Mean Calculator
Calculate mean, median, mode, range, and standard deviation from any list of numbers.
Reviewed & updated for 2026 by Rakesh Choudhary, PhDMethodology & sources · How we calculate
How each statistic is calculated
Five descriptive statistics tell you everything about a dataset's center and spread:
- Mean (arithmetic average): Sum of all values divided by count. Formula: Σx ÷ n. Sensitive to outliers, a single billionaire in a small sample dramatically inflates the mean. Best for symmetric, normally distributed data.
- Median: The middle value when sorted ascending. With odd count: the literal middle entry. With even count: average of the two middle entries. Resistant to outliers, used for income, home prices, and salaries because the highest earners pull the mean upward.
- Mode: The most frequent value. Can be multiple modes (bimodal) or none (every value unique). Most useful for categorical data: which shoe size sells most, which color is most popular.
- Range: Maximum minus minimum. Simple but only uses two data points. A poor measure of spread when outliers exist.
- Standard deviation: A typical distance of each value from the mean. The calculator's main figure is the population formula, √(Σ(x − mean)² / n); the sample figure below it divides by n − 1 instead, the definition in NIST's e-Handbook of Statistical Methods. Used in finance (volatility), quality control, and any field measuring consistency.
Worked example: test scores
Class test scores: 65, 72, 78, 78, 85, 88, 91, 95, 100.
- Count = 9, Sum = 752
- Mean = 752 ÷ 9 = 83.6
- Median = middle value (5th of 9) = 85
- Mode = 78 (appears twice)
- Range = 100 − 65 = 35
- Population standard deviation ≈ 10.6 (sample: 11.3). 5 of the 9 scores (78, 78, 85, 88, 91) lie within one population SD of the mean, 72.9 to 94.2.
Mean vs median: which to use?
For most fair representation of a "typical" value, use median when:
- Income / salary data: A few high earners pull the mean well above the median, so the median is closer to what a typical household earns.
- Home prices: Same reason, luxury sales inflate mean. Realtors report median.
- Wealth / net worth: Wealth is even more concentrated than income, so a mean net worth sits far above the median and says little about a typical family.
- Survey response times: A few slow responses (or system outages) skew mean. Median is fairer.
Use mean when data is symmetric, sample is large, and there are no extreme outliers, like measuring product weights from a factory line, or weather temperatures over a month.
Sample vs population statistics
A subtle but important distinction in statistics: are you measuring the entire population or just a sample? The formulas for mean, median, and mode are identical either way. Standard deviation and variance are not.
When calculating standard deviation for a population, you divide by N (the total count). When calculating it for a sample, you divide by N−1. This is called Bessel's correction and it makes the sample standard deviation an unbiased estimate of the true population standard deviation.
| Scenario | Use population SD (÷N) | Use sample SD (÷N−1) |
|---|---|---|
| Test scores | Every student in your class | 25 students sampled from the school |
| Stock returns | All historical days you care about | 100 days sampled to estimate volatility |
| Survey results | Census data, every responder | 1,000 surveys representing a city |
In Excel, the two functions are =STDEV.P() (population) and =STDEV.S() (sample). Pick wrong and your SD will be slightly off, which matters only when N is small: the sample SD is always √(N ÷ (N − 1)) times the population SD, so it is 11.8% larger with 5 points and 1.7% larger with 30.
FAQs
What is the mean?
The mean is the arithmetic average: add every value, then divide by how many there are (Σx / n). For 5, 7 and 9 that is 21 / 3 = 7. In everyday speech it is simply called the average.
Mean vs median vs mode?
Mean: arithmetic average (sum / count). Median: middle value when sorted. Mode: most frequent value. Example for [1, 2, 3, 3, 7]: Mean = 16/5 = 3.2. Median = 3. Mode = 3. Median is preferred when there are outliers, which is why income statistics are usually reported as medians.
What is standard deviation?
Standard deviation measures how spread out the values are around the mean. Divide the summed squared deviations by n for a whole population, or by n − 1 for a sample (NIST's definition), then take the square root. Larger SD = more spread. Smaller SD = clustered near mean.
When should I use mean vs median?
Use MEAN for: symmetric data, when every value matters equally. Use MEDIAN for: data with outliers (income, home prices), skewed distributions, when you want a 'typical' value resistant to extremes. Use MODE for: categorical data, finding most common.
What's the range?
Range = Maximum − Minimum. The simplest measure of spread. Example: numbers 2, 4, 7, 10. Range = 10 − 2 = 8. Useful for quick spread estimate but sensitive to outliers.