Mean Calculator

Mean = (x₁ + x₂ + … + xₙ) ÷ n = Σx ÷ n

xᵢ

One number in the list. Values may be negative — a temperature, a balance, a deviation from a target — and they may be decimals or fractions; only the total and the count go into the formula, so nothing has to be sorted first

Σx

The sum of every value, printed on the results panel. It is the numerator, and it is the number that carries the units: a sum of minutes is measured in minutes, and so is the mean that comes out of it

n

How many values there are, also printed. It is the denominator, and it is where this average gets its meaning — the sum is shared out equally among the values, so the answer is what each one would be if they all had the same amount

Σx ÷ n

The one division, written out on the panel with the actual numbers in it. Checking it is the point of showing the sum and the count separately: if the answer looks wrong, this is the line that says whether the sum was mistyped or the count was

Use the mean for quantities that add up, and where every entry should count the same. That covers most of the everyday cases: the average mark in a class, the average price of a basket of goods, the average number of minutes a task takes, the average temperature over a month. The mean is also the number that makes the total come out right — the sum of the deviations from it is exactly zero — which is why it is the natural summary of a total, and why it is the input to nearly everything else in statistics. Use something else when the entries are not equally important, which is a weighted average; when the values multiply instead of adding, such as growth rates and ratios, which is a geometric mean; when the values are rates of doing something, which is a harmonic mean; and when a few extreme values would drag the answer away from the typical case, which is a median. The mean is sensitive to outliers by construction, since every value pulls on it in proportion to its distance, and that sensitivity is sometimes the reason to use it and sometimes the reason not to.

1. Five test scores

   1. Add the values: 85 + 90 + 78 + 92 + 88 = 433
   2. Count them: there are 5
   3. Divide: 433 ÷ 5 = 86.6

   The panel writes that division out in full — (85 + 90 + 78 + 92 + 88) / 5 = 86.6000 — so the answer can be checked without redoing the addition, and so the denominator is visibly the count rather than something else. The sum of 433 is worth a look on its own here: it is the total the class earned, and the mean is the mark every student would have had if the total were shared out equally. This same five-number list is the default on all four pages of this group, so entering one page from another gives the same data under a different average: 86.6 is the arithmetic mean, and the weighted, geometric and harmonic means are all lower than it for this list.

2. A list with negative values

   1. Add the values: −5 + (−1) + 3 = −3
   2. There are 3 values
   3. Divide: −3 ÷ 3 = −1

   Negative values are ordinary inputs: temperatures below zero, balances in the red, readings relative to a baseline. The panel writes the sum as (−5 − 1 + 3) rather than (−5 + −1 + 3), which is the same thing but reads the way a person would say it. Note that the answer, −1, sits between the smallest value and the largest, as a mean always must — a mean that falls outside the range of the data is a sign that something was mistyped. It also need not be one of the values in the list, and for this list it is not.

3. Two decimal values

   1. Add the values: 0.3 + 0.45 = 0.75
   2. There are 2 values
   3. Divide: 0.75 ÷ 2 = 0.375

   Decimals are entered with a decimal point here, and a value may be written with any number of digits — nothing is rounded on the way in. The answer is shown to four decimal places, which for this pair adds a trailing zero: 0.3750 rather than 0.375. That padding is deliberate and it is not a claim of precision; it is what makes the result panel line up with the division written above it, and it is the number of places every result on this page uses.

4. Eight values, written as a Σ

   1. Add the values: 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40
   2. There are 8 values
   3. Divide: 40 ÷ 8 = 5

   This list is longer than six values, so the step-by-step line switches to the shorthand: Σx / 8 = 5.0000 instead of writing all eight numbers out. The Σ is the same symbol the formula block uses, and it means exactly what the expanded line meant — add them all up — so nothing is lost, and a two-hundred value list does not produce a line two hundred numbers long. The switch happens above six values and not at some other count: below it the expanded line is short enough to be the more useful of the two, and at six values it still is.

The list has to contain at least one value and no more than two hundred, and a token that is not a number is refused rather than skipped — a list of 1, 2 and abc is rejected rather than averaged over the two numbers that parse. A comma between digits is refused as well, so 1,500 has to be written 1500 or 1.5, because a comma is the decimal separator in much of the world and guessing which convention was meant would silently change the answer by a factor of a thousand. Nothing is rounded on the way in: the values are used exactly as typed, and only the printed result is fixed at four decimal places, so a long list of many-digit values produces a correct answer that may print the same as a neighbouring one. The mean has no upper or lower bound to appeal to — it will report an answer for a list of temperatures and for a list of prices without knowing which it is looking at, and it cannot tell that the two do not belong in the same list. Finally, this average is not robust: a single very large value moves it a long way, and every value pulls on it in proportion to how far it sits from the answer.

How do I find the mean of a list of numbers?

Add the numbers up, count how many there are, and divide the total by the count. For the five scores on this page the total is 433 and there are 5 of them, so the mean is 433 ÷ 5 = 86.6. Both the total and the count are printed on the results panel beside the answer, which means the division can be checked without repeating the addition yourself.

What is the difference between the mean and the average?

In everyday use they are the same number, and this page uses the two words interchangeably. In more careful writing, average is the general term for any middle value — the mean, the median and the mode are all averages — while the mean names the specific one computed by adding the values and dividing by the count, sometimes called the arithmetic mean to distinguish it from the geometric and harmonic means.

Can the mean be a number that is not in the list?

Yes, and usually it is. Nothing in the calculation requires the answer to be one of the values: the mean of 1 and 2 is 1.5, which is not in that list, and the mean of the five scores here is 86.6, which is not one of the five marks either. What is guaranteed is that the answer falls between the smallest value and the largest, and lands on one of them only when the list happens to work out that way.

How do I find the mean of negative numbers?

Exactly the same way as for positive ones: add them up, then divide by the count. Signs take care of themselves in the addition, so −5, −1 and 3 add up to −3, and dividing by 3 gives −1. The panel writes the sum with minus signs between the terms rather than plus signs before negative numbers, which is the same arithmetic written the way it would be said out loud.

Why does the result show four decimal places?

Because every result on this page is printed to a fixed four places, and a fixed number of places is what makes the panel comparable from one entry to the next — 0.375 is shown as 0.3750 so that it lines up with the division written above it. It is not a claim that the answer is accurate to four places; the underlying value carries the full precision of the calculation, and only its display is fixed.

When is the mean the wrong average to use?

When the entries are not equally important, when they multiply rather than add, or when a few extreme values would drag the answer away from the typical case. Unequal importance calls for a weighted average; growth rates and ratios, which compound, call for a geometric mean; rates such as speeds over a fixed distance call for a harmonic mean; and a skewed list where the typical case matters more than the total calls for a median.

• Arithmetic Mean — MathWorld (the definition, the zero-sum property of the deviations, and the inequality with the geometric and harmonic means) — Wolfram MathWorld
• Measures of Location — e-Handbook of Statistical Methods, section 1.3.5.1 (the mean as a balance point, and how a single extreme value moves it) — National Institute of Standards and Technology (NIST)