One Number for All of Them
Nobody reads 500 numbers. They ask one thing first: where is the middle? There are six honest answers — and choosing the wrong one makes your summary quietly lie.
The Arithmetic Mean
Add everything, divide by how many. Five quiz scores — 72, 85, 90, 68, 95 — sum to 410, so x̄ = 82. It is the balance point of the data.
The Weighted Mean
Section A: 10 students, average 90. Section B: 40 students, average 70. Averaging the two averages gives 80 — but 40 of the 50 students sit in the 70 group.
The Geometric Mean
Gain 50%, then lose 40%. The arithmetic mean says +5% a year. Reality: 1.50 × 0.60 = 0.90, so you are down 10%. The geometric mean says −5.13%, and 0.9487² = 0.90.
The Harmonic Mean
60 km out at 30 km/h, 60 km back at 60 km/h. That is 120 km in 3 hours — 40 km/h, not the 45 the arithmetic mean claims. You spend more time at the slow speed.
Three Means, One Ranking
For positive values that are not all equal, harmonic < geometric < arithmetic, always. Here they are on 30 and 60 — and it is a free check on your arithmetic.
The Median
Sort, then take the middle. Five houses at 240, 255, 270, 290, 310 have mean 273 and median 270. Swap the last for a 2,000 mansion: the mean jumps to 611, the median never moves.
The Mode
The most frequent value — the only centre available for nominal data, where there is nothing to add and no order to have a middle of.
- Email 46 — the mode of 100 contact preferences
- Phone 31, SMS 23 — no mean exists for these
- Two peaks usually mean two populations mixed
Mean vs Median Under Skew
The mean is pulled toward the long tail. The median stays put. So their difference tells you the shape for free.
- Right-skewed — long right tail, mean > median (income)
- Left-skewed — long left tail, mean < median (an easy exam)
- Symmetric — the two coincide
Drag the Skew
Which One, and Why
The data decides, not convention:
- Names → mode · ranks → median
- Symmetric numbers → mean · skewed → median
- Growth factors → geometric · rates → harmonic
- Unequal groups → weighted mean
What you learned
Six ways to say “the middle”. The mean balances, the median resists, the mode counts. Multiply → geometric; rate → harmonic; unequal groups → weighted. Under skew the mean chases the tail.