Average Calculator

Learn the concept

These guides explain the idea this tool computes, with worked examples and the distinctions readers most often trip over.

Other names for this tool

An average finder and an average calculator are the same thing, and this is it. Paste in a list and it returns the average of numbers you entered, along with the median, the mode and the range so you can see whether that average is representative or is being dragged by an outlier.

How to calculate an average

The most common kind of average is the arithmetic mean: the sum of all values divided by how many there are.

Formula: Mean = (sum of all values) ÷ (count of values)

Example for 5, 10, 15, 20, 25:

• Sum: 5 + 10 + 15 + 20 + 25 = 75
• Count: 5
• Mean: 75 ÷ 5 = 15

Working it out by hand

Take the list the calculator loads with, 5, 10, 15, 20 and 25. Every figure below is the one the tool returns for it.

1. Sum: 5 + 10 + 15 + 20 + 25 = 75.
2. Count: 5 values.
3. Mean: 75 ÷ 5 = 15.
4. Median: sorted, the middle of five values is the third, so the median is 15 as well. Mean and median agreeing is what evenly spread data looks like.
5. Mode: every value appears once, so nothing is more typical than anything else and the calculator reports no mode.
6. Range: 25 − 5 = 20.

The two narrower averages come out lower, as they always do on positive data that is not all one number: the geometric mean is 13.025855 and the harmonic mean is 10.948905.

A weighted average by hand

Scores of 90, 80 and 70 carrying weights of 3, 4 and 3. Multiply each score by its weight: 90 × 3 = 270, 80 × 4 = 320, 70 × 3 = 210. Add the products: 270 + 320 + 210 = 800. Add the weights: 3 + 4 + 3 = 10. Divide: 800 ÷ 10 = 80. The unweighted mean of the same three scores is 80 too, because the weights happen to be symmetric; change the 4 to a 6 and they part company.

A harmonic average by hand

Driving 60 mph out and 40 mph back over the same distance. Add the reciprocals: 1/60 + 1/40 = 1/24. Divide the count by that total: 2 ÷ (1/24) = 48 mph. The arithmetic mean would say 50, which is wrong here because you spend longer at the slower speed.

Mean, median and mode answer different questions

• Mean. Sum divided by count. It uses every value, which makes it the right summary for roughly symmetric data and the wrong one when a few extreme values dominate.
• Median. The middle value once the list is sorted, or the average of the middle two when the count is even. At most half the data lies above it and at most half below, so an outlier resists moving it: pushing one value further out only changes which observation is central, rather than dragging the figure by the size of the outlier. Resistant is not the same as immune. Adding a seventh value to a six-value list moves the median from a pair to a single observation whatever that value is.
• Mode. The value that occurs most often. It is the only one of the three that works for categories as well as numbers, though this calculator accepts numbers only, and a list can have one mode, several, or none.

Example for salaries of $40K, $45K, $50K, $50K, $55K, $200K:

• Mean: about $73.3K, pulled upward by the single $200K figure
• Median: $50K, which describes a typical salary here far better
• Mode: $50K, the only repeated value

What one outlier does to the mean

This is the single most useful thing to know about averages, and it is easier to see on a number line than in a table. Six values sit below, five of them clustered between 40 and 55 and one far out at 200. The median stays inside the cluster where the data actually is. The mean is dragged out to 73.3, a figure no value in the set is anywhere near.

Neither figure is wrong. They answer different questions: the mean is what everyone would get if the total were shared out equally, and the median is what a typical member of the set looks like. When those two numbers are far apart, the gap between them is telling you the data is skewed, and reporting only one of them hides that.

When a list has no mode

This calculator reports no mode when every distinct value occurs the same number of times. In 1, 2, 3, 4 each value appears once; in 1, 1, 2, 2 each appears twice. In both lists nothing is more typical than anything else, so naming a mode would invent a distinction the data does not contain.

This is a convention rather than a law, and you may meet the alternative one, under which every value in such a list is called a mode. The reading that matters is the same either way: no single value stands out. When there is a genuine mode the calculator lists every tied value in ascending order, so the same data always reads the same way.

Weighted average

A weighted average lets some values count more than others. Multiply each value by its weight, add the products, and divide by the total weight.

Formula: Weighted mean = Σ(value × weight) ÷ Σ(weights)

Because the formula divides by the sum of the weights, the weights are relative. They do not need to total 1 or 100. Credit hours of 3, 4 and 3 give exactly the same answer as proportions of 0.3, 0.4 and 0.3.

Example. Exams worth 60% with a score of 80, homework worth 40% with a score of 95:

• Products: 80 × 0.60 = 48 and 95 × 0.40 = 38
• Total weight: 0.60 + 0.40 = 1.00
• Weighted mean: (48 + 38) ÷ 1.00 = 86

Weights must be zero or positive. A weight of zero excludes that value, which is useful for dropping an entry without editing your list. A negative weight is rejected, because it can push the result outside the range of your data and the answer would no longer be an average.

Geometric and harmonic means

Two further averages answer narrower questions, and both are shown only when every value is strictly positive.

Geometric mean. The nth root of the product of n values. It is the correct average for quantities that compound, such as year-on-year growth factors. An investment growing 10%, then 20%, then falling 5% has growth factors of 1.10, 1.20 and 0.95, and the cube root of their product is about 1.0784, or roughly 7.8% a year. Averaging the three factors arithmetically overstates the result.

Harmonic mean. The count divided by the sum of the reciprocals. It is the correct average for a rate when the quantity on top of that rate is held constant across the legs being averaged. The familiar case is speed over equal distances: driving 60 mph out and 40 mph back over the same distance gives 2 ÷ (1/60 + 1/40) = 48 mph, not 50.

That condition matters. If you instead spend equal time at each speed, the arithmetic mean of 50 mph is the correct answer and the harmonic mean is not. Check which quantity is held constant before choosing, rather than treating the harmonic mean as the general rule for rates.

Edge cases

• One value can own the mean. For 40, 45, 50, 50, 55, 60 the mean and the median are both 50. Change that last 60 to a 200 and the mean becomes 73.333333 while the median stays at 50. Nothing else in the list moved.
• In Weighted mode, only the headline mean is weighted. The weights apply to the weighted mean and to nothing else. The sum, median, mode, range, minimum, maximum and the geometric and harmonic means below it are all still computed from your values alone, exactly as they are in Standard mode. There is no weighted median here, and a row sitting under a weighted headline is not one.
• Averaging averages is a different sum. If 80% of 10 people and 50% of 100 people pass, the overall rate is (8 + 50) ÷ 110 = 52.727273%, not (80 + 50) ÷ 2 = 65%. The plain average silently treats the two groups as the same size. Weighted mode does this correctly: enter the rates as values and the group sizes as weights.
• An empty list has no average. The calculator waits rather than showing a figure, because the mean divides by the count and the count is zero. A single value is its own mean, its own median and its own range of 0.
• A token it cannot read stops everything. A stray letter or currency symbol is named on screen and nothing is calculated until it is fixed. A partial answer from a silently shortened list would be wrong in a way you could not see, because the count is part of the arithmetic.
• Zero is a value. It counts toward the count and pulls the mean down, and it makes the geometric mean unavailable rather than zero. Whether a zero belongs in your list is a question about your data, not about the calculator.
• Negative values narrow what is available. The mean, median, mode and range handle them normally. The geometric and harmonic means are reported only when every value is strictly positive, and the calculator says why rather than printing a figure you should not use.
• Eight significant figures, and real arithmetic underneath. Results are rounded for display only. Sums use compensated addition, so a long list or a mix of very large and very small values keeps its accuracy, but binary floating point still cannot represent every decimal exactly.

Your numbers stay in the page

The list you paste and any weights you enter are processed by this page in your browser. They are not sent to a server, not stored after you close the tab, and not visible to anyone else. There is no account and nothing to sign up for.

Common mistakes

• Confusing mean and median. Published averages are sometimes one and sometimes the other, and for skewed data they differ a lot. Check which was used before comparing.
• Dropping values silently. A blank, a stray letter or a currency symbol changes the count and so the mean. This calculator names anything it cannot read rather than averaging what survives.
• Averaging percentages from unequal groups. Weight by group size instead. 80% of 10 people and 50% of 100 people is an overall 52.7%, not 65%.
• Using the arithmetic mean for compounding growth. Use the geometric mean, which accounts for the multiplication.
• Assuming zero should be excluded. A zero is a value and changes the mean. Whether it belongs depends on the question: a skipped run counts as 0 miles for weekly distance, but not for average pace on the days you ran.

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.1 Measures of Location for the definitions of the mean and median used here.
• Neumaier, A. (1974), Rundungsfehleranalyse einiger Verfahren zur Summation endlicher Summen, ZAMM 54(1), 39 to 51 is the origin of the compensated summation used here, and compares its error bound with the ordinary running total.
• Higham, N. J. (1993), The Accuracy of Floating Point Summation, SIAM Journal on Scientific Computing 14(4), 783 to 799 surveys how summation order and compensation affect the error, which is why an ordinary running total loses accuracy on long lists and mixed magnitudes.