A new value moves the mean

The idea

Including a new value greater than the mean increases the mean, and including a value smaller than the mean decreases it, so that the distances on both sides stay balanced.

A new weight arrives

Five weights at 4, 6, 8, 10 and 12 balance the beam with the pivot at 8, the mean. Now a new weight is placed at 14, more than the mean.

What will happen to the mean?

• It stays 8
• It goes up
• It goes down

Include a new value

Choose a new value and press Include. The beam tips, then the pivot slides to the new mean. Try a value more than, less than and equal to the old mean.

The mean moves towards the new value

Including a new value greater than the mean increases the mean, and including a value smaller than the mean decreases it, so that the distances on both sides stay balanced.

The data 4, 6, 8, 10, 12 has mean 8. Include 14: (40 + 14) ÷ 6 = 9 The mean increased, 8 → 9. The new weight is far to the right, so the pivot must slide right to balance.

Include 2 instead: (40 + 2) ÷ 6 = 7 The mean decreased, 8 → 7. Include 8, the mean itself, and nothing moves: the mean stays 8.

After the pivot has moved, the distances to the left and to the right are equal again.

Notes

Including a new value greater than the mean increases the mean, and including a value smaller than the mean decreases it, so that the distances on both sides stay balanced.

Check yourself

The data 4, 6, 8, 10, 12 has mean 8. A new value 20 is included. What is the new mean?

Answer: 10

The old total is 4 + 6 + 8 + 10 + 12 = 40. The new total is 40 + 20 = 60 and there are 6 values: 60 ÷ 6 = 10. The mean went up from 8 to 10.

The mean of some data is 15. A new value 9 is included. What happens to the mean?

The mean of 5 numbers is 20. A sixth number is included and the mean becomes 22. What is the sixth number?

Answer: 32

5 numbers with mean 20 have total 100. 6 numbers with mean 22 have total 132. The sixth number is 132 − 100 = 32.

Why does a very big new value make the mean go up?

• It increases.. Any value below the mean pulls the mean down. 9 is below 15.
• It becomes 9.. One new value does not become the mean. The mean only moves towards it.
• It stays 15.. It stays the same only if the new value is exactly the mean, 15.
• It decreases, but stays above 9. — correct. Yes! 9 is below the mean, so the mean decreases. It moves towards 9 but never reaches it.
• Because adding any number makes the mean bigger.. A small new value makes the mean smaller. It depends on whether the value is above or below the mean.
• Because there are more values now.. More values alone do not raise the mean. A value equal to the mean changes nothing.
• Because the new weight adds distance on the right, so the pivot must move right to balance. — correct. Yes! The right side is heavier, so the pivot slides to the right until the distances are equal again.
• Because the mean always moves to the biggest value.. The mean moves towards the new value, but it never reaches the biggest value.