Enter the numbers you want to average, separated by commas (,). Along with the average (arithmetic mean), the sum, count, median, geometric mean, maximum, minimum and range are calculated too.
Enter 2 or more numbers separated by commas (,), for example 10, 2, 38. Decimals and negative numbers are OK.
Result
Enter your numbers separated by commas in the field on the left and press "Calculate". The result will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formula
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter numbers separated by commas (,) and you get the average (arithmetic mean) \(\bar{x}\) on the spot
- Questions like "What is the average of the test scores 82, 76, 90, 65, 88?" are answered in one step
- The sum, count, median, geometric mean, maximum, minimum and range are calculated at the same time
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
This page calculates the ordinary average (arithmetic mean), which gives every value the same weight. It cannot be used for an average where values have different weights (a weighted average). The geometric mean can only be calculated when all the numbers are greater than 0.
What is this calculation used for?
Finding your average test score (school)
If your scores on five tests are 82, 76, 90, 65 and 88, your average is \((82+76+90+65+88) \div 5 = 401 \div 5 = 80.2\) points.
Tracking your average over the semester or comparing it with the class average, the mean is the most familiar way to sum up your grades in one number.
Finding a monthly average to set a budget (money)
If your electric bills for three months are $84, $72 and $99, the monthly average is \(255 \div 3 = 85\) dollars.
Spending that changes from month to month becomes a budget number, "about this much every month", once you take the average. The "monthly average" in budgeting apps is this same calculation.
Using the median so the "average" does not mislead you (reading statistics)
Suppose five households have savings of $10,000, $20,000, $30,000, $40,000 and $400,000. The mean is $100,000, but the median is only $30,000.
A few large values pull the mean up, so for lopsided data such as savings or income, the mean and the median are often far apart. When a news report says "the average is X dollars", checking the median too helps you compare it with real life.
Climate "normals", the averages behind weather forecasts (weather)
When a forecast says a day is "warmer than normal", the "normal" is based on the average of 30 years of observations (in the US, NOAA updates these Climate Normals every 10 years).
The very standard for judging whether today's temperature is ordinary is a long-term average. It is an everyday use of the mean that society relies on.
Using the geometric mean for average growth rates (business and investing)
If sales grew to 1.21 times in 2 years, the growth per year is found not by adding and dividing but with the geometric mean, \(\sqrt{1.21} = 1.1\) (1.1 times per year, or 10% growth per year).
For amounts that build up by multiplication, such as investment returns and sales growth, use the geometric mean. This is standard practice in business.
Formula
Average (arithmetic mean)
Standard notation
(the usual math form)
\(\bar{x}\)
\(=\)
\((x_1 + x_2 + \cdots + x_n)\)
\(\div\)
\(n\)
In words
(symbols replaced with words)
③
\(\bar{x}\): mean
\(=\)
①
sum of the data
\(\div\)
②
\(n\): number of values
The formula in words
①
Take the
sum of the data
②
divide it by the
\(n\): number of values
③
and you get the
\(\bar{x}\): mean
Quick example
The average of five quiz scores out of 10, which are 8, 6, 9, 7 and 10, is
\(\bar{x}\): mean
\(=\)
sum (8 + 6 + 9 + 7 + 10 = 40)
\(\div\)
count (5 quizzes)
\((8 + 6 + 9 + 7 + 10) \div 5 = 40 \div 5 = 8\)
Key idea
The mean tells you how much each person would get if the total were shared out equally. You can also think of it as the value you get when you level out uneven data.
Be careful: the mean is pulled strongly by extremely large (or small) values, called outliers. For example, if 5 people have $0, $0, $0, $0 and $10,000, the mean is $2,000, but you cannot say that "everyone has about $2,000". When the data may be lopsided, it helps to look at the median (next formula) as well.
Median
Standard notation
(the usual math form)
\(\tilde{x}\)
\(=\)
\(x_{\left(\,(n+1)/2\,\right)}\)
\(\tilde{x}\)
\(=\)
\(\left( x_{(n/2)} + x_{(n/2+1)} \right)\)
\(\div\)
\(2\)
In words
(symbols replaced with words)
②
\(\tilde{x}\): median (odd number of values)
\(=\)
①
the middle value after sorting from smallest to largest
⑤
\(\tilde{x}\): median (even number of values)
\(=\)
③
sum of the two middle values
\(\div\)
④
\(2\)
The formula in words
①
Sort the data from smallest to largest. When there is an odd number of values,
the value exactly in the middle
②
is the
\(\tilde{x}\): median
③
When there is an even number of values, there are two middle values. Take the
sum of the two middle values
④
divide it by
\(2\)
(this is the mean of the two middle values)
⑤
and you get the
\(\tilde{x}\): median
Quick example
For the data 3, 5, 8, 10 (an even number of values, 4), the median is the mean of the two middle values (5 and 8), so
\(\tilde{x}\): median
\(=\)
sum of the two middle values (5 + 8 = 13)
\(\div\)
2
\((5 + 8) \div 2 = 13 \div 2 = 6.5\)
Key idea
In the formula, \(x_{(k)}\) means "the \(k\)th value when the data is sorted from smallest to largest" (for example, \(x_{(1)}\) is the minimum).
The median is the middle by rank, so its strength is that extremely large or small values (outliers) hardly affect it. When the mean and the median are far apart, it is a sign that the data is being pulled by a few large (or small) values.
Geometric mean
Standard notation
(the usual math form)
\(G\)
\(=\)
\(\left( x_1 \times x_2 \times \cdots \times x_n \right)\)
\(1/n\)
In words
(symbols replaced with words)
③
\(G\): geometric mean
\(=\)
①
product of all the data (everything multiplied together)
②
to the power \(\dfrac{1}{n}\) (the \(n\)th root)
The formula in words
①
Take the
product of all the data multiplied together
②
raise it
to the power \(\dfrac{1}{n}\) (the \(n\)th root: the number that gives the product when multiplied by itself \(n\) times)
③
and you get the
\(G\): geometric mean
Quick example
The geometric mean of 2 and 8 is the square root of their product 16 (the number that gives 16 when squared), so
\(G\): geometric mean
\(=\)
product of the two (2 × 8 = 16)
to the power \(\dfrac{1}{2}\) (square root)
\(G = \sqrt{2 \times 8} = \sqrt{16} = 4\)
Key idea
The geometric mean is used to average amounts that build up by multiplication. For example, if your savings grew to 1.21 times in 2 years, the growth per year is \(\sqrt{1.21} = 1.1\) (1.1 times per year on average). The key is to multiply and take a root, not to add and divide as in the ordinary mean.
The geometric mean is only defined when all the numbers are greater than 0. It cannot be used for data that includes 0 or negative numbers, so this calculator does not show it in that case.
Range
Standard notation
(the usual math form)
\(R\)
\(=\)
\(x_{\max}\)
\(-\)
\(x_{\min}\)
In words
(symbols replaced with words)
③
\(R\): range
\(=\)
①
\(x_{\max}\): maximum
\(-\)
②
\(x_{\min}\): minimum
The formula in words
①
Take the
maximum (the largest value)
②
subtract the
minimum (the smallest value)
③
and you get the
\(R\): range
Quick example
For the data 10, 2, 38, 23, 38, 23, 21, the range is the maximum 38 minus the minimum 2, so
\(R\): range
\(=\)
maximum (38)
\(-\)
minimum (2)
\(38 - 2 = 36\)
Key idea
The range is the quickest way to see how widely the data is spread out. But it only looks at two values, the maximum and the minimum, so a single outlier can change it a lot. To measure the spread more accurately, the standard deviation and similar measures are used.
The average (arithmetic mean) is basically "sum ÷ count". Look at the median as well when the data may have outliers, and use the geometric mean for amounts that build up by multiplication (growth rates, ratios). Choosing the right one makes it easier to see what the data really says.
Symbols and terms
Symbols
| \(x_1, x_2, \ldots, x_n\) | x sub 1, x sub 2, …, x sub n | The individual values you want to average. (Example - the test scores of 5 students) |
| \(n\) | n | The number of values. (Example - for 5 students, \(n = 5\)) |
| \(\bar{x}\) | x-bar | The mean (arithmetic mean). A bar over the letter is the standard way to write a mean in statistics. |
| \(\tilde{x}\) | x-tilde | The median. Textbooks often just write the word "median" instead of a symbol, and some write it as M or Med. |
| \(x_{(k)}\) | x sub k in parentheses | The \(k\)th value when the data is sorted from smallest to largest. (Example - \(x_{(1)}\) is the minimum) |
| \(G\) | G | The geometric mean, from the first letter of "geometric". |
| \(R\) | R | The range, the maximum minus the minimum. It comes from the first letter of "range". |
Terms
| arithmetic mean | The ordinary "average". Add up all the data and divide by the number of values. When people just say "average" or "mean", this is usually what they mean. |
| measure of center | A single number that sums up where the data is centered. The three main ones are the mean, the median and the mode. In US schools they are usually taught in Grade 6. |
| median | The value exactly in the middle when the data is sorted from smallest to largest. With an even number of values, take the mean of the two middle values. It is a measure of center that outliers hardly affect. |
| mode | The value that appears most often in the data. It is one of the measures of center, along with the mean and the median (the calculator on this page does not cover it). |
| geometric mean | The average found by multiplying all the data together and taking the \(n\)th root. It is used to average amounts that build up by multiplication, such as growth rates and ratios. It is only defined when all the numbers are greater than 0. |
| range | The maximum minus the minimum. It is the quickest measure of how spread out the data is. |
| outlier | A value that is extremely far from the other values. The mean is pulled strongly by outliers, but the median is hardly affected. |
| weighted average | An average where each value gets its own "weight". It is used, for example, when a grade counts the midterm exam as 40% and the final exam as 60% (the calculator on this page gives all values the same weight). |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| Addition and division (Grades 3–4) |
|
| The mean (Grade 6) |
|
| Division with decimal answers (Grade 5) |
|
| Statistics and measures of center (Grade 6) |
|
| Square roots (Grade 8) [for the geometric mean] |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the average (arithmetic mean)
| Value 1 | 10 |
| Value 2 | 2 |
| Value 3 | 38 |
| Value 4 | 23 |
| Value 5 | 38 |
| Value 6 | 23 |
| Value 7 | 21 |
| Sum | =SUM(B1:B7) |
| Count | =COUNT(B1:B7) |
| Average (arithmetic mean) | =AVERAGE(B1:B7) |
Table to find the median
| Value 1 | 10 |
| Value 2 | 2 |
| Value 3 | 38 |
| Value 4 | 23 |
| Value 5 | 38 |
| Value 6 | 23 |
| Value 7 | 21 |
| Median | =MEDIAN(B1:B7) |
Table to find the geometric mean
| Value 1 | 10 |
| Value 2 | 2 |
| Value 3 | 38 |
| Value 4 | 23 |
| Value 5 | 38 |
| Value 6 | 23 |
| Value 7 | 21 |
| Geometric mean | =GEOMEAN(B1:B7) |
Table to find the range (max − min)
| Value 1 | 10 |
| Value 2 | 2 |
| Value 3 | 38 |
| Value 4 | 23 |
| Value 5 | 38 |
| Value 6 | 23 |
| Value 7 | 21 |
| Maximum | =MAX(B1:B7) |
| Minimum | =MIN(B1:B7) |
| Range (max − min) | =B8-B9 |
After pasting, B1 to B7 are the cells for your numbers, and the cells in bold green are calculated automatically.
In the first table, B8 shows the sum 155, B9 shows the count 7, and B10 shows the average, about 22.14.
To use a different number of values, add (or remove) rows of numbers, then change "B1:B7" in the formulas to your actual data range (for 10 values, B1:B10).
The geometric mean (GEOMEAN) gives an error if the data includes a number that is 0 or less.
In the first table, B8 shows the sum 155, B9 shows the count 7, and B10 shows the average, about 22.14.
To use a different number of values, add (or remove) rows of numbers, then change "B1:B7" in the formulas to your actual data range (for 10 values, B1:B10).
The geometric mean (GEOMEAN) gives an error if the data includes a number that is 0 or less.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the average (arithmetic mean)
| Value 1 | 10 |
| Value 2 | 2 |
| Value 3 | 38 |
| Value 4 | 23 |
| Value 5 | 38 |
| Value 6 | 23 |
| Value 7 | 21 |
| Sum | =SUM(B1:B7) |
| Count | =COUNT(B1:B7) |
| Average (arithmetic mean) | =AVERAGE(B1:B7) |
Table to find the median
| Value 1 | 10 |
| Value 2 | 2 |
| Value 3 | 38 |
| Value 4 | 23 |
| Value 5 | 38 |
| Value 6 | 23 |
| Value 7 | 21 |
| Median | =MEDIAN(B1:B7) |
Table to find the geometric mean
| Value 1 | 10 |
| Value 2 | 2 |
| Value 3 | 38 |
| Value 4 | 23 |
| Value 5 | 38 |
| Value 6 | 23 |
| Value 7 | 21 |
| Geometric mean | =GEOMEAN(B1:B7) |
Table to find the range (max − min)
| Value 1 | 10 |
| Value 2 | 2 |
| Value 3 | 38 |
| Value 4 | 23 |
| Value 5 | 38 |
| Value 6 | 23 |
| Value 7 | 21 |
| Maximum | =MAX(B1:B7) |
| Minimum | =MIN(B1:B7) |
| Range (max − min) | =B8-B9 |
The same functions as in Excel (AVERAGE, MEDIAN, GEOMEAN, MAX, MIN) work as is. Copy the whole table, paste it into cell A1, and replace B1 to B7 with your own numbers.
How to calculate it in Python
import statistics
numbers = [10, 2, 38, 23, 38, 23, 21] # the numbers to average
average = statistics.mean(numbers) # average (arithmetic mean)
median = statistics.median(numbers) # median
geometric_mean = statistics.geometric_mean(numbers) # geometric mean (only when all numbers are positive)
largest = max(numbers) # maximum
smallest = min(numbers) # minimum
value_range = largest - smallest # range
print(f"Average: {average}")
print(f"Sum: {sum(numbers)} / Count: {len(numbers)}")
print(f"Median: {median}")
print(f"Geometric mean: {geometric_mean}")
print(f"Maximum: {largest} / Minimum: {smallest} / Range: {value_range}")
Runs with the standard library only (the statistics module). Replace the numbers list at the top with your own numbers and run it. The geometric mean (geometric_mean) is available in Python 3.8 and later, but it raises an error for data that includes 0 or negative numbers, so delete the geometric mean lines in that case.
How to write it in LaTeX and other math languages (copy and paste)
Average (arithmetic mean)
x̄ = (x₁ + x₂ + ⋯ + xₙ) ÷ n
\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mover accent="true"><mi>x</mi><mo>¯</mo></mover>
<mo>=</mo>
<mfrac>
<mrow>
<msub><mi>x</mi><mn>1</mn></msub>
<mo>+</mo>
<msub><mi>x</mi><mn>2</mn></msub>
<mo>+</mo>
<mo>⋯</mo>
<mo>+</mo>
<msub><mi>x</mi><mi>n</mi></msub>
</mrow>
<mi>n</mi>
</mfrac>
</mrow>
</math>
bar x = (x_1 + x_2 + cdots + x_n) / n
Mean[{x1, x2, x3}]
xbar := add(x[i], i = 1 .. n)/n;
xbar = mean(x);
x̄ = (x_1 + x_2 + ⋯ + x_n)/n
Median
\tilde{x} = \begin{cases} x_{(\,(n+1)/2\,)} & (n:\ \mathrm{odd}) \\[6pt] \dfrac{x_{(n/2)} + x_{(n/2+1)}}{2} & (n:\ \mathrm{even}) \end{cases}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mover accent="true"><mi>x</mi><mo>~</mo></mover>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mtable columnalign="center left">
<mtr>
<mtd><msub><mi>x</mi><mrow><mo>(</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn><mo>)</mo></mrow></msub></mtd>
<mtd><mrow><mo>(</mo><mi>n</mi><mo>:</mo><mtext>odd</mtext><mo>)</mo></mrow></mtd>
</mtr>
<mtr>
<mtd><mfrac>
<mrow>
<msub><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>)</mo></mrow></msub>
<mo>+</mo>
<msub><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msub>
</mrow>
<mn>2</mn>
</mfrac></mtd>
<mtd><mrow><mo>(</mo><mi>n</mi><mo>:</mo><mtext>even</mtext><mo>)</mo></mrow></mtd>
</mtr>
</mtable>
</mrow>
</mrow>
</math>
tilde x = {(x_(((n+1)/2)), if n odd), ((x_(n/2) + x_(n/2+1))/2, if n even):}
Median[{x1, x2, x3}]
Statistics:-Median([x1, x2, x3]);
med = median(x);
x̃ = (x_(n/2) + x_(n/2+1))/2 (for an even number of values; for an odd number, x̃ = x_((n+1)/2))
Geometric mean
G = (x₁ × x₂ × ⋯ × xₙ)^(1/n)
G = \left( x_1 x_2 \cdots x_n \right)^{1/n}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>G</mi>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<msub><mi>x</mi><mn>1</mn></msub>
<msub><mi>x</mi><mn>2</mn></msub>
<mo>⋯</mo>
<msub><mi>x</mi><mi>n</mi></msub>
<mo>)</mo>
</mrow>
<mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow>
</msup>
</mrow>
</math>
G = (x_1 x_2 cdots x_n)^(1/n)
GeometricMean[{x1, x2, x3}]
G := (x1 * x2 * x3)^(1/3);
G = prod(x)^(1/numel(x));
G = (x_1 x_2 ⋯ x_n)^(1/n)
Range
R = xₘₐₓ − xₘᵢₙ
R = x_{\max} - x_{\min}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>R</mi>
<mo>=</mo>
<msub><mi>x</mi><mi>max</mi></msub>
<mo>−</mo>
<msub><mi>x</mi><mi>min</mi></msub>
</mrow>
</math>
R = x_(max) - x_(min)
Max[{x1, x2, x3}] - Min[{x1, x2, x3}]
R := max(L) - min(L);
R = max(x) - min(x);
R = x_max − x_min
How to have ChatGPT do the calculation
You are a statistics calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). For the following data, find each of the values below. Data: 10, 2, 38, 23, 38, 23, 21 1. The average (arithmetic mean) 2. The sum and the count 3. The median 4. The geometric mean 5. The maximum, the minimum and the range (maximum − minimum) Show the formulas you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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