Mean, average, arithmetic mean¶

Level: 101 · for anyone who was taught to add them up and divide by how many

One line: In a classroom, average, mean and arithmetic mean all name one calculation, the sum divided by the count, but they are three sizes of word rather than three synonyms: average is any single value that stands for a set, mean is one family of averages, and arithmetic picks out the member of that family that adds.

One calculation, a pile of names¶

A typical school definition reads: the mean, also called the arithmetic average, is a calculated central value of a set of numbers; to find it, add all the numbers, then divide the sum by how many there are. That sentence uses two names, and the word everyone actually says, average, is a third. Keep reading and more turn up:

Name	Where you meet it
average	everyday speech, the news, and the spreadsheet function `AVERAGE`
mean	statistics books, and Python's `statistics.mean`
arithmetic mean, arithmetic average	wherever another kind of mean is also in play
x̄, read "x-bar"	the mean of a sample of data
μ, the Greek letter mu	the mean of a whole population
`AVG`	SQL
expected value, E[X]	probability: the long-run mean of a quantity that depends on chance

On a plain list of numbers every one of them is the same calculation: add them up, divide by how many. Section 1 of the program hands five test scores to Python's statistics.mean and to SQL's AVG, and both return the 78 that adding and dividing gives by hand.

So why so many names? There are two reasons, and they are worth keeping apart.

Everyday word, technical word. Average is the word of ordinary speech: average height, average temperature, the average commute. Mean is the word mathematicians and statisticians use for the same thing. Neither is more correct; they belong to different registers, the way total and sum do.

More than one mean. Mean is not a single calculation. There are others, the geometric mean and the harmonic mean among them, and each is the right answer to a different question. The adjective arithmetic names the one the school definition describes, the one that adds. It exists for the moments when another mean is in the room, and in a first course on averages no other mean ever is. That is why a textbook can introduce "arithmetic average" as a synonym and never explain the adjective: in that classroom it has nothing to tell apart.

There is one more fact that the usual explanation leaves out, and it is the one that can actually mislead you: "average" is a wider word than "mean". In statistics the median and the mode are averages too, and English schools, for one, teach the mean, the median and the mode as three kinds of average. So average is not only the everyday word for the mean. Sometimes it names a different number altogether.

Three sizes of word¶

Put the names in order of size and each one sits inside the one before it:

average            any single value that stands for the whole set
├── median         the middle value
├── mode           the most common value
└── mean           keeps a total unchanged; the adjective says which total
    ├── arithmetic     keeps the sum                 ← "the average" in school
    ├── geometric      keeps the product
    ├── harmonic       keeps the sum of reciprocals
    └── quadratic      keeps the sum of squares

The school definition sits at the bottom of the tree, on the arithmetic mean. In a classroom where nothing else on the tree is ever used, every name above it points at the same number. The names come apart as soon as you meet a second branch.

What the program prints¶

Verified output of mean_vs_average.py — regenerated by tools/run_examples.py, never hand-typed.

1. ONE CALCULATION, SEVERAL NAMES
   Five test scores: 72, 85, 90, 64, 79
     add them up           72 + 85 + 90 + 64 + 79 = 390
     divide by how many    390 / 5 = 78
   The same five scores, handed to software under its own names:
     Python  statistics.mean(scores)        78
     SQL     SELECT AVG(score) FROM scores  78.0
   Average, mean, arithmetic mean, arithmetic average, x-bar:
   on these scores every one of them is this calculation, and gives 78.
   Nobody scored 78. The mean is calculated, not picked from the list.

2. WHAT THE ARITHMETIC MEAN KEEPS: THE SUM
   Give every student 78 instead of their own score:
     78 + 78 + 78 + 78 + 78 = 390          the total is still 390
   That is what the mean is: the one number that can stand in for every
   value without changing the total. So the distances from it cancel:
     72 - 78 =  -6
     85 - 78 =  +7
     90 - 78 = +12
     64 - 78 = -14
     79 - 78 =  +1
     sum of the distances = 0              above and below balance exactly
   That balance is the "central" in "central value".

3. WHY "ARITHMETIC": GROWTH KEEPS THE PRODUCT
   Invest 1,000.00. Year 1: +100%. Year 2: -50%.
     1,000.00 -> 2,000.00 -> 1,000.00      back where it started
   Arithmetic mean of the two returns: +25% a year
     1,000.00 x 1.25 x 1.25 = 1,562.50     WRONG: the account holds 1,000.00
   Growth multiplies, so the mean has to keep the PRODUCT of the factors:
     geometric mean of x2 and x1/2 = square root of (2 x 1/2) = 1
     a factor of 1 is +0% a year
     1,000.00 x 1 x 1 = 1,000.00           right

4. WHY "ARITHMETIC": SPEED KEEPS THE TIME
   Drive 60 km at 30 km/h, and the 60 km back at 60 km/h.
     60/30 + 60/60 = 2 + 1 = 3 hours for 120 km
     120 km in 3 hours is 40 km/h
   Arithmetic mean of the speeds: (30 + 60) / 2 = 45 km/h
     at 45 km/h, 120 km takes 2 h 40 min   WRONG: it took 3 h
   Time is distance / speed, so the mean has to keep the sum of 1/speed:
     harmonic mean = 2 / (1/30 + 1/60) = 40 km/h
     at 40 km/h, 120 km takes 3 h          right

5. "AVERAGE" IS A WIDER WORD THAN "MEAN"
   Ten salaries at a small firm; the last one is the owner's:
      32,000   34,000   34,000   34,000   36,000
      38,000   40,000   42,000   45,000  400,000
   Three numbers, each of them honestly called "the average salary":
     mean      73,500    the payroll shared out equally
     median    37,000    the middle: half earn less, half earn more
     mode      34,000    the salary the most people are paid
   9 of the 10 earn less than the mean. "Average" alone did not say which.

What makes a mean a mean¶

Section 2 is what makes the adjectives make sense. Replace every score by 78 and the total is still 390. That is the arithmetic mean, put exactly: the one number that can stand in for every value without changing the sum. The definition's "add, then divide" is just how you find that number.

It also explains the definition's other two words. Central: because the total is kept, the distances above the mean and below it cancel exactly, −6 + 7 + 12 − 14 + 1 = 0, so the mean is the balance point of the data. Put five equal weights on a ruler at 72, 85, 90, 64 and 79 and it balances at 78. Calculated: nobody scored 78. The mean is worked out from the list, not picked from it, and it need not be one of the numbers at all.

Every other mean keeps a different total, and the adjective says which one:

Mean	Keeps unchanged	Of two numbers a and b	Right for
arithmetic	the sum	(a + b) / 2	amounts that add: scores, salaries, lengths
geometric	the product	√(ab)	rates that multiply: growth, interest, inflation
harmonic	the sum of reciprocals	2 / (1/a + 1/b)	rates over equal amounts: speeds over equal distances
quadratic, or root mean square	the sum of squares	√((a² + b²) / 2)	sizes that matter squared: the rated value of an AC voltage

On the same two numbers they disagree. For 2 and 8 the arithmetic mean is 5, the geometric mean 4 and the harmonic mean 3.2. For positive numbers that is always the order, arithmetic ≥ geometric ≥ harmonic, with equality only when all the numbers are equal. So using the arithmetic mean where the geometric or harmonic mean belongs is wrong in a predictable direction: it always comes out too high.

When the adjective matters¶

Growth multiplies (section 3). An investment doubles one year and halves the next. The returns are +100% and −50%, and their arithmetic mean says +25% a year, which would turn 1,000 into 1,562.50. The account holds 1,000, exactly where it started. Returns are applied by multiplying, so the mean that describes them has to keep the product of the growth factors: √(2 × ½) = 1, a growth factor of 1, which is 0% a year. This is why investors quote a compound annual growth rate, which is a geometric mean, and why an arithmetic mean of returns overstates what the money did whenever the returns are not all equal.

Speed divides (section 4). Drive 60 km at 30 km/h and the 60 km back at 60 km/h. The arithmetic mean of the two speeds is 45 km/h, but the trip took 2 hours plus 1 hour for 120 km, which is 40 km/h. Time is distance divided by speed, so the mean that keeps the total time has to keep the sum of the reciprocals of the speeds: the harmonic mean, 2 / (1/30 + 1/60) = 40. You spend twice as long on the slow leg as on the fast one, and the arithmetic mean counts the two legs as if they took equal time.

In both cases the arithmetic mean was calculated correctly. It answered a question nobody asked.

When "average" misleads¶

Section 5 is where the loose word does damage. Ten people work at a small firm; nine earn between 32,000 and 45,000, and the owner takes 400,000. Three different numbers can each honestly be called "the average salary":

the mean, 73,500, is the payroll shared out equally. Nine of the ten earn less than it.
the median, 37,000, is the middle: half earn less, half earn more.
the mode, 34,000, is the salary the most people are paid.

None of them is wrong. They answer different questions: what would each person get if the payroll were shared equally? and what does a typical person here earn? One large value drags the mean a long way and leaves the median where it was. That is why official income figures are so often medians; the headline figure from the US Census Bureau, for example, is median household income. When a sentence says "the average" and the numbers are lopsided, as incomes, house prices and waiting times usually are, ask which average it means.

Which word to use¶

"Average" is fine in everyday writing, as long as nobody could reasonably ask "which one?".
"Mean" is the word to use when the median is also in play, or when the data are lopsided and the difference matters.
"Arithmetic mean" is worth its extra word only when a geometric or harmonic mean is also in the room. Anywhere else the adjective has nothing to tell apart, and that is exactly why it sounds like a mystery.

In software¶

The tools use the same names, and mostly agree on them. A spreadsheet's AVERAGE is the arithmetic mean, and GEOMEAN, HARMEAN, MEDIAN and MODE sit beside it. Python's statistics module has mean, geometric_mean, harmonic_mean, median and mode. SQL has only AVG, the arithmetic mean. NumPy has both numpy.mean and numpy.average, and they compute the same thing unless you give average a list of weights, which mean does not accept.

Run it yourself¶

From the root of your clone of this repository:

python3 05_Statistics/mean_vs_average/examples/mean_vs_average.py

Auf Deutsch: Stichwörter¶

„Mittelwert“, „Durchschnitt“ und „arithmetisches Mittel“ sind drei Namen, die ineinander geschachtelt sind, nicht gleichbedeutend; dazu geometrisches, harmonisches und quadratisches Mittel.

Stichwörter: arithmetisches Mittel, Durchschnitt (average), Mittelwert (mean), geometrisches Mittel, harmonisches Mittel, quadratisches Mittel, Median, Modus, gewichtetes Mittel.