Working Backwards from a Mean

A coach knows that seven wrestlers have mean mass 56 kg, but one entry in the record is missing. The missing number does not make the problem impossible. The mean tells us the total mass that all seven entries must have together.

Working backwards turns an average into a useful constraint. First recover the total, then compare it with the information already known. We will use the same idea for missing entries, recording errors, groups of different sizes, and number puzzles.

Finding the Unknown

The equation mean = total ÷ count can be reversed by multiplication. If n values have mean m, their total is n × m. This total includes the missing observation, so subtracting the known subtotal leaves exactly the missing value.

Always check what the count refers to. If a question says seven values have a certain mean and lists six of them, the required total uses seven, not six. Then check your answer by adding all seven values and dividing by seven.

Correcting an Incorrect Record

A correction replaces one observation; it does not add a new observation. Therefore the count remains fixed. Start with the old total, remove the wrong entry, and insert the correct entry. If the recorded entry was 3 too large, the total must decrease by 3.

You can also calculate the change in mean directly. Divide the change in the total by the unchanged count. This shortcut is especially helpful when the original individual values are unavailable.

Combining Unequal Groups

Suppose one group has two observations and another has eight. Taking the ordinary mean of the two group means gives the groups equal influence, even though one contains four times as many observations. To find the overall mean, combine their totals and their counts.

There is one useful shortcut when the groups have equal counts: then the mean of their two means does give the combined mean. But the count condition must be checked, rather than assumed.

Constructing Data with a Target Mean

A target mean usually allows many datasets. For four values with mean 7, the total must be 28. You could choose 4, 6, 8, 10 or 1, 7, 9, 11. Additional conditions, such as distinct values or a chosen median, narrow the possibilities.

Number grids add several total conditions at once. If each row of a three-by-three grid must have mean 10, each row must total 30. If each column and both long diagonals must also have that mean, those lines must total 30 too. Start by checking sums rather than repeatedly dividing.

In this grid, every row, every column, and both corner-to-corner diagonals total 30. Each line contains three numbers, so each line has mean 10. All nine entries are distinct. Adding the same amount to every cell would shift every line’s mean by that amount.

Build and Check a Dataset

Try making five positive whole numbers with mean 8 and median 7.

1. The five numbers must total 40.
2. Place 7 in the third position of an ordered list.
3. Choose two values at most 7 and two values at least 7.
4. Adjust the choices until their total is 40, then verify both conditions.

One answer is 3, 5, 7, 11, 14. The total is 40, giving mean 8, and the third value is 7. A correct mean alone does not guarantee the requested median.