How to Find the Missing Frequency When the Mean Is Given

What is the method?

Replace the missing frequency with x and use mean = Σfx/Σf. Include x in both the weighted total and the total frequency, then solve the equation. Finally, check that the answer is a non-negative whole number when frequency counts individual observations.

Worked table

The mean is 12. Find the missing frequency.

Values and frequencies for the worked example

Value	Frequency	Value × frequency
5	2	10
10	x	10x
15	6	90
Total	x + 8	10x + 100

Set up (10x + 100)/(x + 8) = 12. Multiply by x + 8 to get 10x + 100 = 12x + 96. Rearranging gives 4 = 2x, so x = 2.

Check the full table: total frequency = 2 + 2 + 6 = 10. Weighted total = 10 + 20 + 90 = 120. The mean is 120/10 = 12, so the answer satisfies the original data.

Why the denominator matters

The unknown frequency changes the number of observations. Dividing by only the known frequencies, 2 + 6 = 8, would treat the missing observations as though they contributed to the total but not the count. Always write both column totals before solving.

The stated mean also matters. If it is rounded, treating it as exact may produce a fractional or inconsistent result. Check the wording before deciding that the table contains an error.

When is there no unique answer?

Suppose the values are 5, 10 and 15 with frequencies 2, x and 2, and the mean is 10. The equation is (40+10x)/(4+x) = 10, which simplifies to 40+10x = 40+10x. Every non-negative integer x satisfies it. You need more information, such as the total frequency, to find a unique value.

For grouped data, the calculation uses class midpoints. Unless the actual observations are known, the resulting mean is based on those representatives rather than an exact reconstruction of every value.

Practice

1. Values 2, 4, 6; frequencies 3, x, 5; mean 4.5. x = 0, since 36/8 = 4.5.
2. Values 10, 20, 30; frequencies 2, x, 4; mean 22.5. x = 2, since 180/8 = 22.5.

For a different mean problem, read correcting a wrong observation. When asking for paid AI help in the study space, include the complete table and whether the mean is exact or rounded.