Descriptive Statistics
Weighted Mean Calculator
The weighted mean lets each value pull in proportion to the weight on its own line, and the page also shows each weight as a share of the total.
Worksheet
Press Calculate to see the result.
The weighted mean lets each value pull in proportion to the weight on its own line, and the page also shows each weight as a share of the total.
What this calculator answers
Three section averages are 80 with weight 2, 90 with weight 3, and 70 with weight 5. Treating them as three equal numbers would pretend the section of 5 is the same size as the section of 2. The weighted mean multiplies each average by its weight, adds those products, and divides by the total weight. The shares then show the vote each line actually cast: 70 casts half of the votes in this example, which is why it pulls the result down from a simple mean of the three averages.
The headline is the weighted mean. Each line also contributes its weight divided by the total weight, so the votes are visible and not only the final quotient. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Weighted arithmetic mean uses this relationship.
weighted mean = (sum of value times weight) / (sum of weights)
- value. The number on a line that is being averaged.
- weight. How many votes that value casts. Must be greater than zero.
- share. That weight divided by the sum of the weights.
- sum of products. Each value multiplied by its own weight, then added.
Equal weights would reduce this to an ordinary mean. Unequal weights scale each value before the division, which is the same as repeating the value once per unit of weight when the weights are counts. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
There are 3 lines of value and weight. Each product is 80 times 2 = 160, 90 times 3 = 270, 70 times 5 = 350. The products add to 780.
The weights add to 10. The weighted mean is 780 / 10 = 78. The weight shares are 80 gets 0.2; 90 gets 0.3; 70 gets 0.5.
A share is that line's weight divided by 10, not the value divided by the mean. Lines with a larger share pull the 78 harder than lines with a smaller share.
| Weighted mean | 78 |
|---|---|
| Total weight | 10 |
| Sum of value times weight | 780 |
| Weight shares | 80 gets 0.2; 90 gets 0.3; 70 gets 0.5 |
A second situation
The score of 70 keeps its value but its weight falls from 5 to 1
There are 3 lines of value and weight. Each product is 80 times 2 = 160, 90 times 3 = 270, 70 times 1 = 70. The products add to 500.
The weights add to 6. The weighted mean is 500 / 6 = 83.3333. The weight shares are 80 gets 0.3333; 90 gets 0.5; 70 gets 0.1667.
A share is that line's weight divided by 6, not the value divided by the mean. Lines with a larger share pull the 83.3333 harder than lines with a smaller share.
| Weighted mean | 83.3333 |
|---|---|
| Total weight | 6 |
| Sum of value times weight | 500 |
| Weight shares | 80 gets 0.3333; 90 gets 0.5; 70 gets 0.1667 |
Cutting the weight on 70 from 5 to 1 changes the products from 160, 270, and 350 to 160, 270, and 70. The weighted mean rises from 78 to 83.3333 because the low score lost most of its vote. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
The weight on 70 changed from 5 to 1. The three scores stayed the same. The headline result went from 78 to 83.3333.
The nudged inputs
There are 3 lines of value and weight. Each product is 80 times 2 = 160, 90 times 3 = 270, 70 times 1 = 70. The products add to 500.
The weights add to 6. The weighted mean is 500 / 6 = 83.3333. The weight shares are 80 gets 0.3333; 90 gets 0.5; 70 gets 0.1667.
A share is that line's weight divided by 6, not the value divided by the mean. Lines with a larger share pull the 83.3333 harder than lines with a smaller share.
| Weighted mean | 83.3333 |
|---|---|
| Total weight | 6 |
| Sum of value times weight | 500 |
| Weight shares | 80 gets 0.3333; 90 gets 0.5; 70 gets 0.1667 |
How to read the result
Say the weighted mean together with the total weight. A mean of 78 from total weight 10 is not the same claim as a mean of 78 from three unweighted section averages. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- Each line pairs the value with the weight that should multiply it.
- Weights are greater than zero.
- The values are in one unit.
- Weights are the votes you intend, not a second measurement that needs a different formula.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Averaging the values and ignoring the weights.
- Averaging the weights themselves.
- Using a negative weight, which this page rejects.
- Mixing a percent weight on one line with a headcount weight on the next.
Where a different calculator is the honest choice
Use the mean calculator when every row should vote once. Use the grouped data mean calculator when you have class intervals and frequencies instead of raw values. Use the average calculator when a high-low trim is the adjustment you want. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Mean Calculator, Arithmetic Mean Calculator, Average Calculator.
Check yourself
Scores 80, 90, and 70 have weights 2, 3, and 5. What is the weighted mean?
The weighted mean is 78.
Using the result in a sentence
Course grades with credit weights, section averages with class sizes, and price indexes with expenditure weights all use this pattern. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
A large weight on a bad measurement dominates the result. The shares show that dominance, but they do not check whether the weight was the right count. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Weighted Mean Calculator commit to?
This page uses Weighted arithmetic mean: weighted mean = (sum of value times weight) / (sum of weights). Equal weights would reduce this to an ordinary mean. Unequal weights scale each value before the division, which is the same as repeating the value once per unit of weight when the weights are counts. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Value and weight on each line, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the weighted mean is 78. Say the weighted mean together with the total weight. A mean of 78 from total weight 10 is not the same claim as a mean of 78 from three unweighted section averages. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Averaging the values and ignoring the weights. A large weight on a bad measurement dominates the result. The shares show that dominance, but they do not check whether the weight was the right count. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the mean calculator when every row should vote once. Use the grouped data mean calculator when you have class intervals and frequencies instead of raw values. Use the average calculator when a high-low trim is the adjustment you want. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The weighted mean is 78. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.