The mean is a balance point
There are three averages: the mean, the median and the mode. Each one gives a single value to stand for a whole data set, and each one is found in a different way. The mean adds the values and divides by how many there are. The median is the middle value when the values are in order. The mode is the value that occurs most often.
Picture the values 2, 4 and 6 as equal weights on a beam, each placed at its own value along it. Their mean is (2 + 4 + 6) ÷ 3 = 12 ÷ 3 = 4, and the beam balances on a support placed at 4. The 2 is 2 below the mean and the 6 is 2 above it, so the distances on the two sides are equal and they cancel.
The 2 sits 2 to the left of the balance point and the 6 sits 2 to the right. The 4 sits on the balance point itself.
One extreme value moves the mean
Now swap the 6 for a 24. The mean becomes (2 + 4 + 24) ÷ 3 = 30 ÷ 3 = 10, so the balance point slides from 4 all the way to 10. Check the balance: the 24 is 14 above 10, and the 2 and the 4 are 8 and 6 below it, and 8 + 6 = 14.
The median has not moved. In order the values are 2, 4, 24, and the middle one is still 4. The median depends only on which value is in the middle, not on how far away the largest value is. So one extreme value, a value far from all the others, pulls the mean toward itself and leaves the median where it was.
The new mean, 10, is larger than two of the three values. An average that most of the data lies below does not describe a typical value well.
With 24 in place of 6, the beam balances at 10. The one distance on the right, 14, equals the two on the left together, 8 + 6.
the median is the middle value once the data is in order, so it depends on rank and not on size — moving x₅ past the others never moves it
Drag the last value away from the rest to make it an outlier
The values 2, 3, 4 and 5 stay where they are, and the gold point is a fifth value. At 8 the mean is 22 ÷ 5 = 4.4. Drag the gold point out to 20: the mean rises to 34 ÷ 5 = 6.8, and the median stays at 4 the whole way.
Choosing for a real data set
Five people work in a small office. Their yearly salaries, in thousands of dollars, are 22, 24, 25, 26 and 180; the 180 is the owner’s. The mean is (22 + 24 + 25 + 26 + 180) ÷ 5 = 277 ÷ 5 = 55.4 thousand dollars. The median is the middle salary, 25 thousand dollars.
Four of the five people earn less than half of the mean. Someone who asks what a typical person in the office earns should be told the median, 25 thousand dollars, because the one large salary pulls the mean far above everyone else.
The mean still has a use here. The mean times the number of values gives back the total: 55.4 × 5 = 277 thousand dollars, the whole salary bill. The office budget needs that total, and the median cannot give it: 25 × 5 = 125 thousand dollars is not what the office pays.
The mode, for categories
Twenty-five students name their favorite color: 9 say blue, 7 red, 5 green and 4 yellow. There is no mean favorite color, because colors cannot be added, and no median, because colors have no order from smallest to largest. The only average is the mode, the most common answer: blue.
The mode also suits numbers that come in fixed steps. Shoes are made only in set sizes, so a mean shoe size of 7.6 names a size nobody makes. A shop deciding what to order wants the size it sells most, which is the mode.
Which average to use
Use the mean when the values are spread fairly evenly with no extreme values, and whenever the total matters, because the mean can be turned back into a total.
Use the median when one or more values lie far from the rest, as with house prices or salaries, because the median depends only on the middle position.
Use the mode for categories, such as colors, and whenever the question asks for the most common value, such as the shoe size to order.
Averages from a frequency table
When the same values repeat, data often comes as a frequency table. A survey counts the people in 10 cars: 2 cars have 1 person, 5 cars have 2 people and 3 cars have 3 people. The numbers of cars, 2, 5 and 3, are the frequencies.
For the mean, first find the total number of people. Multiply each value by its frequency and add: 1 × 2 + 2 × 5 + 3 × 3 = 2 + 10 + 9 = 21 people. Then divide by the total frequency, the number of cars, 2 + 5 + 3 = 10. The mean is 21 ÷ 10 = 2.1 people in a car.
For the median, count along the frequencies. With 10 cars the median is halfway between the 5th and the 6th car. The first 2 cars have 1 person, and the next 5, the 3rd to the 7th, have 2 people. The 5th and the 6th cars both have 2 people, so the median is 2. The largest frequency is 5, beside 2 people, so the mode is 2 people as well.
Worked example: Shoe Sizes Sold in a Week, and the Average a Shop Can Actually Order From
Question A shoe shop records the sizes it sold in a week: size 5, 4 pairs; size 6, 8 pairs; size 7, 14 pairs; size 8, 20 pairs; size 9, 10 pairs; size 10, 4 pairs. (a) Find the mean, the median and the modal size. (b) The shop will order 120 pairs for next week in the same proportions. How many of them should be size 8, and which of the three averages is of use in placing the order?
1.Total the pairs: 4 + 8 + 14 + 20 + 10 + 4 = 60 pairs. Then multiply each size by its frequency: 5 × 4 = 20, 6 × 8 = 48, 7 × 14 = 98, 8 × 20 = 160, 9 × 10 = 90 and 10 × 4 = 40, which total 456.
The week's sales: 60 pairs in all, and 456 once each size is weighted by the pairs sold. 2.The mean size is 456 ÷ 60 = 7.6.
The mean size is 456 ÷ 60 = 7.6, and the arrow shows where that falls: between the sizes, not on one. 3.(a) For the median, 60 pairs put it halfway between the 30th and the 31st. Counting along the frequencies, 4, then 12, then 26, then 46 pairs are reached, so the 27th to the 46th pairs were all size 8 and the median is size 8. The largest frequency is 20, at size 8, so the modal size is 8 as well. The mean is 7.6, the median is 8 and the mode is 8.
(a) The 30th and 31st pairs are both size 8, so the median is size 8, and the tallest bar is size 8, so that is the modal size. 4.Size 8 took 20 of the 60 pairs sold, which is 2060 = 13 of the week's sales. Keeping the same proportions, the order should hold 13 of 120 pairs in size 8.
Size 8 took 20 of the 60 pairs sold, which is 13 of the week. 5.(b) 13 × 120 = 40 pairs of size 8. The mode is the average of use here, because 7.6 is not a size any shoe is made in and no pair of that size could be ordered. Check: ordering every size in the same proportion doubles each frequency, and 2 × 20 = 40.
(b) 13 × 120 = 40 pairs of size 8. The mode is the average of use, because no shoe is made in size 7.6.
Answer: (a) over the 60 pairs sold the mean is 7.6, the median is size 8 and the modal size is 8; (b) 40 pairs of size 8, and the mode is the average of use, because no shoe is made in size 7.6
Common mistakes
- Ordering to the mean and asking for size 7.6, or rounding it to size 8 and calling that the answer. A mean of sizes is an average of numbers, not a size, and rounding it only happens to land on the right size here. The mode is the size the shop actually sold most of.
- Taking the median as size 7 because 7 is the middle of the six sizes listed. The list of sizes is not the data; the data is 60 pairs of shoes, and the median is the middle of those 60 pairs, which the frequencies have to be counted along to find.