Weighted Average Calculator

What Is a Weighted Average?

One idea does all the work here: some numbers deserve more say in the answer than others, and a weight is how you hand it to them. Everything below follows from that.

A Weight Is a Multiplier, Not Another Value

A weight is not an extra data point, something added to the pile of numbers being averaged. It is a multiplier applied to its value before any averaging happens, and the sum of those multipliers becomes the divisor. Give the final exam a weight of 50 and its score enters the total fifty times over; give homework a weight of 20 and it enters twenty times. Nothing is added to the count of items.

The division is what puts the answer back on a usable scale. Σ(value × weight) for a 90/78/85 grade weighted 20/30/50 is 8,390 — a figure on no scale anyone recognizes. Dividing by the total weight of 100 returns 83.90, which is a grade again, expressed in the same units as the three scores that produced it.

Weighted Average vs Simple Average

A simple average is the special case in which every weight is identical. Enter scores of 90, 78 and 85 with weights of 1, 1 and 1 and the panel returns 84.33 weighted average above Simple (unweighted) average: 84.33 — the same number twice, because equal weights cancel out of the arithmetic entirely. Switch the weights to 20, 30 and 50 and the headline drops to 83.90 while the comparison line stays at 84.33.

Nothing about the student's work changed between those two runs. The 90 is the strongest score and it carries the smallest share of the course, a fifth of the total weight; the 78 is the weakest and carries three tenths. Whether a gap of that size matters is a question about your data rather than about the formula — 0.43 of a mark rarely decides a grade, while the weighted-against-simple divergence in the inventory example below is worth $195 across the 1,000 units sitting in stock.

When every value genuinely does count the same, the plain mean, median and mode are quicker to reach in the Average Calculator.

Why the Weights Do Not Have to Add to 100

Weights of 20, 30 and 50 are convenient because they read as percentages, but the formula never asks them to total any particular figure. It divides by whatever they add up to, which normalizes any scale automatically. The same three scores return an identical answer on four different weight scales:

Only the ratios between the weights reach the result. Use whichever scale is natural for the data — percentages off a syllabus, credit hours off a transcript, unit counts off an invoice — and skip the conversion step entirely.

The one thing the scale does change is the second line of the output, and that line is useful. If a syllabus adds to 100 and the panel reports Total weight: 90, a component is missing or one weight box was left empty.

What Is the Weighted Average Formula?

A single line covers every use of it, from a report card to a stock index to a physicist's center of mass.

The Weighted Average Formula

Weighted average = (v₁ × w₁ + v₂ × w₂ + … + vₙ × wₙ) ÷ (w₁ + w₂ + … + wₙ)

Two properties are worth committing to memory. Equal weights collapse the expression into the ordinary mean, because a constant multiplier cancels top and bottom. And the weights are self-normalizing, so multiplying every weight by the same number leaves the answer untouched.

What Each Part Means

• v — a value: the score, the price, the return, the measurement. This is the quantity being averaged.
• w — the weight on that value: its share of the syllabus, its credit hours, the units bought, the size of the position.
• v × w — one product per row, and the only place the two ever meet. The calculator keeps a running total of these.
• Σw — the total of the weights, and the divisor. The panel prints it so you can check it against what you meant to type.
• The quotient — always on the scale of the values, never on the scale of the weights. Averaging prices with unit counts gives a price, not a count.

The Same Formula in Sigma Notation

Written compactly, with i running from 1 to n over the rows:

x̄ = Σ (wᵢ × vᵢ) ÷ Σ wᵢ

Wolfram MathWorld states the discrete definition in its pre-normalized form — x̄ = Σ wᵢvᵢ, written there with xᵢ for the values — where each weight is nonnegative and the weights are required to sum to 1, so the division has already been absorbed into them. Dividing by Σw, as above, is what lifts that requirement and lets the weights arrive on any scale. MathWorld also lists where the construction turns up in physics: finding a center of mass, which is a weighted average of positions with mass as the weight, and moments of inertia.

The Shortcut When Weights Already Add to 100

If the weights are percentages totaling 100, the division is by 100, which you can absorb by writing each weight as a decimal and dropping the denominator:

0.20 × 90 + 0.30 × 78 + 0.50 × 85
= 18.0 + 23.4 + 42.5
= 83.9

Written that way the division disappears into the weights, and a set of weights that already totals 1 can look like a requirement rather than a convenience. It is not one. The calculator does not need the conversion either: enter 20, 30 and 50 and it divides by 100 for you.

How Do You Calculate a Weighted Average?

By hand it is five operations, and the calculator runs them in the same order. Below: the steps, a worked course grade, what the input rows accept, and the spreadsheet equivalent.

Calculating a Weighted Average Step by Step

• List every item with its value and its weight. The order of the rows makes no difference to the answer.
• Multiply each value by its weight. Keep the full figure — rounding a product here pushes the error into the result.
• Add the products together. That total is the numerator, and on its own it means nothing.
• Add the weights together. That total is the denominator, and it is the figure worth sanity-checking.
• Divide the first total by the second, and round once, at the very end.

Step two is where an early rounding does its damage. Rounding 3.7 × 3 to 11 rather than keeping 11.1 looks harmless, and it moves the GPA in the transcript further down this page from 3.43 to 3.42 — enough to matter on a form that asks for two decimals.

Worked Example: A Course Grade

A syllabus splits a course into homework worth 20%, a midterm worth 30% and a final worth 50%. The student scores 90, 78 and 85. Each score is multiplied by its weight before anything is added:

Eight thousand three hundred and ninety, divided by a total weight of 100, is 83.9. Typed into three rows as 90/20, 78/30 and 85/50, the panel returns 83.90 weighted average, with Total weight: 100 and Simple (unweighted) average: 84.33 beneath it. The unweighted figure is the higher of the two because the strongest score carries the smallest weight.

Every score has to be on one scale before it is weighted — 42 out of 60 cannot be averaged against 85 out of 100 until both are percentages, which is the job of the Grade Percentage Calculator.

How the Input Rows Work

Each row is three boxes: Item Name, Value and Weight. Only the last two reach the formula — Item Name is a label for your own benefit, and the arithmetic never reads it. Value and Weight are number fields, so letters cannot be typed into them; a box that refuses what you are typing is registering as empty rather than as text.

The page opens with the counter at 1 and a single blank row, and nothing is computed until you press the button — pressing it on that opening row returns the amber Check your inputs notice reading Weights must be greater than zero. Raise the counter to add rows. It is marked 1 to 20, the row builder accepts up to 50 and caps anything higher, and a blank or zero count falls back to one row. Changing the number rebuilds every row empty, so set it before you type. Reset returns the form to a single blank row.

What each row then contributes depends on what is in it:

One rule summarizes the table: the weighted average ignores a row the instant its weight stops being a positive number, and the simple average ignores nothing at all. The fifth line is the trap worth a second look — leaving a component's score blank while its weight still stands does not skip that component, it scores it zero. A student who has not sat the final and leaves the score box empty while 50 stays in the weight box sees 41.40 weighted average — the same figure the per-point table further down this page gives for a final of zero. Clearing the weight box as well drops the row instead and returns 82.80 with Total weight: 50, which is what homework and the midterm have earned between them.

The Same Calculation in Excel or Sheets

SUMPRODUCT multiplies two ranges element by element and adds the products in one pass, which is precisely the numerator. Microsoft documents it as returning the sum of the products of corresponding ranges, and Google Sheets implements the same function under the same name.

=SUMPRODUCT(B2:B4, C2:C4) / SUM(C2:C4)

With the three scores in B2:B4 and their weights in C2:C4 that returns 83.9. The SUM in the denominator is what makes the weights self-normalizing, and dropping it leaves the numerator standing on its own: with weights of 2, 3 and 5 the bare SUMPRODUCT returns 839, and with percentage weights it returns 8,390. Both look like answers, and neither is a grade.

Weighted Average Examples

Four fields running one formula under four different names. Every set of numbers below has been put through the calculator, and the output is quoted as it appears.

GPA: Grade Points Weighted by Credit Hours

A grade point average is a weighted average whose values are grade points and whose weights are credit hours. The University of Illinois registrar sets out the procedure in exactly those terms: multiply each grade's points by the course's credit hours to get quality points, add the quality points, add the hours, and divide one by the other.

Entered as four rows, the panel returns 3.43 weighted average with Total weight: 12. The simple average of the four grade points is 3.17, and the difference is the one-credit seminar: a straight mean lets that 2.0 pull as hard as the four-credit calculus course, while the GPA gives it a twelfth of the say.

The last decimal deserves a warning. 41.1 ÷ 12 is 3.425, which this calculator rounds up to 3.43. Not every institution rounds — Georgia Tech's registrar states that grade point averages are truncated after two decimal places, which turns the same transcript into a 3.42. Check your own institution's rule before quoting that second decimal anywhere it counts.

Inventory: Unit Prices Weighted by Quantity

The accountant's weighted average cost method is this formula applied to purchases: total cost of goods available divided by total units available. The IFRS Foundation's summary of IAS 2 permits it alongside first-in, first-out for items that are ordinarily interchangeable. Three purchases at three prices:

Prices go in the Value boxes and unit counts in the Weight boxes. The panel returns 4.66 weighted average with Total weight: 1000 — the exact unit cost is $4.655, shown rounded to two decimals.

The simple average of the three prices is $4.85, and the distance between the two figures is not a rounding artifact. Valuing the same 1,000 units at $4.85 apiece puts $4,850 on the balance sheet against a true cost of $4,655: an overstatement of $195, produced entirely by treating a 500-unit purchase as equal in size to a 200-unit one.

Portfolio Return: Returns Weighted by Position Size

A portfolio's return is each holding's return weighted by the money sitting in it. The weights here are dollars, and there is no need to convert them into percentages first — dividing by the total does that.

Three rows of 8.4/60000, 3.1/25000 and −2.5/15000 return 5.44 weighted average with Total weight: 100000. The simple average of the three returns is 3.00, and it describes a portfolio nobody owns — one holding equal money in all three funds.

This example also settles a question the boxes raise. Negative figures are fine in the Value column, and a losing position is averaged in properly; it is the Weight that has to be positive, because a negative weight would mean a holding of less than no money.

Survey Data: Responses Weighted to the Population

Survey weighting is the same operation applied to people. Where a sample under-represents a group, each response from that group is given a larger weight so the group counts in proportion to its real share of the population. Pew Research Center's methods work compares raking, matching and propensity weighting as competing ways of choosing those weights.

Entered as 7.2/850 and 5.6/150, the panel returns 6.96 weighted average against Simple (unweighted) average: 6.40.

That 0.56 difference is the whole argument for weighting. Averaging the two group means directly would treat 150 people as equal in weight to 850, and the further apart the group sizes are, the further the unweighted figure drifts from the population it claims to describe.

Weighted Average Chart: What the Weights Are Worth

Both tables below hold the values completely still and move only the weights. It is the clearest way to see what a weight actually buys.

The Same Three Scores Under Eight Weightings

Homework 90, midterm 78, final 85 — one student, one set of marks, eight ways a syllabus could divide the course between them:

Five full points separate the best split from the worst, 85.40 against 80.40, on identical work. The simple average is 84.33 for every row in the table, and the nearly even 34/33/33 split lands beside it at 84.39, which is what near-equal weights always do.

Two readings are worth taking away. Weight follows the score: any split that moves weight onto the 90 lifts the total, and any split that moves it onto the 78 drags the total down. And a zero weight removes a component from the course altogether — the bottom row is a grade computed from homework and the midterm alone, which is exactly why it sits lowest.

What One Point on the Final Is Worth

With homework at 90 and the midterm at 78 already banked, this table shows what each possible final score does to the course grade under two syllabuses. In the left column the final is worth 50 and the other two components take 20 and 30; in the right column the final is worth 20 and homework and the midterm take 40 each.

Ten marks on the final move the course grade by exactly 5.00 points in the left column and by exactly 2.00 in the right. That exchange rate is the weight divided by the total weight and nothing else: 50 ÷ 100 is half a course point per exam point, 20 ÷ 100 is a fifth of one.

It is also why the two columns cross. At a final of 85 the heavily weighted syllabus is the worse deal, 83.90 against 84.20, because the final is only the second-best of the three marks. Below that the gap widens fast in the other direction: dropping from that 85 to a zero costs 42.50 course points under the first syllabus and only 17.00 under the second.

Working Backwards to the Score You Need

Rearranged, the same formula answers the reverse question. Multiply the grade you want by the total weight, subtract the points already banked, and divide by the weight still outstanding:

Score needed = (target × total weight − points already banked) ÷ remaining weight

(80 × 100 − 4,140) ÷ 50 = 3,860 ÷ 50 = 77.2

Banked points are the products already fixed: 90 × 20 + 78 × 30 = 4,140. Putting 77.2 into the final's row beside those two components returns 80.00 weighted average, which is the check that the algebra held.

The same line shows what is out of reach. A 90 average needs (90 × 100 − 4,140) ÷ 50 = 97.2, which a paper marked out of 100 can technically deliver; a 95 average needs 107.2, which it cannot. Learning that in week ten is more useful than learning it in week fifteen.

How to Read Your Result

Every successful run prints three lines. None of them is decoration, and the third carries a quirk worth knowing about.

The Three Lines the Panel Returns

Running the course-grade example produces these, in this order:

• 83.90 weighted average — the headline, rounded to two decimals, on the same scale as the values you entered.
• Total weight: 100 — the sum of the weights the calculator actually used, printed plainly with no thousands separator, so a hundred thousand appears as 100000.
• Simple (unweighted) average: 84.33 — the ordinary mean of the values in the rows, offered for comparison.

There is no fourth line, and the figures do not update as you type: the calculator runs when you press Calculate and then holds that answer, so an edited row keeps showing the previous result until you press it again. Copy result puts all three lines on the clipboard in one go.

What the Gap Between the Two Averages Tells You

The distance between the first line and the third measures how lopsided the weighting is, and its direction carries information of its own.

• They match exactly when every weight is identical — three scores at weight 1 each return 84.33 weighted average and Simple (unweighted) average: 84.33.
• The weighted figure sits lower when the weight is on your weaker values: 83.90 against 84.33 in the course grade, because the 90 carries the smallest share.
• The weighted figure sits higher when the weight is on your stronger values: 5.44 against 3.00 in the portfolio, because most of the money is in the fund that returned 8.4%.

A wide gap is not a sign of an error. It answers the question of whether the weighting matters here at all, and when the two lines agree to within a few hundredths the honest conclusion is that it does not — the weights you carefully entered are doing nothing to the result.

How much the weighting matters is one question; how far the values sit from the average is a separate one, answered by the Standard Deviation Calculator.

Use Total Weight as Your Error Check

Glance at the second line before the first. It is the part of the output that catches a mistyped entry, and three failures show up there:

• A syllabus that adds to 100 should report Total weight: 100. A total of 90 means a component is missing or one weight box is empty.
• A weight typed as 0, left blank or entered negative is dropped without comment, so the total falls short by exactly that weight.
• A weight typed into a Value box — or the reverse — usually produces a total far larger or smaller than anything you intended, long before the headline looks odd.

One note on the rounding. Both averages print to two decimals, and a figure landing exactly on a half-cent can display its lower neighbor: a result of 2.675 shows as 2.67, because that value has no exact binary representation. It affects what you see, never what was computed.

Limits: When a Weighted Average Misleads

The arithmetic is honest about whatever it is handed. These are the situations in which what it is handed is wrong for the job.

Rows the Calculator Drops, and Rows It Counts Anyway

The tool's own boundary comes first, because it is silent. Any row whose weight is not above zero is dropped from the weighted average — and still counted by the simple average, which divides by the number of rows on the screen rather than the number of rows it used.

Three filled rows plus two left empty return 83.90 weighted average, which is right, above Simple (unweighted) average: 50.60 — the 253 those three scores total, divided by 5 rather than by 3. Add a third empty row and that comparison line reads 42.17.

Trust the headline: the weighted average is computed only from rows carrying a positive weight, and nothing left blank can move it. The third line is comparable only when the row count equals the number of items you entered, so set How many values? to that number before typing and the two lines will describe the same set of data.

A Weighted Average Cannot Leave the Range of Its Values

With positive weights the result always lands between the smallest and the largest value, however extreme the weighting. A 78 at weight 1 against a 90 at weight 999 returns 89.99 weighted average — close to 90, and still inside the range. Weights redistribute influence; they cannot manufacture an answer that no value supports.

That makes a useful self-test. If a result comes back above your best figure or below your worst, a weight went into a Value box or a value into a Weight box. Negative weights could break the guarantee mathematically, which is one reason every row carrying one is discarded before the division happens.

You Cannot Average Two Averages Without Their Weights

This is the error that survives into published figures. A class of 30 averages 84.33 and a class of 10 averages 71.67; the mean of those two numbers is 78.00, and it is wrong. Weighting each class average by its headcount — 84.33 at weight 30, 71.67 at weight 10 — returns 81.17 weighted average with Total weight: 40, and the panel's own third line reports the 78.00 as the unweighted comparison.

The two displayed figures differ by 3.17 marks, and nothing produced that gap except discarding the group sizes. Whenever you average numbers that are themselves averages, the weights are the counts behind them, and those counts have to travel with the numbers.

Rates and Growth Need a Different Mean

A weighted average of the correct values can still be the wrong statistic. Rates sharing a numerator — miles per hour over a fixed distance, price-to-earnings ratios across a portfolio — call for a harmonic mean. Multi-period growth calls for a geometric mean, because returns compound rather than add: a year of +50% followed by a year of −50% averages to 0% by this formula, while $100 actually becomes $75.

The figure that reproduces that $75 is the geometric mean, −13.4% a year, and it is nowhere near zero. Both alternatives are weighted averages in disguise — of reciprocals in the first case, of logarithms in the second — which is exactly why feeding raw rates into this formula returns something that looks plausible and is not.

A Weight Has to Be a Real Quantity

The formula assumes every weight measures one thing on one scale: credit hours against credit hours, units against units, dollars against dollars. Mix the scales — one row weighted by revenue, the next by headcount — and the total is a number with no meaning, even though every step of the arithmetic completes without complaint.

Nothing in the calculation can detect that. It divides by the total of whatever was typed and returns a figure to two decimals either way, which is why the weights deserve more scrutiny than the values do.