Arithmetic average is one of 1 averages formula in the arithmetic section of this library, and it is used at middle school · high school level.
Why arithmetic average works
Adding everything up and splitting the total evenly answers the question how much would each one get if they all shared alike. The result also sits at the balance point of the values: the amounts above it stick out by exactly as much as the amounts below it fall short.
What each symbol means
$x_i$ are observations and $n$ is their count.
Arithmetic average: when it holds
$n>0$; the observations should represent quantities that can sensibly be averaged.
When it stops applying
One extreme value can drag the mean somewhere no one actually lives. If four people earn 30,000 dollars and one earns 500,000, the average is 124,000, which describes none of the five, and the median of 30,000 is the fairer summary.
Arithmetic average: a worked example
The average of $6,8,10$ is $24/3=8$.
The mistake to avoid
What people do: Students average two group averages by taking the middle of the two numbers.
Why it goes wrong: That only works when the groups are the same size. A class of 10 averaging 80 and a class of 30 averaging 60 do not combine to 70, because the bigger class should count for more.
Do this instead: Add up the actual totals and divide by the whole count: (800 + 1800)/40 = 65. Whenever the groups are different sizes, go back to totals rather than averaging averages.
Arithmetic average: 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 observations should represent quantities that can sensibly be averaged.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most arithmetic slips.
Where this formula fits
- Subject
- Arithmetic formulas — 13 entries in this library
- Topic
- Averages
- Level
- Middle school · High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where arithmetic average 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.
- Arithmetic Calculator — check your substitution and the value it produces.
- Study arithmetic — the subject guide that explains the ideas these formulas compress.
- Arithmetic Practice — questions that make you retrieve the formula instead of recognising it.
- All 13 arithmetic formulas — the full grouped reference, or the complete formula library.
Questions about arithmetic average
Is the average the same as the middle value?
No, that one is the median. For 6, 8, and 10 they happen to agree at 8, but for 1, 2, and 30 the mean is 11 while the median is 2, so they part company as soon as the data is lopsided.
What happens if I add a value equal to the mean?
Nothing at all; the mean stays put. Adding a number above the mean pulls it up and adding one below pulls it down, which is a quick way to predict the effect of a new score.
Can I average percentages?
Only when they come from equally sized groups. Scoring 50 percent on a 10 question quiz and 90 percent on a 90 question test is not 70 percent overall; count the actual right answers instead.
Does the average have to be one of the values?
No. The average of 6 and 7 is 6.5, and the average family really can have 2.4 children. The mean is a summary of the group, not a member of it.