Mean and median — how they differ and how to find each

Top left on Board 1 is the list of eight scores. The 81 at the bottom of the list is their median: the menu at the top of the card is set to median. The card on the right, "Mean (total ÷ count)", lays out the steps to the mean for the same eight scores. The chart at the bottom sorts the scores into ranges and counts how many fall in each (a histogram). Changing a score in the list updates all three cards.

The mean: the total divided by the count

The eight scores add up to 626, shown in the "Sum Σx" row of the card on the right. Divided by the 8 people, 626 ÷ 8 gives 78.25.

The mean is the score everyone would have if the total were shared out evenly. If all eight people had 78.25, the total would still be 626.

The card also shows deviations, the variance and the standard deviation. A deviation is each score minus the mean. The variance and the standard deviation measure how spread out the scores are; this article does not use them.

The median: the middle of the order

The median is the value that sits exactly in the middle once the values are in order. The scores on Board 1 start out already sorted from lowest to highest.

With an odd number of values there is one in the middle. With an even number, like eight, there are two: the 4th score, 78, and the 5th, 84. The median is then halfway between them: (78 + 84) ÷ 2 = 81.

The order of the list does not matter. If an edited score is out of order, the card still sorts the values before finding the median.

When one score is far from the rest

With Person 1's 62 changed to 0 on Board 1, one person sits far away from everyone else.

The mean drops from 78.25 to 70.5, down 7.75. The median stays at 81. In the histogram, one bar now stands on its own at the far left.

The mean dropped because of the total. Turning 62 into 0 takes the total from 626 down to 564, 62 less. Shared across eight people, that is 62 ÷ 8 = 7.75 off the mean.

The median held because of the order. At 0, Person 1 is still the lowest score, so the two in the middle are still 78 and 84.

So the median did not hold because the value was extreme. It held because the middle of the order did not change. Even a value at one end moves the median if it crosses over to the other side of the middle.

Data with a few very large values

Board 2 is an example of ten households' savings. The numbers are made up for this example; they are not survey data. The amounts are in thousands, in any currency. The cards are the same as on Board 1.

In this example the ten households hold 8,000 in total, so the mean is 800. The median is halfway between the 5th value, 350, and the 6th, 450: 400. The mean is twice the median.

Seven of the ten households have less than the mean of 800. What pulls the mean up is Household J, with 3,700. The histogram shows it too: eight households in the bar on the left, and one on its own at the far right.

When a few large values pull the total up, as in this example, the mean can come out above the median. The mean alone does not show that most households are below it. When a report gives both the mean and the median, the gap between them hints at whether the data is lopsided like this.

Which one to use

The two describe different things. The choice depends on what is being described.

The mean reflects every single value. The mean is useful when the total or the amount per person matters: the mean multiplied by the count gives the total back (78.25 × 8 = 626).

The median shows where the middle of the group sits. If an end value becomes even larger or smaller but the middle of the order stays the same, the median does not change. The total, however, cannot be recovered from the median.

When the two are far apart, there may be values at one end that sit far from the rest. The gap itself says something about the shape of the data.

Other numbers

The numbers in the lists on Boards 1 and 2 can be edited directly.

With Person 1 left at 0, changing Person 4's 78 to 70 lowers the mean by just 1, from 70.5 to 69.5: the total drops by 8, and 8 ÷ 8 = 1. The median drops by 3.5, from 81 to 77.5. This time the median moved more, because the middle pair changed from 78 and 84 to 71 and 84.

Changing Person 1's 62 to 90 moves the lowest score past the middle. In order the scores are now 65, 71, 78, 84, 85, 88, 90 and 93; the middle pair is 84 and 85, and the median is 84.5.

Raising Person 8's 93 to 100 lifts the mean to 79.125, while the median stays at 81. A bigger top value does not change the middle of the order.

With Household J set to 7,700 or to 1,000, the median stays at 400; only the mean changes. Either way Household J is still at the top of the order, so the middle pair does not change.

Removing Household J with the × at the end of its row leaves nine households. With an odd count the median is simply the 5th value, 350, and the mean falls to about 477.8.

Read next — Average speed for a round trip — total distance over total time It calculates the average speed for a trip out and back at different speeds as total distance ÷ total time, which differs from the plain average of the two speeds.

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