Track 2 · Describing Data · Lesson 3 of 11

Weighted and geometric means

Calculate weighted means and geometric means, and decide when each one — rather than the ordinary mean — answers the question.

30 minFoundation8 graded exercisesFree

A fund gains 50% one year and loses 50% the next. Its “average return” is 0%, yet an investor who put in 10 000 SAR now holds 7 500 SAR. Averages of rates and of values that count unequally need the right kind of mean.

The ordinary (arithmetic) mean treats every value as equally important and combines them by adding. Two common situations break one of those assumptions:

the values do not count equally — an exam worth 50% of a grade, a purchase of 400 units against one of 50 → use a weighted mean;
the values combine by multiplying — growth rates and returns compound year on year → use a geometric mean of growth factors.

\bar{x}_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}

Weighted mean

$x_i$: the $i$ -th value (a score, a price, a return)
$w_i$: its weight: how much it counts (a percentage of the grade, a quantity, an amount invested); weights are non-negative
$\sum w_i$: the total weight; when the weights are percentages that add to 100%, dividing by it is the same as multiplying by the weights as decimals

Worked example

A weighted course grade

A trainee scores 80 on coursework (worth 20%), 65 on the midterm (30%) and 90 on the final exam (50%). What is the course grade?

Multiply each score by its weight: $0.20 \times 80 = 16$ , $0.30 \times 65 = 19.5$ , $0.50 \times 90 = 45$ .
Add and divide by the total weight ( $0.20 + 0.30 + 0.50 = 1$ ): $\bar{x}_w = (16 + 19.5 + 45)/1 = 80.5$ .
Compare with the unweighted mean $(80 + 65 + 90)/3 = 78.3$ . The final exam counts most and was the best score, so the weighted grade is higher.

The course grade is 80.5.

Growth compounds. If sales grow 10% one year and 20% the next, they are multiplied by $1.10 \times 1.20 = 1.32$ — a 32% rise, not 30%. Because growth multiplies, the right average is the constant rate which, applied every period, gives the same end result. That is the geometric mean of the growth factors $1 + r$ .

G = \left(\prod_{i=1}^{n} x_i\right)^{1/n}, \qquad \bar{r}_g = \left(\prod_{i=1}^{n} (1 + r_i)\right)^{1/n} - 1

Geometric mean, and the average growth rate per period

$\prod$: “the product of”: multiply all the terms together
$x_i$: positive values (the geometric mean is defined only for values greater than 0)
$r_i$: the growth rate or return in period $i$ as a decimal ( $-0.20$ for a 20% fall), so $1 + r_i$ is the growth factor ( $0.80$ )
$\bar{r}_g$: the average (compound) growth rate per period

Worked example

Three years of revenue growth

A company's revenue changed by +10%, −20% and +15% over three years. What is its average annual growth rate?

Convert rates to growth factors: 1.10, 0.80, 1.15. (A rate of −20% becomes the factor 0.80 — never feed the negative rate itself into a geometric mean.)
Multiply: $1.10 \times 0.80 \times 1.15 = 1.012$ , so revenue ended 1.2% above where it started.
Take the $n$ -th root: $1.012^{1/3} = 1.00398$ , then subtract 1: $\bar{r}_g = 0.00398$ , about 0.40% a year.
Check: $1.00398^3 = 1.012$ ✓. The arithmetic mean of the rates, $(10 - 20 + 15)/3 = 1.67\%$ , would claim four times as much growth: $1.0167^3 = 1.051$ , a 5.1% rise that never happened.

Average growth ≈ 0.40% per year. The arithmetic mean of the rates (1.67%) overstates it.

Dataset

Daily returns of three funds

750 rows · one row = one trading day · synthetic data

About this dataset Open in the lab

Date	Income fund return (%)	Balanced fund return (%)	Growth fund return (%)
2022-01-02	0.127	-0.19	0.935
2022-01-03	0.145	0.287	2.042
2022-01-04	-0.097	-0.218	-0.934
2022-01-05	0.004	-0.305	0.727
2022-01-06	0.295	0.228	-0.136
2022-01-09	-0.199	-0.216	-0.017

Daily returns, in percent, of three (synthetic) funds over 750 trading days. A value of −0.5 means the fund lost 0.5% that day: growth factor 0.995.

The gap matters even with tiny daily numbers. In this dataset the balanced fund's arithmetic mean daily return is slightly positive (about +0.0014% a day), yet compounding its 750 daily growth factors leaves the investor about 0.09% down: its geometric mean daily return is slightly negative (about −0.0001% a day). Volatility turned a positive average into a loss.

Quick check

Over two years a salary rises by 4% and then by 6%. Which is the average annual rise?

Exercises

The lesson is complete when every exercise is answered correctly. Hints are free; viewing a worked solution before solving means the exercise won't count towards mastery.

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Exercise 1 · Guided, difficulty 1 of 3

Weighted course grade

Not started

A professional certificate is graded on coursework (worth 20%), a midterm test (30%) and a final exam (50%). A candidate scores 59 on coursework, 75 on the midterm and 86 on the final.

What is the candidate's overall grade? Give it to 1 decimal place.

Exercise 2 · Practice, difficulty 2 of 3

Average price paid per unit

Not started

A hospital's procurement team bought the same disposable item in four orders:

Order	Units	Price per unit (SAR)
1	87	15.74
2	219	14.58
3	42	14.82
4	279	14.13

What was the average price paid per unit across all four orders? Give it in SAR to 2 decimal places.

Exercise 3 · Practice, difficulty 2 of 3

Average annual growth

Not started

A logistics company's shipment volume changed by +6%, +4%, +20%, −8% in four consecutive years.

What was the average annual growth rate over the four years? Give it as a percentage to 2 decimal places (negative if volume fell on average).

Exercise 4 · Practice, difficulty 2 of 3

Build the growth-rate formula

Not started

A value grows from a starting level $V_0$ to a final level $V_n$ over $n$ years. Write an expression for the average annual growth rate (as a decimal, so 0.05 means 5%) in terms of V0, Vn and n.

Exercise 5 · Application, difficulty 3 of 3

Daily returns of the growth fund

Not started

The column growth_fund in asset_returns holds 750 daily returns in percent. Calculate (a) the arithmetic mean daily return, (b) the geometric mean daily return (from the growth factors $1 + r/100$ ), both in percent to 4 decimal places, and (c) the total change in value over the whole period, in percent to 2 decimal places.

Data: asset_returns · Open in the lab to explore and calculate.

Exercise 6 · Practice, difficulty 2 of 3

Which mean?

Not started

For each question, choose the kind of mean that answers it.

Average yearly growth of a city's population, given five yearly growth rates

A student's grade point average when courses carry different credit hours

Average daily temperature over a week

Return of a portfolio holding 70% in one fund and 30% in another over the same month

Average annual return of a fund over ten years, for someone who stayed invested throughout

Average number of calls handled per agent in a contact centre

Exercise 7 · Interpretation, difficulty 2 of 3

The advertised average return

Not started

A fund's annual returns over four years were +30%, −20%, +25% and −15%. Its brochure states: “Average annual return: 5%.” What is the most accurate comment?

Exercise 8 · Interpretation, difficulty 3 of 3

Positive average, negative result

Not started

In the asset_returns dataset the balanced fund's arithmetic mean daily return is slightly positive, yet an investor who held it for all 750 days ended slightly down. In two or three sentences, explain how both can be true.

What you can now do

Calculate a weighted mean and recognise when values count unequally
Average growth rates with the geometric mean of growth factors 1 + r
Explain why volatile returns make the arithmetic mean overstate compound growth
Choose between the arithmetic, weighted and geometric mean for a question