Foundations · Averages
Weighted Mean Calculator
A weighted mean lets some values count for more than others — credit hours toward a GPA, each exam’s share of a final grade, the size of each holding in a portfolio. It is the balance point of the data once every value pulls with its own weight, and it can sit well away from the plain average. This computes it, shows what each value contributes, and draws exactly where the result balances.
Weighted mean = Σ(weight × value) ∕ Σ(weight). Weights can be credit hours, percentages, counts, probabilities — anything that says how much each value should count. They need not add to 1; the calculator normalises them.
Result
In plain English
A plain average treats every value as equally important. A weighted average asks: how much should each one count? Multiply each value by its weight, add those up, and divide by the total weight. The result leans toward whichever values carry the most weight — which is exactly why a GPA reflects a five-credit course more than a one-credit elective, and why a class’s overall pass rate is dominated by its biggest section.
- weighted mean
- Σ(wᵢ · xᵢ) ∕ Σwᵢ — the sum of each value times its weight, divided by the total weight. The everyday average is the special case where every weight is equal.
- weight
- How much a value counts: a credit hour, a percentage share, a frequency, a probability. Only the relative sizes matter, so doubling every weight changes nothing.
- contribution
- Each value’s share of the result, wᵢ ∕ Σw of the way toward xᵢ. The table below splits the weighted mean into these pieces so you can see what is driving it.
- balance point
- Physically, the weighted mean is the centre of mass: place each value on a beam with a weight hung at it, and the beam balances at the weighted mean. Heavier values pull the balance toward themselves.
- weighted vs plain
- They agree only when the weights are all equal (or, by coincidence, when the values are uncorrelated with their weights). The bigger the spread in weights, the more the two can diverge.
- expected value
- A weighted mean where the weights are probabilities is exactly the expected value of a random variable — the same formula wearing a different hat.
Frequently asked
How do you calculate a weighted mean?
Multiply each value by its weight, add up those products, and divide by the sum of the weights: weighted mean = Σ(wᵢ · xᵢ) ∕ Σwᵢ. For values 90, 80, 70 with weights 1, 2, 3 that is (90·1 + 80·2 + 70·3) ∕ (1 + 2 + 3) = 460 ∕ 6 = 76.67 — lower than the plain average of 80, because the smallest value carries the most weight. The weights do not need to sum to 1; dividing by their total normalises them.
What is the difference between a weighted average and a regular average?
A regular (arithmetic) average gives every value the same importance; a weighted average lets you say that some values count for more. The regular average is just the weighted average with all weights equal. They differ whenever the weights vary and are related to the values — a GPA where you did well in your small classes and poorly in your big ones will sit below the simple average of your grades. If the weights happen to be unrelated to the values, the two come out close.
How do I calculate my GPA with a weighted mean?
Put your grade points (4.0, 3.7, 3.0, …) in as the values and the credit hours of each course as the weights. The weighted mean is your GPA: Σ(grade · credits) ∕ Σcredits. A 4.0 in a 1-credit seminar barely moves it; a 2.0 in a 4-credit core course drags it down hard. That credit-weighting is the whole point — it stops a string of easy one-credit A’s from masking a poor result in a course that actually counted.
Can the weighted mean fall outside the range of the values?
No — as long as the weights are non-negative, the weighted mean always lies between the smallest and largest value, just like a plain average. It is a balance point of the data, and a balance point cannot sit beyond the ends. (Negative weights can push it outside, but a negative weight rarely makes sense for an average.) What it can do is sit much closer to one end than the simple average does, if that end carries most of the weight.