skewed data such as incomes or house prices: the median.
  • Data with known errors or extreme outliers: the median.
  • When further calculation is needed, such as standard deviation: the mean, because the formulas are built on it.

Worked examples

Example 1

Question

Five salaries in thousands: 30, 32, 35, 38, 250. Find the mean and the median, and say which represents the group better.

Answer

Mean 77, median 35. The median is the fair description.

Steps

  1. 1Sum: 30 + 32 + 35 + 38 + 250 = 385.
  2. 2Mean: 385 / 5 = 77 thousand.
  3. 3The data is already sorted, and with five values the middle one is the third: 35 thousand.
  4. 4Four of the five people earn less than the mean, so it describes nobody in the group.
  5. 5The median at 35 sits in the middle of what people actually earn, so it is the honest summary here.

This is exactly why income statistics are reported as medians. One very large value pulls a mean somewhere no one lives.

Example 2

Question

Test scores: 68, 71, 74, 77, 80. Find the mean and the median.

Answer

Both are 74

Steps

  1. 1Sum: 68 + 71 + 74 + 77 + 80 = 370.
  2. 2Mean: 370 / 5 = 74.
  3. 3Sorted already, so the median is the third value: 74.
  4. 4They agree because the data is symmetric with no outliers.
  5. 5When mean and median agree, either one is a fair summary.

Example 3

Question

Six delivery times in minutes: 20, 22, 25, 27, 30, 95. Find both averages.

Answer

Mean 36.5, median 26

Steps

  1. 1Sum: 20 + 22 + 25 + 27 + 30 + 95 = 219.
  2. 2Mean: 219 / 6 = 36.5 minutes.
  3. 3With six values the median is the average of the third and fourth: (25 + 27) / 2 = 26.
  4. 4The single 95-minute delivery adds over ten minutes to the mean and two to the median.
  5. 5For a typical wait time, quote 26 minutes; for total driver hours, the mean is the one you need.

Common mistakes

Finding the median without sorting the data first.

Sort every time. The middle of an unsorted list is not the median.

Taking one middle value when the count is even.

Average the two middle values. With six items that is the third and fourth.

Quoting the mean for skewed data such as house prices.

Use the median when a few very large values are pulling the average away from typical.

Assuming the median is always the better choice.

The mean uses all the information and is required for variance, standard deviation, and totals.

Practice it

Work these out first, then open the answer. Stuck halfway? Select the problem and ask Solvecoon for the step you are missing.

  • 1. Find the mean and median of 2, 3, 3, 4, 100.

    Show answer

    Mean 22.4, median 3. The median describes this data far better.

  • 2. Find the median of 5, 9, 2, 8.

    Show answer

    6.5. Sort to 2, 5, 8, 9 and average the middle two.

  • 3. A data set has mean 50 and median 35. What does that suggest?

    Show answer

    It is skewed to the right: a few large values are pulling the mean above the middle of the data.

Questions people ask

Which average is the real average?
Both are. Average is a general word; mean and median are specific measures, and a question should say which it wants.
What does it mean when the mean is much higher than the median?
The data is skewed to the right, with a tail of large values. Incomes, house prices, and wait times usually look like this.
Where does the mode fit in?
The mode is the most frequent value. It is the only one of the three that works for categories, like the most common eye colour.
Can the median be a value not in the data set?
Yes, whenever the count is even, because it is the average of the two middle values.
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