Average Calculator

Average = Σ xᵢ ÷ n

xᵢ

One value in the list. Every value carries the same weight — there is no way to say that one entry counts more than another

n

How many values there are, up to 200 here. It is printed on the results panel so the division can be checked without recounting the list by hand

Σ xᵢ

The sum of every value, also printed. Reading the sum and the count together is usually enough to spot a typo: 433 over 5 values is plausible, 4330 over 5 is not

min / max

The minimum and maximum values in the list — the smallest and the largest. They bound the average, which can never fall outside them, and they are how you tell a tight list from one with a value pulling the answer away

Use the average when every value should count the same and the total is meaningful: the average mark on a test, the average spend per customer, the average of repeated measurements. It is the right summary for a rate or a total-per-item, and the easiest one to explain. Use something else when the list has extremes you do not want to let dominate — one runaway value shifts the average by its full distance divided by the count, so a house price or an income average is usually better reported as a median. When the average sits close to the middle of the smallest and largest values, the list is roughly balanced and the average describes it well; when the average is near one of the two extremes, most of the list is on the other side and the average is being pulled.

1. Five test scores

   1. Sum: 85 + 90 + 78 + 92 + 88 = 433
   2. Count: 5 scores
   3. Average: 433 ÷ 5 = 86.6
   4. Smallest 78, largest 92 — the average sits between them, near the middle

   The average of 86.6 is close to the middle of the range 78 to 92, which is what a balanced list looks like: no single score is dragging the answer. Notice that the average is not one of the five scores — it does not have to be, and a list of five values whose average is a sixth number is completely ordinary. The sum and the count are printed so you can see that 86.6 came from 433 over 5 rather than taking it on trust.

2. A list with negative values

   1. Sum: −5 + (−1) + 3 = −3
   2. Count: 3
   3. Average: −3 ÷ 3 = −1
   4. Smallest −5, largest 3

   Negative values are accepted because plenty of real lists have them: temperatures through a week, gains and losses, a reading expressed as a deviation from a baseline. Two things stay true with negatives in the list. The average can land outside the range the positive values alone would suggest, and the smallest is still the most negative number rather than the one with the largest size — −5 is below −1 even though 5 is a bigger number than 1. Getting that ordering backwards is the usual way a minimum and a maximum come out swapped.

3. A single value

   1. Sum: 42
   2. Count: 1
   3. Average: 42 ÷ 1 = 42
   4. Smallest and largest are both 42, since there is only one value

   One value is a legitimate list and the answer is the value itself, which surprises people who expect a minimum of two. Nothing here needs a second number: the sum and the count are both 42 and 1, the division is defined, and the average of a single measurement is that measurement. The minimum and maximum collapse onto the same number, and that is the honest reading rather than a sign the page gave up.

This is the plain unweighted average: a value entered once counts once, so a frequency table or a weighted survey needs a different calculation, and a list where some entries stand for many observations will give a misleading answer here. The average is also the most easily dragged of the summary measures — a single value far from the rest moves it by that distance divided by the count, which is why the smallest and largest values are printed beside it rather than left to be worked out. It says nothing about the spread of the middle of the list: two lists can share an average and share both extremes and still be nothing alike, one clustered in the middle and one split into two groups at the ends. The list is capped at 200 values, and a token like 1,500 is refused rather than guessed at, because a comma between digits is a decimal comma in much of the world; write 1500 or 1.5.

How do I calculate the average?

Add every value together, then divide by how many values there are. For 85, 90, 78, 92 and 88 the sum is 433 and there are 5 values, so the average is 433 ÷ 5 = 86.6. Both of those operands are printed on the results panel, which makes the calculation checkable without recounting the list by hand.

What is the difference between the average and the mean?

Nothing, in ordinary use — they are two names for the same number. The average of a list is the sum divided by the count, and that is exactly what the arithmetic mean is. The word average is sometimes used loosely for other kinds of average, such as a median or a weighted average, and that looseness is worth watching for when someone quotes an average without saying which one they mean.

Can I use negative numbers?

Yes. Temperatures across a week, gains and losses, or readings expressed as a deviation from a baseline are all legitimate lists, and the arithmetic is unchanged. One thing to watch is the smallest and largest values: with negatives in the list, the smallest is the most negative number, not the one with the largest size, so −5 is below −1.

Why does the calculator show the smallest and largest values?

Because the average alone does not say whether it describes the list well. A set of scores averaging 86.6 that runs from 78 to 92 is a consistent group; one that averages 86.6 but runs from 40 to 100 is a group with two very different halves, and the average hides that. The two extremes bound the answer and show how far it sits from either end.

Can one extreme value change the average much?

It changes it by its distance from the average divided by the count, so the larger the list the smaller the effect of any one entry. In a list of 5 values, replacing a 92 with a 100 raises the average by 1.6; in a list of 50 values the same change would raise it by 0.16. That is why an average is a fair summary of a long list and a fragile one for a short list with an obvious outlier.

• Measures of the Center of the Data — Introductory Statistics 2e, section 2.5 — OpenStax, Rice University
• Measures of Scale — e-Handbook of Statistical Methods, section 1.3.5.6 (the mean as the balance point of the data, and why a single extreme value moves it) — National Institute of Standards and Technology (NIST)