Averages
How to Find the Average
To calculate the average, add the numbers and divide by how many there are. That is the mean. But average can also mean the median, the mode or the midrange, and each answers a different question. Here is one dataset run through all four.
How to calculate the average: the four formulas
Add every value and divide by the count. For 4, 7, 7, 9, 12, 15 and 88, that is 142 ÷ 7 = 20.29. That is the mean, and it is what most people intend by “average”. Three other measures also claim the word, and on this dataset they give 9, 7 and 46.
The other three are just as legitimate, and each one summarizes the set differently.
Nothing here is a trick question. All four are correct answers to “what is the average”, and the only real skill is knowing which one describes your data honestly. When someone quotes an average without saying which, that is worth asking about, because the gap between them can be enormous.
One dataset, all four averages
Take seven values: 4, 7, 7, 9, 12, 15, 88. They are already sorted, which is the first step for two of the four calculations.
The mean
No value in the set is anywhere near 20.29. Five of the seven numbers are below 10.
The median
Three values sit below it and three above. The size of the 88 is irrelevant — it only has to be above 9.
The mode
The midrange
The midrange uses two numbers and discards the other five.
So the average is 20.29, or 9, or 7, or 46. All four are arithmetically correct. The median of 9 is the most honest description of this set, because six of the seven values cluster between 4 and 15 and one number is off on its own.
When the count is even
Drop the 88 and you have six values: 4, 7, 7, 9, 12, 15. The two central ones are the 3rd and 4th, which are 7 and 9, so the median is (7 + 9) ÷ 2 = 8. The median calculator sorts the list and picks the middle for you, which removes the two places this goes wrong: forgetting to sort, and miscounting which positions are central.
Which average to use
The choice is not a matter of taste. Each measure has a job it does well and a situation where it misleads.
| Average | Formula | Value here | Best used for | Weakness |
|---|---|---|---|---|
| Mean | sum ÷ count | 20.29 | Roughly symmetric data; any figure you will do further arithmetic on, such as a total or a rate | A single extreme value drags it a long way |
| Median | middle value when sorted | 9 | Skewed data: incomes, house prices, response times, anything with a long tail | Ignores how far away the outer values are; needs sorting first |
| Mode | most frequent value | 7 | Categories and discrete choices: shoe sizes, survey answers, the most-ordered item | May not exist, and a set can have several |
| Midrange | (min + max) ÷ 2 | 46 | A fast center for a range: a daily temperature span, a forecast band | Uses only two values, so it is the most outlier-sensitive of the four |
The practical rule: use the mean when the values are reasonably clustered, and switch to the median when they are not. This is why reports on income quote the median. A handful of very large earners pull the mean well above what a typical household actually earns, while the median stays where the middle of the distribution is.
The mode is the only one of the four that works on things that are not numbers. You cannot take the mean of a color or a shirt size, but you can name the one that comes up most. It is also the only average that must be a value actually present in the data.
The average calculator reports the mean, median, mode and midrange together for any list you enter, which is the quickest way to see how far apart they are before you quote one.
How outliers drag the mean
An outlier is a value far from the rest of the set. The mean feels every one of them in proportion to its size, because every value goes into the sum. The median barely notices, because it only cares about position.
The same set with the outlier removed
Replace the 88 with an 18, leaving 4, 7, 7, 9, 12, 15, 18.
Changing one number halved the mean and cut the midrange by 35. The median and mode did not move at all.
That is the whole argument for the median in one comparison. Ten data points and one typo, one measurement error, or one genuinely exceptional case can move a mean far enough to make it useless as a summary, while the median stays anchored to the middle of the data.
None of which makes outliers deletable. An 88 in a set of small numbers may be a mistake, or it may be the most important observation you have. Look at it before you decide. The right move is usually to report both the mean and the median and let the gap between them speak: when they are close, the data is fairly symmetric, and when they are far apart, something is pulling on one tail.
Why the average of averages is usually wrong
Averaging a set of averages treats each group as though it were the same size. When the groups differ, the answer is wrong, and it can be wrong by a lot.
Two classes take the same test
Class A has 30 students and averages 80%. Class B has 10 students and averages 90%.
The true average is 82.5%, not 85%. Class A has three times as many students, so its 80% should count three times as heavily.
The fix has a name: a weighted mean. Multiply each group’s average by its size, add the products, and divide by the total size.
A plain mean is the special case where every weight is equal. The weighted average calculator handles the general case, and course grading is the most common everyday use of it — see the guide on how to calculate your final grade.
Averaging percentages when the groups differ
Percentages are averages already, so this trap is easy to fall into. Suppose one team converts 45 of 50 leads and another converts 12 of 200.
Forty-eight percent against 22.8%. Team B handled four fifths of the leads, and averaging the two rates pretended it handled half. Whenever you average percentages, add the numerators and add the denominators instead.
The same problem with money
Buying 100 shares at $40 and 300 shares at $20 does not give an average cost of $30. The correct figure weights each price by the number of shares: (100 × 40 + 300 × 20) ÷ 400 = 10,000 ÷ 400 = $25. The stock average calculator does this across any number of purchases.
Finding the mean from a frequency table
When data arrives as counts rather than as a list, you do not need to write every value out. Multiply each value by how many times it occurs, add those products, and divide by the total frequency.
Take a survey asking 22 households how many cars they own.
| Cars owned (value) | Households (frequency) | Value × frequency | Running total of households |
|---|---|---|---|
| 1 | 3 | 3 | 3 |
| 2 | 5 | 10 | 8 |
| 3 | 8 | 24 | 16 |
| 4 | 4 | 16 | 20 |
| 5 | 2 | 10 | 22 |
| Total | 22 | 63 |
All four averages from the table
The running-total column is what makes the median findable. Work down it until you pass the halfway position. Here that is between the 11th and 12th observations, and both land inside the row for 3 cars.
Two errors account for most wrong answers here. The first is dividing by the number of rows instead of the total frequency: 63 ÷ 5 = 12.6, which is more cars than any household owns. The second is taking the mean of the frequency column, which answers a question nobody asked. Always divide the sum of the products by the sum of the frequencies.
Common mistakes
- Dividing by the wrong count. The divisor is how many values there are, not how many rows in your table or how many groups you started with.
- Not sorting before taking the median. The median of 12, 4, 9 is 9, not 4. Sort first, every time.
- Averaging averages. Two class averages of 80% and 90% do not give 85% unless both classes are the same size.
- Averaging percentages from different-sized groups. Add the numerators and the denominators instead.
- Quoting the mean on skewed data. On incomes, house prices or response times, the median describes the typical case far better.
- Assuming a mode exists. If every value appears once there is no mode, and if two values tie there are two.
- Confusing midrange with median. Midrange is the average of the smallest and largest. Median is the middle value. On 4, 7, 7, 9, 12, 15, 88 they are 46 and 9.
- Rounding partway through. Carry the decimals and round once, at the end.
- Including a zero that should be a blank. A missing measurement recorded as 0 pulls the mean down. Leave it out of both the sum and the count.
Frequently asked questions
How do you calculate the average of a set of numbers?
Add every value, then divide by how many values there are. For 4, 7, 7, 9, 12, 15 and 88, the sum is 142 and the count is 7, so the mean is 142 ÷ 7 = 20.29. That is the arithmetic mean. The average calculator also returns the median, mode and midrange for the same list.
What is the average value formula?
The mean is sum of values ÷ number of values. Every value counts once and counts equally. If some values should count more than others — larger groups, bigger purchases, heavier syllabus weights — the formula becomes a weighted mean instead: multiply each value by its weight, add, and divide by the sum of the weights.
How do you find the midrange?
Add the smallest and largest values and divide by 2. For 4, 7, 7, 9, 12, 15 and 88, that is (4 + 88) ÷ 2 = 46. Only those two numbers matter, so the midrange is the most sensitive of the four averages to a single extreme value. It is useful for temperature ranges and little else.
What is the difference between the mean and the median?
The mean adds everything and divides by the count. The median is the middle value once the list is sorted. On 4, 7, 7, 9, 12, 15, 88 the mean is 20.29 and the median is 9. The mean is dragged upward by the 88; the median is not, which makes it the better summary of skewed data.
What is the mean of 4, 7, 7, 9, 12, 15 and 88?
It is 20.29. The sum is 4 + 7 + 7 + 9 + 12 + 15 + 88 = 142, and 142 ÷ 7 = 20.2857…, which rounds to 20.29. Notice that no value in the set is close to it. The median of 9 describes this data much more fairly.
Why is the average of averages wrong?
Because it treats every group as the same size. A class of 30 averaging 80% and a class of 10 averaging 90% do not combine to 85%. The correct figure is (30 × 80 + 10 × 90) ÷ 40 = 82.5%. Weight each average by its group size, which is what a weighted mean does.
How do you calculate an average percentage?
Add the numerators and add the denominators, then divide. Converting 45 of 50 leads and 12 of 200 gives 57 ÷ 250 = 22.8%. Averaging the two rates of 90% and 6% gives 48%, which is badly wrong because the second group was four times larger. Averaging percentages is safe only when the groups are equal in size.
How do you find the mean from a frequency table?
Multiply each value by its frequency, add the products, then divide by the total frequency. For values 1 to 5 with frequencies 3, 5, 8, 4 and 2, the products sum to 63 across 22 observations, so the mean is 63 ÷ 22 = 2.86. Do not divide by the number of rows.
What is a weighted mean?
An average in which each value carries an importance factor. Multiply every value by its weight, add the results, and divide by the sum of the weights. Buying 100 shares at $40 and 300 at $20 gives a weighted average cost of 10,000 ÷ 400 = $25, not the $30 a plain mean returns.
Can a set of numbers have more than one mode?
Yes. If two values tie for the highest frequency the set is bimodal, and more than two makes it multimodal. A set where every value appears exactly once has no mode at all. That is the mode’s main limitation: unlike the mean, median and midrange, it is not guaranteed to exist.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/how-to-find-the-average/