Enter your data values separated by commas (,). The mean (arithmetic mean), median, mode and range are calculated together, along with the data sorted from smallest to largest.
Enter 2 or more numbers separated by commas (,), for example 10, 2, 38, 23, 38, 23, 21. 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 get four measures at once: the mean (arithmetic mean), the median, the mode and the range
- The mode (the value that appears most often) is counted automatically. It also handles data with two or more modes (bimodal or multimodal) and data where every value appears once, so there is no mode
- The data sorted from smallest to largest, the maximum, the minimum, the sum and the count are shown too, so you can check the median and the range yourself
- A plain-language guide to "When should I use the mean, the median or the mode?" and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
The mode is found by counting whether exactly the same value appears two or more times. With data that is finely spread out, such as height or weight, the same value rarely appears twice. In that case it is usual to make a frequency distribution table (group the data into class intervals) first and look at the class interval with the highest frequency.
What is this calculation used for?
Stocking the best-selling sizes (in retail, the mode matters)
Suppose a store sold 7 pairs of shoes in one day, in sizes 9, 9.5, 9, 10, 9.5, 9 and 11. The mean is about 9.6, but what the store should stock more of is the size that sells most, which is the mode: size 9 (3 pairs).
When you decide what to order or keep in stock based on "which one sells most", such as sizes, colors or models, the mode, not the mean, is the number that matters.
Making sense of "the average is X dollars" in the news (mean or median?)
Suppose five households have savings of $30,000, $30,000, $40,000, $50,000 and $400,000. The mean is $110,000, but the median is $40,000 and the mode is $30,000.
For lopsided data such as savings or income, a few large values pull the mean up, so the median or the mode is closer to what a typical household has. That is why the US Census Bureau reports household income as both a median and a mean. Knowing the difference keeps you from being misled when you read about an "average" in the news.
Finding the most popular choice in a survey or vote (categorical data)
In a vote on favorite school lunches by a class of 30 students, with pizza 12 votes, tacos 10 and chicken nuggets 8, the typical choice is the mode, "pizza". There is no way to calculate a mean or median of lunch menus.
The mode is the only measure of center you can use for data that is not numbers (categorical data). From product surveys to election results, it is one of the most used basics of summarizing data.
Quality control on a production line (watching the spread with the range R)
In factories, a few products are regularly pulled from the line and measured for size or weight, and the range (maximum − minimum) of each small sample is recorded and watched. This is called an R chart, and it is widely used (control charts are one of the seven basic tools of quality).
If the range suddenly gets wider, it is a sign that something is wrong, such as a loose machine part or uneven material. Because one subtraction shows the spread, it is used every day on the factory floor.
Seeing how test scores are really spread out (for teachers and parents)
Even if the class average is 60 points, the class may be split into a group that did well (around 80) and a group that struggled (around 40), with almost no students near 60 (a bimodal distribution).
Instead of judging by the mean alone, also look at the mode, the median and the spread of the scores (the range). Then you can plan help that fits the real situation of the class.
Formula
Mode
Standard notation
(the usual math form)
\(M_o\)
\(=\)
\(x \ \ \bigl(\, f(x) = f_{\max} \,\bigr)\)
In words
(symbols replaced with words)
②
\(M_o\): mode
\(=\)
①
\(x\): the value that appears most often
The formula in words
①
Count how many times each value appears in the data (its frequency), and pick
\(x\): the value that appears most often
②
That value is the
\(M_o\): mode
Quick example
For the scores of 5 tests, 60, 70, 70, 80, 95, only 70 appears twice, the most of any value, so the mode is
\(M_o\): mode
\(=\)
the value that appears most often (70)
\(M_o = 70 \ \ \bigl(\, f(70) = 2 \,\bigr)\)
Key idea
The mode is a measure of center that you find just by counting, with no adding or dividing as in the mean. In the formula, \(f(x)\) is "how many times the value \(x\) appears in the data (its frequency)", and \(f_{\max}\) is "the largest of those frequencies".
If two or more values share the highest count, all of them are modes (data with two peaks is called bimodal). If every value appears only once, there is no mode.
The biggest strength of the mode is that it is the only measure of center that also works for data that is not numbers (categorical data), such as a vote on favorite school lunches.
Mean (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 mean of the scores of 5 tests, 60, 70, 70, 80, 95, is
\(\bar{x}\): mean
\(=\)
sum (60 + 70 + 70 + 80 + 95 = 375)
\(\div\)
count (5 tests)
\((60 + 70 + 70 + 80 + 95) \div 5 = 375 \div 5 = 75\)
Key idea
The mean tells you how much each person would get if the total were shared out equally. It is the only measure of center that uses the value of every data point in the calculation. Because of that, its weak point is that extremely large (or small) values, called outliers, pull it strongly. In the example above, the best score of 95 pulls the mean of 75 above three of the five scores (60, 70, 70).
In statistics, the mean of a sample is usually written \(\bar{x}\) (x-bar), and the mean of a whole population is written \(\mu\) (mu).
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
\(2\): number of middle values
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 the
\(2\): number of middle values
(this is the mean of the two middle values)
⑤
and you get the
\(\tilde{x}\): median
Quick example
Sorted from smallest to largest, the scores of 5 tests are 60, 70, 70, 80, 95 (an odd number of values, 5). The median is the value exactly in the middle (the 3rd one), so
\(\tilde{x}\): median
\(=\)
3rd value from the smallest (70)
\(\tilde{x} = x_{(3)} = 70\)
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. In the example above, the mean is 75 but the median is 70, which is not pulled up by the single score of 95. 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.
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 scores of 5 tests, 60, 70, 70, 80, 95, the range is the maximum 95 minus the minimum 60, so
\(R\): range
\(=\)
maximum (95)
\(-\)
minimum (60)
\(95 - 60 = 35\)
Key idea
The range is the quickest way to see how widely the data is spread out. The mean, the median and the mode describe the center of the data, while the range is the one measure here that describes the spread. Keep this difference in mind.
But the range 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 word "range" can mean other things in some fields, but this calculator uses the most basic definition, "maximum − minimum".
The mean, the median and the mode are the three main measures of center, each summing up data in a single number. The mean reflects every small difference, the median resists outliers, and the mode is found just by counting and also works for categorical data. Each has its own strength, so choose the one that fits the shape of your data. The range (maximum − minimum) is the quickest measure of how wide the spread is.
Symbols and terms
Symbols
| \(x_1, x_2, \ldots, x_n\) | x sub 1, x sub 2, …, x sub n | The individual data values. (Example - the scores of 5 tests) |
| \(n\) | n | The number of values. (Example - for 5 tests, \(n = 5\)) |
| \(\bar{x}\) | x-bar | The mean (arithmetic mean). A bar over the letter is the standard way to write a mean in statistics. The mean of a whole population is written \(\mu\) (mu) instead. |
| \(\tilde{x}\) | x-tilde | The median. Textbooks often just write the word "median" instead of a symbol, and some write it as M or Med. |
| \(M_o\) | M sub o (mode) | The mode, a short form of the word "mode". It is also written Mo. Textbooks usually just write the word instead of a symbol. |
| \(f(x)\) | f of x | How many times the value \(x\) appears in the data (its frequency). (Example - if 70 appears twice, \(f(70) = 2\)) |
| \(f_{\max}\) | f max | The largest frequency (the highest count). A mode is a value whose frequency equals this number. |
| \(x_{(k)}\) | x sub k in parentheses | The \(k\)th value when the data is sorted from smallest to largest. (Example - \(x_{(1)}\) is the minimum) |
| \(x_{\max},\ x_{\min}\) | x max, x min | The maximum and the minimum of the data. They are the inputs to the range formula. |
| \(R\) | R | The range, the maximum minus the minimum. It comes from the first letter of "range". |
Terms
| 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. |
| arithmetic mean | All the data added up and divided by the number of values. This is what people usually mean by "average". Because it uses every value, it is easily pulled by outliers. |
| 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. If several values share the highest frequency, all of them are modes; if every value appears once, there is no mode. It is the only measure of center that also works for categorical data. |
| range | The maximum minus the minimum. It is the quickest measure of how spread out the data is. |
| frequency | How many times a value (or a class interval) appears in the data. The mode is the value with the highest frequency. |
| frequency distribution table | A table that splits the data into ranges (class intervals) such as "60 to under 70" and counts the frequency of each. For finely spread data such as height, the mode is thought of as the class interval with the highest frequency. |
| class interval | In a frequency distribution table, one of the ranges such as "60 to under 70" that the data is split into (also called a class or a bin). The value in the middle of a class interval is its midpoint, and for finely spread data the mode is taken as the midpoint of the class interval with the highest frequency. |
| bimodal (multimodal) | When the distribution of the data has two (or more) peaks. Having more than one mode is a sign of this, and the data may be a mix of two groups with different characteristics. |
| outlier | A value that is extremely far from the other values. The mean and the range are pulled strongly by outliers, but the median and the mode are hardly affected. |
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, subtraction and division (Grades 3–4) |
|
| The mean (Grade 6) |
|
| Statistics and measures of center (Grade 6) |
|
| Frequency tables and histograms (Grade 6) |
|
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 mode
| Value 1 | 60 |
| Value 2 | 70 |
| Value 3 | 70 |
| Value 4 | 80 |
| Value 5 | 95 |
| Mode | =MODE.SNGL(B1:B5) |
Table to find the mean (arithmetic mean)
| Value 1 | 60 |
| Value 2 | 70 |
| Value 3 | 70 |
| Value 4 | 80 |
| Value 5 | 95 |
| Sum | =SUM(B1:B5) |
| Count | =COUNT(B1:B5) |
| Mean (arithmetic mean) | =AVERAGE(B1:B5) |
Table to find the median
| Value 1 | 60 |
| Value 2 | 70 |
| Value 3 | 70 |
| Value 4 | 80 |
| Value 5 | 95 |
| Median | =MEDIAN(B1:B5) |
Table to find the range (max − min)
| Value 1 | 60 |
| Value 2 | 70 |
| Value 3 | 70 |
| Value 4 | 80 |
| Value 5 | 95 |
| Maximum | =MAX(B1:B5) |
| Minimum | =MIN(B1:B5) |
| Range (max − min) | =B6-B7 |
After pasting, B1 to B5 are the cells for your numbers, and the cells in bold green are calculated automatically.
The first table shows the mode 70 in B6. The second shows the sum 375 in B6, the count 5 in B7 and the mean 75 in B8. The third shows the median 70 in B6, and the fourth shows the range 35 in B8.
MODE.SNGL returns only the first mode when there are several. To get all of them, use "=MODE.MULT(B1:B5)" (the results spill into several cells). When every value appears once, you get the "#N/A" error, which means there is no mode.
To use a different number of values, add (or remove) rows of numbers, then change "B1:B5" in the formulas to your actual data range (for 7 values, B1:B7).
The first table shows the mode 70 in B6. The second shows the sum 375 in B6, the count 5 in B7 and the mean 75 in B8. The third shows the median 70 in B6, and the fourth shows the range 35 in B8.
MODE.SNGL returns only the first mode when there are several. To get all of them, use "=MODE.MULT(B1:B5)" (the results spill into several cells). When every value appears once, you get the "#N/A" error, which means there is no mode.
To use a different number of values, add (or remove) rows of numbers, then change "B1:B5" in the formulas to your actual data range (for 7 values, B1:B7).
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 mode
| Value 1 | 60 |
| Value 2 | 70 |
| Value 3 | 70 |
| Value 4 | 80 |
| Value 5 | 95 |
| Mode | =MODE(B1:B5) |
Table to find the mean (arithmetic mean)
| Value 1 | 60 |
| Value 2 | 70 |
| Value 3 | 70 |
| Value 4 | 80 |
| Value 5 | 95 |
| Sum | =SUM(B1:B5) |
| Count | =COUNT(B1:B5) |
| Mean (arithmetic mean) | =AVERAGE(B1:B5) |
Table to find the median
| Value 1 | 60 |
| Value 2 | 70 |
| Value 3 | 70 |
| Value 4 | 80 |
| Value 5 | 95 |
| Median | =MEDIAN(B1:B5) |
Table to find the range (max − min)
| Value 1 | 60 |
| Value 2 | 70 |
| Value 3 | 70 |
| Value 4 | 80 |
| Value 5 | 95 |
| Maximum | =MAX(B1:B5) |
| Minimum | =MIN(B1:B5) |
| Range (max − min) | =B6-B7 |
Almost the same functions as in Excel (AVERAGE, MEDIAN, MAX, MIN) work as is. Only the mode function is named "MODE" (it works the same as MODE.SNGL in Excel; to get several modes at once, use MODE.MULT).
Copy the whole table, paste it into cell A1, and replace B1 to B5 with your own numbers.
Copy the whole table, paste it into cell A1, and replace B1 to B5 with your own numbers.
How to calculate it in Python
import statistics
numbers = [10, 2, 38, 23, 38, 23, 21] # the data to look at
mean_value = statistics.mean(numbers) # mean (arithmetic mean)
median_value = statistics.median(numbers) # median
mode_values = statistics.multimode(numbers) # mode (a list with all modes if there are several)
value_range = max(numbers) - min(numbers) # range (max - min)
print(f"Mean: {mean_value}")
print(f"Median: {median_value}")
print(f"Mode: {mode_values}")
print(f"Range: {value_range}")
print(f"Sorted: {sorted(numbers)}")
Runs with the standard library only (the statistics module). Replace the numbers list at the top with your own numbers and run it. For the mode, multimode (Python 3.8 and later) returns a list with all the modes if there are several. In the example above it is [38, 23] (38 and 23 each appear twice). When every value appears once, it returns all the values; read this as "no mode".
How to write it in LaTeX and other math languages (copy and paste)
Mode
Mo = x (f(x) = f_max)
M_o = x \quad \bigl(\, f(x) = f_{\max} \,\bigr)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>M</mi><mi>o</mi></msub>
<mo>=</mo>
<mi>x</mi>
<mspace width="1em"/>
<mrow>
<mo>(</mo>
<mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo>
<mo>=</mo>
<msub><mi>f</mi><mi>max</mi></msub>
<mo>)</mo>
</mrow>
</mrow>
</math>
M_o = x (f(x) = f_(max))
Commonest[{x1, x2, x3}]
Statistics:-Mode([x1, x2, x3]);
M = mode(x);
M_o = x (f(x) = f_max)
Mean (arithmetic mean)
x̄ = (x₁ + x₂ + ⋯ + xₙ) ÷ n
\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{1}{n}\sum_{i=1}^{n} x_i
<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)))
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 mean (arithmetic mean) 2. The median 3. The mode. If there are several, list all of them. If every value appears once, answer "no mode" 4. The range (maximum − minimum) 5. The data sorted from smallest to largest 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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