Arithmetic mean is one of 2 descriptive statistics formulas in the statistics section of this library, and it is used at high school · ap · university level.
Why arithmetic mean works
The mean is what you get by pouring every value into one pile and sharing it back out equally. Adding preserves the total and dividing splits it into n identical shares, which is why multiplying the mean by the count always returns the original total.
What each symbol means
$x_i$ are observations and $n$ is their count.
Arithmetic mean: when it holds
$n>0$; the mean is sensitive to extreme values.
When it stops applying
It assumes the values are roughly balanced around a centre. In the set 1, 2, 3, 4, 100 the mean is 22 even though four of the five values are below 5, so a single extreme observation drags the summary away from anything typical. The median of 3 describes that set far better.
Arithmetic mean: a worked example
For $2,4,9$, $\bar x=(2+4+9)/3=5$.
The mistake to avoid
What people do: Averaging two group averages directly, calling 70 and 90 an overall average of 80.
Why it goes wrong: That treats both groups as the same size. With 10 students at 70 and 30 at 90, the real overall mean is 3400 divided by 40, which is 85, not 80.
Do this instead: Rebuild the totals first: multiply each group mean by its own count, add those totals, and divide by the combined number of observations.
Arithmetic mean: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $x_i$ are observations and $n$ is their count.
- Check the conditions before substituting. $n>0$; the mean is sensitive to extreme values.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most statistics slips.
Where this formula fits
- Subject
- Statistics formulas — 21 entries in this library
- Topic
- Descriptive statistics
- Level
- High school · AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where arithmetic mean comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Mean, Median, and Mode — the lesson behind this formula: choose a summary that matches the distribution.
- Statistics Calculator — check your substitution and the value it produces.
- Study statistics — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 21 statistics formulas — the full grouped reference, or the complete formula library.
Questions about arithmetic mean
When should I report the median instead?
Whenever the data are skewed or contain extreme values, which is why house prices and incomes are almost always reported as medians. The median only cares about position, so one huge value cannot move it.
Can the mean be a number that never actually occurs?
Yes, and often it is. A mean of 2.4 children per family is a perfectly good summary even though no family has 2.4 children, because it stands for the shared-out total.
What happens to the mean if I remove one extreme value?
It can jump a long way. Dropping the 100 from 1, 2, 3, 4, 100 moves the mean from 22 down to 2.5, which is why you should never delete a value just because it is inconvenient.
What is the mean of a data set where every value is identical?
It is that value, and every deviation from it is zero. In general the deviations from the mean always add to exactly zero, which is the balance-point property that defines it.