To find a missing number from a known average, multiply the average by the total number of values to find the required sum. Add the known values, then subtract their total from the required sum. The difference is the missing number.
Enter a list of numbers to calculate the mean, sum, count, median, mode, range, minimum, and maximum.
The known average and number of values reveal what the complete sum must be.
The Reverse Average Method
An arithmetic mean is normally found by dividing a total by a count.
When the average is already known, reverse the division by multiplying the average by the count.
The Average Calculator can verify the result after the missing value has been found.
Missing-Number Formula
Multiply the known average by the total number of values.
Add all known values.
Subtract the known total from the required total.
Worked Example
Suppose four numbers have an average of 20, and three known values are 10, 15, and 25.
The required total is 20 multiplied by four, which equals 80.
The known values total 50, so the missing number is 80 minus 50, or 30.
Verify the Missing Value
Add the recovered value to the known values.
The complete set is 10, 15, 25, and 30, which totals 80.
Dividing 80 by four confirms the average of 20.
Example with a Decimal Average
Suppose five values average 7.5 and four known values total 29.
The required total is 7.5 multiplied by five, which equals 37.5.
The missing value is 37.5 minus 29, or 8.5.
Example with a Negative Missing Value
Suppose three values average two and two known values are five and four.
The required total is six, while the known total is nine.
The missing value is six minus nine, which equals minus three.
Finding Multiple Missing Values
When more than one value is missing, the average reveals only their combined total.
Additional information is needed to identify each missing value separately.
For example, knowing their difference or ratio may make the individual values solvable.
Finding a Required Future Score
The same method can determine the score needed to reach a target average.
Multiply the target average by the final number of scores and subtract the current total.
The result is the required next score, assuming all scores are equally weighted.
Weighted Missing Values
The simple reverse method assumes equal weighting.
When the missing value has a different weight, calculate the required weighted total and divide the remaining contribution by the missing weight.
Read How to Calculate a Weighted Average first.
Common Mistakes
Do not multiply by the number of known values; use the total count including the missing value.
Do not subtract individual known numbers from the average itself.
Do not apply the equal-weight formula to weighted observations.
Conclusion
Find the required total by multiplying the average by the total count.
Subtract the sum of the known values to recover the missing number.
Enter the completed list into the Average Calculator to verify the result.
FAQs
How do I find a missing number from an average?
Multiply the average by the total count and subtract the sum of the known values.
Does the count include the missing value?
Yes. Use the total number of values after the missing value is included.
Can the missing value be negative?
Yes. It can be negative when the known total exceeds the required total.
Can I find two missing values?
You can find their combined total, but more information is needed to identify them separately.
How do I verify the result?
Add the missing value to the list and calculate the average again.