Math
Statistics Calculator: Mean, Median and Mode
Paste your data and get the mean, the mode and the standard deviation
Paste your data into this statistics calculator and get the mean, median and mode in seconds, along with the range, the variance and the population and sample standard deviation.
Population variance: 3.938 · Sample variance: 4.500
Population standard deviation: 1.984 · Sample standard deviation: 2.121Five-number summary: minimum 3 · Q1 4.5 · median 6.5 · Q3 8 · maximum 9Interquartile range (IQR): 3.5 · Outliers (beyond 1.5 × IQR, below -0.75 or above 13.25): noneSum: 50 · Standard error of the mean: 0.750 · Coefficient of variation: 33.9%Sorted: 3, 4, 5, 6, 7, 8, 8, 9
| Value | Frequency | Relative | Cumulative |
|---|---|---|---|
| 3 | 1 | 12.5% | 1 |
| 4 | 1 | 12.5% | 2 |
| 5 | 1 | 12.5% | 3 |
| 6 | 1 | 12.5% | 4 |
| 7 | 1 | 12.5% | 5 |
| 8 | 2 | 25.0% | 7 |
| 9 | 1 | 12.5% | 8 |
The key idea
The mean is not the typical value: it is a summary that extremes pull around
With 4, 8, 6, 5, 3, 8, 9 and 7, the mean is 6.25 and the median is 6.5: almost the same. If you change the 9 to 90, the mean jumps to 16.375 and the median stays at 6.5. That is why you should always look at both, and at the standard deviation too.
How it is calculated
Mean = Σx / n Variance = Σ(x − mean)² / n (sample: / (n − 1)) Standard deviation = √Variance
- Mean: the sum of the values divided by how many there are.
- Median: the middle value once the data is sorted (or the average of the two middle values).
- Mode: the value that appears most often.
- Variance: the average of the squared distances from the mean. The sample variance divides by n − 1 instead of n.
- Standard deviation: the square root of the variance, in the same units as the data.
- Quartiles: Q1 is the median of the lower half of the sorted data and Q3 the median of the upper half, leaving out the overall median when there is an odd number of values. It is the method most textbooks and graphing calculators use; spreadsheets like Excel may interpolate and give slightly different values.
- Outliers: values more than 1.5 times the interquartile range (Q3 − Q1) below Q1 or above Q3.
Worked example
Data: 4, 8, 6, 5, 3, 8, 9, 7 (n = 8). Sum = 50, mean = 6.25. Sorted: 3, 4, 5, 6, 7, 8, 8, 9 → median = (6 + 7) / 2 = 6.5. Mode = 8. Population standard deviation = 1.984; sample standard deviation = 2.121.
Common questions
Frequently asked questions
Should I use the population or the sample standard deviation?
Use the population one if you have every value of the group you are studying, and the sample one if your data is a sample from a larger group. The sample version divides by n − 1 and comes out a bit larger: with the example data, 2.121 versus 1.984.
Mean or median?
The median, if there are extreme values; the mean, if the data is balanced. For incomes or home prices, the median usually reflects the typical case better.
How do I find the median with an even number of values?
Sort the values and average the two in the middle. With 3, 4, 5, 6, 7, 8, 8, 9, the middle values are 6 and 7, and the median is 6.5.
What if there are two modes or none?
If two or more values tie for the most repeats, they are all modes and the calculator shows all of them. If no value repeats, there is no mode.
What is the difference between variance and standard deviation?
The standard deviation is the square root of the variance. It is used more often because it is in the same units as the data: if you measure in dollars, the variance is in dollars squared and the standard deviation is in dollars.
How do I enter decimals and large numbers?
Use a period for decimals, for example 4.5. In a list on one line, commas separate values, so write 1500 rather than 1,500. If you paste a column with one value per line, thousands separators like 1,500 are read correctly.
Equations, statistics and probability
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