# Mean, median and mode calculator (average calculator)

> Average calculator for a list of numbers: mean, median, mode, range, count and sum, with the sorted data and the middle values highlighted.

Interactive version: https://www.calcopenly.com/math/mean-median-mode-calculator
Subject: Math calculators

The mean is the sum of the values divided by how many there are. The median is the middle value once the data are sorted: the value at position (n + 1) ÷ 2 when n is odd, and the average of the two middle values when n is even. The mode is the value that occurs most often, and the range is the largest value minus the smallest.

For the 10 default quiz scores the sum is 849, so the mean is 84.9. Sorted, the 5th and 6th values are 85 and 88, which puts the median at 86.5; 92 appears three times and is the mode, and the range is 95 − 70 = 25.

Data can have several modes, all listed here, or none: when every value occurs equally often, no value is more typical than another. Standard deviation, quartiles and outliers are on the descriptive statistics calculator.

## Inputs

- **Data**: Numbers separated by commas, spaces or new lines. The default is 10 quiz scores.

## Results

- Mean (average) — main result
- Median
- Mode
- Range
- Count
- Sum
- Minimum
- Maximum
- Mean as a fraction

## Formula

$$
\bar x = \frac{1}{n}\sum_{i=1}^{n} x_i \qquad \tilde x = \begin{cases} x_{((n+1)/2)} & n \text{ odd} \\ \tfrac12\left(x_{(n/2)} + x_{(n/2+1)}\right) & n \text{ even} \end{cases}
$$

## Worked examples

### Quiz scores (default)

- Data: 85, 92, 78, 92, 88, 76, 95, 81, 92, 70
- **Mean (average): 84.9**
- **Median: 86.5**
- **Mode: 92 (appears 3 times)**
- **Range: 25**
- **Count: 10**
- **Sum: 849**
- **Minimum: 70**
- **Maximum: 95**
- **Mean as a fraction: 849/10 = 84 9/10**
- Checked against: Python 3.8 statistics.mean = 84.9, statistics.median = 86.5, statistics.multimode = [92]; sum 849

### calculator.net example, two modes

- Data: 10, 2, 38, 23, 38, 23, 21
- **Mean (average): 22.143**
- **Median: 23**
- **Mode: 23 and 38 (each appears twice)**
- **Range: 36**
- **Sum: 155**
- **Mean as a fraction: 155/7 = 22 1/7**
- Checked against: calculator.net mean, median, mode, range calculator worked example: mean 22.143, median 23, modes 23 and 38, range 36

### Calculator Soup example, 16 values

- Data: 9, 10, 12, 13, 13, 13, 15, 15, 16, 16, 18, 22, 23, 24, 24, 25
- **Mean (average): 16.75**
- **Median: 15.5**
- **Mode: 13 (appears 3 times)**
- **Range: 16**
- **Count: 16**
- **Sum: 268**
- **Minimum: 9**
- **Maximum: 25**
- Checked against: Calculator Soup mean, median, mode calculator example output: mean 16.75, median 15.5, mode 13, range 16, count 16, sum 268

### NIST StRD NumAcc1

- Data: 10000001, 10000003, 10000002
- **Mean (average): 10,000,002**
- **Median: 10,000,002**
- **Mode: No mode (every value appears once)**
- **Range: 2**
- **Mean as a fraction: not applicable**
- Checked against: NIST StRD NumAcc1 certified mean 10000002; median and range by hand from the sorted values 10000001, 10000002, 10000003

### Every value twice: no mode (edge case)

- Data: 1, 1, 2, 2
- **Mean (average): 1.5**
- **Median: 1.5**
- **Mode: No mode (every value appears twice)**
- **Range: 1**
- Checked against: Calculator Soup convention: if all numbers occur the same number of times there is no mode; mean 6/4 and median (1 + 2)/2 by hand

### Single value

- Data: -4.5
- **Mean (average): -4.5**
- **Median: -4.5**
- **Mode: −4.5 (appears once)**
- **Range: 0**
- **Count: 1**
- Checked against: Definitions (NIST e-Handbook §1.3.5.1): with one value the mean, median and mode equal it and the range is 0

## Questions

### How do you find the mean, median and mode?

Add the values and divide by the count for the mean: 10, 2, 38, 23, 38, 23, 21 sum to 155, and 155 ÷ 7 = 22.14. Sort them (2, 10, 21, 23, 23, 38, 38) and take the middle one, the 4th, for the median: 23. Count repeats for the mode: 23 and 38 each appear twice, so both are modes. The range is 38 − 2 = 36.

### How do you find the median of an even number of values?

Sort the values and average the two in the middle, at positions n ÷ 2 and n ÷ 2 + 1. For the 16 values 9, 10, 12, 13, 13, 13, 15, 15, 16, 16, 18, 22, 23, 24, 24, 25, the 8th and 9th are 15 and 16, so the median is 15.5. This is the definition in the NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.1.

### Can a data set have more than one mode or no mode?

Yes to both. NIST notes that the mode is not necessarily unique: 2, 3, 3, 5, 5, 8 has two modes, 3 and 5, and is called bimodal. When every value appears the same number of times, as in 4, 7, 9 or 1, 1, 2, 2, no value occurs more often than the others and the data have no mode.

### Which average should I use: mean, median or mode?

Use the mean for roughly symmetric data without extreme values, the median when a few values are far from the rest, and the mode for categories or the most common size. One outlier moves the mean but not the median: for 3, 4, 5 the mean and median are both 4, but changing 5 to 50 lifts the mean to 19 while the median stays 4.

### What is the difference between average and mean?

In everyday use "average" means the arithmetic mean, the sum divided by the count, and spreadsheet functions follow that: Excel's AVERAGE returns the mean. In statistics "average" is a looser word for any measure of center, so the median and mode are also called averages. When a report says "average" without saying which, it is almost always the mean.

### How accurate is the mean, median and mode calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 6 worked examples whose answers come from independent sources; for example, “Quiz scores (default)” is checked against Python 3.8 statistics.mean = 84.9, statistics.median = 86.5, statistics.multimode = [92]; sum 849.

### Where does the method come from?

NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.1 Measures of location (mean, median, mode); NIST Statistical Reference Datasets: univariate summary statistics (NumAcc1); Python statistics module: mean, median, multimode.

## Sources

- [NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.1 Measures of location (mean, median, mode)](https://www.itl.nist.gov/div898/handbook/eda/section3/eda351.htm)
- [NIST Statistical Reference Datasets: univariate summary statistics (NumAcc1)](https://www.itl.nist.gov/div898/strd/univ/homepage.html)
- [Python statistics module: mean, median, multimode](https://docs.python.org/3/library/statistics.html)