Average Calculator

Enter your numbers separated by commas and this calculator returns the average, along with the sum, the count, the median, the geometric mean, the largest and smallest values and the range. The set it opens with, 10, 2, 38, 23, 38, 23, 21, adds up to 155 across seven values, so its average is 22.142857142857. The plot underneath redraws as you type, so you can see where each value sits against the average rather than reading it off a table.

What an average is

The word average carries a few meanings in ordinary speech. It can mean typical, or unremarkable, or the one number a whole group gets summed up by. In mathematics it usually means one thing in particular: the arithmetic mean. Add every value together, divide by how many values there were, and what comes out is the average. The idea is old, plain and used everywhere from school reports to national income figures.

The equation below is the definition most people already carry in their heads:

$$\mathrm{Average}=\frac{\mathrm{Sum}}{\mathrm{Count}}$$

The sum is the result of adding all the given numbers, and the count is how many values were added. Take the five numbers 2, 7, 19, 24 and 25. They add to 77, and there are five of them, so the average works out like this:

$$\mathrm{Average}=\frac{2+7+19+24+25}{5}=\frac{77}{5}=15.4$$

Note that 15.4 is not one of the five numbers, and it never had to be. An average summarises a set without belonging to it. A more useful way to picture it is as the level everything would sit at if the total were shared out equally: pour all 77 into five identical glasses and each holds 15.4.

What the average leaves out

One number cannot describe a whole set, and the average hides the thing people most often want to know, which is how spread out the values are. The set 20, 20, 20, 20 has an average of 20. So does 5, 10, 25, 40. The first has a range of zero and the second a range of 35, yet both report the same average. This is why the calculator prints the largest value, the smallest and the range beside the answer, and why the standard deviation calculator exists at all: standard deviation puts a number on that spread instead of leaving it to the eye.

The plot on this page makes the same point visually. Each bar is one of your values, the dashed line across them is the average, and the dotted line is the median. When the two lines sit apart, the set is lopsided.

When the average stops being typical

Averages behave badly when a set contains a few values far from the rest. Suppose nine people in an office earn 30,000 a year and the tenth, who owns the place, earns 1,000,000. The average salary is 127,000. Nobody earns anything close to that. Nine of the ten earn less than the average, and the one figure meant to represent the group represents none of them.

The median handles this better. Sort the ten salaries and the middle of them is 30,000, which is what an employee would actually recognise. This is exactly why house prices, household incomes and salary surveys are almost always reported as medians. One footballer moving into a small town lifts its average income and changes its median hardly at all.

The rule of thumb is short. If the average and the median are close, either will do. If they are far apart, the set is skewed and the median is usually the more honest summary, with the average still worth quoting when what you care about is the total, such as a payroll or a fuel bill.

Mean, median and mode

Three summaries answer three different questions, and picking the wrong one is the most common mistake made with data.

The mean is the total shared equally. Use it for anything additive, such as the total cost of a delivery run divided across the drops. It uses every value, which is its strength and also why one wild value can drag it around.

The median is the middle value once the set is sorted, or the average of the middle two when the count is even. It ignores how extreme the extremes are and only cares about their position, which is what makes it steady. In the default set, sorting gives 2, 10, 21, 23, 23, 38, 38, and the fourth of the seven is 23.

The mode is the value that turns up most often. It is the only one of the three that works on things you cannot add, such as the most common shoe size sold or the most frequent answer on a survey. A set can also have several modes, or none at all. The mean, median, mode and range calculator reports all of them together.

Weighted averages

Sometimes the values do not deserve equal say. A course might set coursework at 30 percent, a presentation at 20, a final exam at 35 and attendance at 15. Scoring 89, 76, 92 and 68 on those, the plain average of the four marks is 81.25, but that is not the grade anyone gets. Multiply each mark by its weight, add those products, and divide by the total weight:

$$x_w=\frac{\sum (\mathrm{weight}\times \mathrm{value})}{\sum \mathrm{weight}}=\frac{8430}{100}=84.3$$

The weighted answer is 84.3, three points above the unweighted one, because the two strongest marks happen to carry the heaviest weights. The weighted average calculator on this page does the same arithmetic for any list, and the grade calculator applies it to a full course.

The plain average is really a weighted average where every weight is the same, so the two are one method rather than two. Weights do not have to be percentages either. Units sold, hours logged and portfolio values all work the same way.

The geometric mean, for things that multiply

Growth rates do not add, they compound, and averaging them the ordinary way gives a wrong answer. Say an investment gains 10 percent, then 50 percent, then loses 30 percent. The arithmetic mean of those three is 10 percent a year, which over three years would turn 1 into 1.331. What actually happened was 1.10 times 1.50 times 0.70, which is 1.155, a total gain of 15.5 percent rather than 33.1.

The geometric mean fixes this by multiplying the values and taking the nth root instead of adding and dividing:

$$\mathrm{GM}=\sqrt[n]{x_1\times x_2\times \dots \times x_n}$$

The cube root of 1.155 is 1.0492, so the honest average is a gain of 4.92 percent a year. Apply that three times and you land back on 1.155, which the arithmetic mean never does. Use the geometric mean for interest rates, population growth, inflation, index numbers and anything else quoted as a ratio or a percentage change.

Two properties are worth remembering. The geometric mean is never larger than the arithmetic mean of the same values, and the two are equal only when every value is identical. And it needs every value to be above zero, since a single zero drags the product to zero and a negative makes the root meaningless. That is why the calculator leaves the row blank when your set contains a zero or a negative number.

The harmonic mean, for rates

A third average handles rates measured per unit of something else, such as speed. Drive 60 kilometres at 60 km/h, then the same 60 kilometres back at 30 km/h. The obvious answer for the average speed is 45 km/h, and it is wrong. The trip out takes one hour and the trip back takes two, so 120 kilometres took three hours, giving 40 km/h.

The harmonic mean produces 40 directly by averaging the reciprocals and flipping the result. It is the right tool whenever the quantity you are averaging sits underneath a division sign, which covers speeds over fixed distances, price-to-earnings ratios across a portfolio and the parallel resistance of a circuit.

Between the three, the harmonic mean is the smallest, the arithmetic mean the largest, and the geometric mean sits between them, whenever the values are positive and not all equal.

Averaging averages

This one catches people constantly. A class of 10 students averaged 60 on a test and a class of 30 averaged 80. The average across both classes is not 70. The 30 students carry three times the weight of the 10, so the real figure is 75, which you get by adding the two totals and dividing by 40. Averaging averages only works when the groups are the same size, and if they are not, you need the weighted method above.

The stronger version of this trap is Simpson's paradox, where a treatment can look better in every subgroup and worse overall once the groups are pooled, purely because of how the group sizes fall. Whenever you see averages combined, ask how many observations sat behind each one.

How many values do you need

An average from a sample is an estimate of the average of the whole population, and its reliability improves with sample size, though slowly. The precision improves with the square root of the count, so cutting your uncertainty in half means gathering four times as much data, and cutting it to a tenth means a hundred times as much. The sample size calculator turns that relationship into an actual number for a survey.

This is also why an average from three readings should be treated gently and one from three thousand can be leaned on. The formula is identical in both cases. What changes is how much weight the answer can bear.

Reading the results on this page

The panel reports seven figures. Sum and count are the two ingredients of the average itself. The median gives you a second opinion that ignores extremes. The geometric mean matters when your numbers are rates. Largest, smallest and range describe the spread the average hides. The step-by-step working underneath shows the addition written out in full for short sets, then the division, so the answer can be checked by hand rather than taken on trust. For quartiles, variance and the rest, the statistics calculator takes the same input and reports the full set.