Weighted mean is one of 2 descriptive statistics formulas in the statistics section of this library, and it is used at high school · ap level.
Why weighted mean works
An ordinary average silently gives every value a weight of one. When some values stand for more cases or count for more of the final result, multiplying each by its weight and dividing by the total weight lets the answer lean toward the heavier values while still landing inside their range.
What each symbol means
$x_i$ are values and $w_i$ their weights.
Weighted mean: when it holds
$\sum w_i\ne0$; use nonnegative weights for the usual averaging interpretation.
When it stops applying
The weights have to mean how much each value counts, and every weight must be measured the same way. Weighting city temperatures by land area answers a question about area, not about people, so the result is still a number but it is no longer the average anyone wanted.
Weighted mean: a worked example
Scores $80,90$ weighted $1,2$ give $(80+180)/3\approx86.67$.
The mistake to avoid
What people do: Dividing by how many values there are rather than by the sum of the weights.
Why it goes wrong: Scores of 80 and 90 with weights 1 and 2 give a weighted total of 260. Dividing by 2 gives 130, an impossible average; dividing by the weight total of 3 gives about 86.67.
Do this instead: Add up the weights first and write that number in the denominator before you compute anything on top.
Weighted mean: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $x_i$ are values and $w_i$ their weights.
- Check the conditions before substituting. $\sum w_i\ne0$; use nonnegative weights for the usual averaging interpretation.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most statistics slips.
Where this formula fits
- Subject
- Statistics formulas — 21 entries in this library
- Topic
- Descriptive statistics
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where weighted mean comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Mean, Median, and Mode — the lesson behind this formula: choose a summary that matches the distribution.
- Statistics Calculator — check your substitution and the value it produces.
- Study statistics — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 21 statistics formulas — the full grouped reference, or the complete formula library.
Questions about weighted mean
What are the weights in a grade point average?
The credit hours for each course. An A worth 4.0 over 3 credits and a B worth 3.0 over 4 credits give 24 divided by 7, which is about 3.43, not the plain average of 3.5.
Do the weights have to add up to 1?
No. Dividing by the total weight rescales them automatically, so weights of 1 and 2 give exactly the same answer as weights of one third and two thirds.
Can a weighted mean land outside the range of the values?
Not when the weights are all zero or positive, because the answer is a blend of the values. Negative weights break that guarantee and can push the result past the largest value.
Is the ordinary mean a special case of this?
Yes. Give every value a weight of 1 and the numerator becomes the plain sum while the denominator becomes n, which is the ordinary mean exactly.