One big value drags the mean

The idea

The mean takes all observations into account, lies between the extremes and allows distributions to be compared, but it is affected by extreme values and may then not represent the data well.

A small workshop

A workshop has 8 workers and an owner. The workers earn (in ₹ thousand a month) 20, 25, 25, 30, 30, 35, 35, 40. Each is a block on the beam, and the pivot always sits at the mean.

The owner is the red block. Right now the owner earns 40 too. Then the owner's pay goes up.

The owner's pay goes up a lot, and the workers' pay stays the same. What happens to the mean?

• It stays about the same
• It goes up a lot
• It goes down

Raise the owner's pay

Use the + button to raise the red block's pay in steps of 5, up to 100 (₹1 lakh). The pivot is the mean. Look at the smallest and largest values, and at the mean between them.

Strengths and a weakness

The mean takes all the observations into account and lies between the smallest and the largest. It also lets us compare two or more distributions. But extreme values affect it, so it may not represent the data well.

By comparing the mean results of different schools in an examination, we can say which school did better. But in your workshop, one very large pay moved the mean up, although no worker earned more.

Notes

The mean uses every observation, lies between the extremes and can compare distributions. Extreme values affect it, and then it may not represent the data well.

Check yourself

The workers earn 20, 25, 25, 30, 30, 35, 35, 40 and the owner earns 100 (all in ₹ thousand a month). Find the mean pay of the nine people, correct to 1 decimal place.

Answer: 37.8

20 + 25 + 25 + 30 + 30 + 35 + 35 + 40 = 240. 240 + 100 = 340, and 340/9 = 37.77…, which is 37.8.

To find which of two schools did better in an examination, we compare…

The marks of a class range from 12 to 95. Which of these could be the mean?

Which situation makes the mean a poor description of the data?

• the mean marks of the two schools — correct. Yes! The mean uses every student's marks, and means can be compared.
• only the highest mark in each school. One mark does not use all the observations.
• only the number of students in each school. That says nothing about how well they did.
• 58 — correct. Yes! The mean always lies between the smallest and the largest observation.
• 8. 8 is below the smallest mark, 12. The mean cannot be below the smallest value.
• 99. 99 is above the largest mark, 95. The mean cannot be above the largest value.
• One value is far away from all the others — correct. Yes! The pivot is pulled toward it, so the mean no longer describes most of the data.
• All values are close to each other. Then the mean describes the data well.
• The data has many observations. The number of observations does not spoil the mean. An extreme value does.