Mean, Median, Mode and Range Calculator
Paste a list, a frequency table or grouped classes and get every average at once, with the working shown and a chart that marks where the mean, median and mode fall.
Average Calculator
Type or paste your data. Results update as you type.
Working appears here.Syntax appears here.Summary appears here.Mean, median, mode, range, midrange, and the geometric and harmonic means from one input.
Lists every mode, says when there is none, and estimates all three from class intervals.
See how each number was reached, so you can check homework instead of just trusting a result.
What Does an Average Calculator Do?
An average calculator finds the typical value of a list of numbers. The word “average” usually means the mean, but statisticians also use the median and the mode, and they can give quite different answers for the same data. This calculator returns all three together, along with the range, the midrange, the sum, the count and the smallest and largest values.
It accepts three kinds of input: a plain list of numbers, a frequency table of values and counts, and grouped data in class intervals such as 60–70. Grouped data is where most online calculators stop, even though it is a standard exam topic. Here it gives an estimated mean, median and mode from the classes, with the working shown.
A chart marks the mean, median and mode on your data, so you can see at a glance whether the numbers are balanced or pulled to one side.
What Are the Mean, Median and Mode?
Each one answers “what is typical?” in a different way.
- Mean: add all the values and divide by how many there are. Mean = Σx / n. It uses every value, so it is sensitive to extreme ones.
- Median: the middle value after sorting. With an odd count it is the middle number. With an even count it is the average of the two middle numbers, so it may not appear in the data. Half the values lie at or below it.
- Mode: the value that appears most often. A data set can have one mode, several modes, or none at all when every value occurs only once.
- Range: the largest value minus the smallest. It measures spread, not center, but nearly every average calculator reports it.
The midrange is the halfway point between the smallest and largest values, (min + max) / 2. It is quick to compute but depends only on the two extremes.
How Do You Use This Mean, Median and Mode Calculator?
- Choose the input style. “List of numbers” takes values separated by commas, spaces, tabs, semicolons or new lines, so a column pasted from Excel works. “Values with frequencies” takes one pair per line, such as
3, 5for the value 3 appearing five times. “Grouped data” takes one class per line, such as60-70, 15. - Read the results. The mean, median and mode are at the top, followed by the range, midrange, sum and count. Modes are listed in full if there are several.
- Check the working. The step list shows how each number was reached, and the table lists each value or class with its count and cumulative total.
- Look at the chart. The orange line is the mean and the green line is the median. If they sit far apart, the data are skewed.
Up to 500 values are accepted in a list and 200 lines in a table. The decimal-places menu changes only the display.
Example 1: How Do You Find the Mean, Median and Mode by Hand?
Nine customers waited 12, 15, 11, 18, 15, 20, 14, 15 and 13 minutes. Sort them first: 11, 12, 13, 14, 15, 15, 15, 18, 20.
All three averages land close together, between 14.8 and 15, which is what you expect from data without extreme values. For nine numbers the median is at position (n + 1) / 2 = 5. For an even count you take positions n / 2 and n / 2 + 1 and average them.
Example 2: What If There Is More Than One Mode, or None?
Ten values: 4, 8, 6, 5, 3, 2, 8, 9, 2, 5. Sorted, they are 2, 2, 3, 4, 5, 5, 6, 8, 8, 9.
When several values tie for the highest count, the data are multimodal and every tied value is a mode. This calculator lists them all rather than picking one. Excel’s MODE.SNGL would report only the first, which is why MODE.MULT exists.
When no value repeats, there is no mode. Take 21, 27, 34, 45 and 52: the mean is 35.8, the median is 34, and the calculator reports “No mode” rather than naming all five. Strictly, you could call every value a mode, but that tells you nothing, so textbooks and software say there is none.
Mean or Median: Which Average Should You Use?
Use the mean for roughly symmetric data without extreme values, and the median when the data are skewed or contain outliers. The mode is the only choice for categories, such as the most common shoe size or the most popular answer.
Here is how sensitive the mean is. Six salaries, in thousands of dollars, are 42, 45, 48, 50, 52 and 55. The mean is 48.6667 and the median is 49, so they agree. Now one executive at 400 joins:
| Salaries ($ thousands) | n | Mean | Median |
|---|---|---|---|
| 42, 45, 48, 50, 52, 55 | 6 | 48.6667 | 49 |
| Same, plus 400 | 7 | 98.8571 | 50 |
The mean doubled. The median moved by 1. That is why income and house-price reports quote medians, and why a mean far above the median is a signal of right skew. The calculator’s chart makes that gap visible: when the orange mean line sits well to the right of the green median line, a few large values are pulling it.
How Can the Mean, Median and Mode Tell You If Data Are Skewed?
The three averages sit in a predictable order that reveals the shape of the data. In a symmetric distribution they coincide. In right-skewed data a long tail of large values drags the mean up, so the order is usually mode < median < mean. In left-skewed data the order reverses.
A rough numerical check is Pearson’s second skewness coefficient, 3 × (mean − median) / standard deviation. For the seven salaries including the executive, the mean is 98.8571, the median is 50 and the sample standard deviation is 132.8614, giving 3 × 48.8571 / 132.8614 = 1.10. A value above about 1 signals strong right skew. For the nine wait times in Example 1, the mean is 14.7778, the median is 15 and the standard deviation is 2.8186, giving −0.24, which is close to symmetric.
These rules are guides, not laws. Small samples and multimodal data can break the usual order, so the chart in the calculator is the better check: look at the shape, then decide which average to report.
How Do You Find the Mean, Median and Mode From a Frequency Table?
When data are already tallied, multiply each value by its frequency instead of retyping every observation. A survey of 38 people asked how many siblings they have:
| Siblings (x) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| People (f) | 8 | 14 | 9 | 4 | 2 | 1 |
| Cumulative | 8 | 22 | 31 | 35 | 37 | 38 |
The cumulative row is the key to the median. Find the first value whose cumulative count reaches half the total. If the half-way point falls exactly between two groups, average the two values on either side.
How Do You Estimate the Mean, Median and Mode of Grouped Data?
When you only know how many values fall in each class, you cannot recover the exact values, so every result is an estimate. The standard method uses the class midpoint as a stand-in for every value in the class.
Exam scores for 50 students: 40–50 (4 students), 50–60 (9), 60–70 (15), 70–80 (12), 80–90 (7) and 90–100 (3).
| Class | Midpoint | f | f × midpoint | Cumulative f |
|---|---|---|---|---|
| 40–50 | 45 | 4 | 180 | 4 |
| 50–60 | 55 | 9 | 495 | 13 |
| 60–70 | 65 | 15 | 975 | 28 |
| 70–80 | 75 | 12 | 900 | 40 |
| 80–90 | 85 | 7 | 595 | 47 |
| 90–100 | 95 | 3 | 285 | 50 |
In the median formula, L is the lower limit of the median class (the one containing the N/2 = 25th value), CF is the cumulative frequency before it, f is its frequency and w is its width. In the mode formula, f₁ is the modal class frequency and f₀ and f₂ are the frequencies of the classes just before and after it.
The mode formula assumes classes of equal width, and the calculator flags it when your widths differ. Estimates from grouped data are only as good as the grouping: narrower classes get closer to the true values.
What Are the Geometric and Harmonic Means?
The arithmetic mean is not always the right average. Two other means appear in practice, and this calculator reports both for lists of positive numbers.
The geometric mean is the nth root of the product, and it is the correct average for growth factors. An investment returns +10%, +20% and −10% in three years, so the yearly growth factors are 1.10, 1.20 and 0.90. The arithmetic mean of the returns is 6.67%, but the money actually grew by a factor of 1.10 × 1.20 × 0.90 = 1.188 over three years. The geometric mean factor is 1.1881/3 = 1.0591, a steady 5.91% per year. The arithmetic mean overstates the result.
The harmonic mean is the correct average for rates over equal distances. Drive to a town at 60 mph and return at 40 mph. The average speed is not 50 mph. It is the harmonic mean, 2 / (1/60 + 1/40) = 48 mph, because you spend more time at the slower speed.
For any list of positive numbers that are not all equal, the harmonic mean is smaller than the geometric mean, which is smaller than the arithmetic mean.
How Do the Mean and Median Relate to Spread?
An average alone can mislead. Two classes can both average 75 on a quiz while one is tightly bunched and the other scattered from 50 to 100. After finding the center, check the spread with the standard deviation calculator or the variance calculator. If your data have outliers, the five number summary calculator reports quartiles and flags extreme values, which suits the median better than the standard deviation does. The mean absolute deviation calculator gives an easier-to-read average distance from the center.
To compare spread across data sets with different units or very different averages, divide by the mean: the coefficient of variation calculator does exactly that.
If some observations should count more than others, as with course grades weighted by credit hours, a plain mean is wrong. Use the weighted mean calculator instead.
Why Do Excel, R and Python Report Different Modes?
The mean and median are standard everywhere, but the mode is where tools disagree, because many do not define it for ties or for data without repeats.
Excel / Sheets =AVERAGE(A2:A10) =MEDIAN(A2:A10) =MODE.SNGL(A2:A10) (first mode only)
=MODE.MULT(A2:A10) returns every mode (array formula)
Python statistics.mean(x) statistics.median(x) statistics.multimode(x)
NumPy / SciPy np.mean(x) np.median(x) scipy.stats.mode(x) # one mode, the smallest if tied
pandas s.mean() s.median() s.mode() # returns all modes, sorted
R mean(x) median(x) # no built-in mode functionFor the ten values in Example 2, scipy.stats.mode returns 2, statistics.multimode returns 8, 5 and 2 (in order of first appearance), and Excel’s MODE.SNGL returns 8. All are valid, since 2, 5 and 8 are tied. Excel also returns #N/A when no value repeats, which this calculator reports as “No mode”. Python’s statistics.multimode returns every value in that case.
R’s built-in mode() reports the storage type of an object, not the statistical mode, which trips up many students. The “Same calculation in other tools” box in the calculator prints working code for your data.
What Mistakes Cause a Wrong Mean, Median or Mode?
- Forgetting to sort before finding the median. The median is the middle of the sorted list, not the middle of the list as typed.
- Using the wrong middle for an even count. With an even number of values there are two middle values, and the median is their average.
- Reporting one mode when there are several. Ties for the highest count make every tied value a mode.
- Inventing a mode when nothing repeats. If every value occurs once, the correct answer is “no mode”.
- Ignoring frequencies. In a frequency table, divide by the total count Σf, not the number of rows.
- Using class limits instead of midpoints when estimating the mean of grouped data.
- Averaging averages. The mean of two group means is the overall mean only if the groups are the same size. Otherwise weight by group size.
- Applying the mean to skewed data. A few large values can drag it far from where most of the data sit.
How Do the Mean, Median and Mode Compare?
| Measure | Works on | Affected by outliers? | Best used when |
|---|---|---|---|
| Mean | Numbers | Yes, strongly | Roughly symmetric data; further calculations |
| Median | Numbers, ranked data | No | Skewed data, incomes, prices |
| Mode | Numbers and categories | No | Most common category or size |
| Midrange | Numbers | Extremely | A quick rough center |
| Geometric mean | Positive numbers | Moderately | Growth rates, ratios |
| Harmonic mean | Positive numbers | Pulled toward small values | Rates, speeds, prices per unit |
In a perfectly symmetric single-peaked distribution such as the normal curve in the normal curve calculator, the mean, median and mode coincide. In right-skewed data the usual order is mode < median < mean, and in left-skewed data it reverses. To see every value, and where the mean and median fall among them, plot the data with the strip chart maker. Browse more spread and center tools in the full list of free statistics calculators.
Frequently Asked Questions
How do you calculate the mean, median and mode?
Add the values and divide by how many there are for the mean. Sort the values and take the middle one (or the average of the middle two) for the median. Find the most frequent value for the mode. For 12, 15, 11, 18, 15, 20, 14, 15 and 13 the mean is 133/9 = 14.7778, the median is 15 and the mode is 15.
What is the difference between mean and median?
The mean is the sum divided by the count and is pulled by extreme values. The median is the middle of the sorted data and ignores how extreme the ends are. For salaries of 42, 45, 48, 50, 52 and 55 thousand dollars they are 48.67 and 49, but adding a salary of 400 raises the mean to 98.86 and the median only to 50.
Can a data set have more than one mode, or no mode?
Yes to both. Values that tie for the highest frequency are all modes, so 4, 8, 6, 5, 3, 2, 8, 9, 2, 5 has modes 2, 5 and 8. If every value appears only once, there is no mode.
How do you find the median of an even number of values?
Sort the list and average the two middle values. For 2, 2, 3, 4, 5, 5, 6, 8, 8, 9 the middle values are 5 and 5, so the median is 5.
How do you find the mean from a frequency table?
Multiply each value by its frequency, add the products, and divide by the total frequency. For the siblings table, 57 / 38 = 1.5.
How do you estimate the mean of grouped data?
Use the midpoint of each class as the value for that class, multiply by the class frequency, add the products and divide by the total frequency. It is an estimate because the exact values inside each class are unknown.
What is the range, and is it an average?
The range is the largest value minus the smallest. It measures spread, not center, so it is not an average, but it is usually reported alongside the mean, median and mode.
When should I use the geometric mean instead of the arithmetic mean?
Use the geometric mean for quantities that multiply, such as growth factors or compound returns. Yearly returns of +10%, +20% and −10% average 6.67% arithmetically but 5.91% geometrically, and only the geometric figure matches the actual compounded growth.
Sources: NIST/SEMATECH e-Handbook of Statistical Methods, section on measures of location; NumPy documentation for numpy.mean and numpy.median; SciPy documentation for scipy.stats.mode, gmean and hmean; Python documentation for the statistics module; Microsoft documentation for AVERAGE, MEDIAN, MODE.SNGL and MODE.MULT.
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