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Mean, Median, Mode, and Standard Deviation Explained
Published May 15, 2026
Descriptive statistics summarize a dataset with a handful of numbers that capture its center, spread, and shape. Whether you are analyzing test scores, sales figures, or scientific measurements, these six values tell most of the story.
The three averages: mean, median, and mode
All three describe the "center" of a dataset, but they measure it differently and lead to very different answers on skewed data.
Mean (arithmetic average)
Mean = Sum of all values / Count of values
Example: {4, 8, 6, 5, 3, 2, 8, 9, 2, 5}
- Sum = 52, Count = 10
- Mean = 5.2
The mean is the most commonly used average and works well when data is roughly symmetric. It is heavily influenced by outliers — a single extreme value can pull it far from the typical value.
Median (middle value)
Sort the data, then pick the middle value. For an even count, average the two middle values.
Example (same data sorted): 2, 2, 3, 4, 5, 5, 6, 8, 8, 9
- 10 values → middle is between positions 5 and 6: (5 + 5) / 2 = 5.0
The median is resistant to outliers. Income distributions almost always use median because a few billionaires make the mean far higher than what a typical person earns. "Median household income" is a more honest figure than "mean household income."
Mode (most frequent value)
The mode is the value (or values) that appears most often.
Example: 2, 2, 3, 4, 5, 5, 6, 8, 8, 9
- Both 2, 5, and 8 appear twice — this is a trimodal dataset.
If all values are unique, there is no mode. If one value dominates, the distribution is unimodal. Modes are especially useful for categorical data (e.g., the most common shoe size or the top-selling product SKU).
Comparing mean, median, and mode
| Scenario | Best average to use |
|---|---|
| Symmetric distribution, no outliers | Mean |
| Skewed distribution or outliers present | Median |
| Categorical data / "most popular" question | Mode |
| Describing exam grades | All three together |
A quick rule of thumb: if the mean and median are far apart, your data is skewed.
Spread: range, variance, and standard deviation
Knowing the center is only half the picture. Spread tells you how tightly the values cluster.
Range
Range = Maximum − Minimum
Simple but fragile — one extreme outlier inflates it dramatically.
Variance
Variance (sample) = Σ(x − x̄)² / (n − 1)
Variance (population) = Σ(x − x̄)² / n
Variance measures the average squared deviation from the mean. Squaring amplifies large deviations, making variance sensitive to outliers. The result is in squared units (e.g., dollars²), which is hard to interpret directly.
Why n−1 for sample variance? Dividing by n−1 (Bessel's correction) gives an unbiased estimate of the true population variance when you only have a sample. Use n only when your data is the entire population.
Standard deviation (SD)
SD = √Variance
Standard deviation is the square root of variance — it brings the unit back to the original scale (e.g., dollars, not dollars²). It is the most widely used measure of spread.
Interpreting SD with the empirical rule (for normally distributed data):
| Range | Approximate % of data |
|---|---|
| Mean ± 1 SD | ~68% |
| Mean ± 2 SD | ~95% |
| Mean ± 3 SD | ~99.7% |
Example: Test scores with mean = 75 and SD = 10.
- 68% of students scored between 65 and 85.
- 95% scored between 55 and 95.
Classic worked example
Dataset: 2, 4, 4, 4, 5, 5, 7, 9 (from Wikipedia)
| Statistic | Value |
|---|---|
| Count | 8 |
| Mean | 5 |
| Median | 4.5 |
| Mode | 4 (appears 3×) |
| Min / Max | 2 / 9 |
| Range | 7 |
| Population variance | 4 |
| Population SD | 2 |
This is a clean example because the population SD comes out to exactly 2, making it easy to verify manually.
Coefficient of variation (CV)
To compare spread across datasets with different scales:
CV = (Standard Deviation / Mean) × 100%
A dataset with mean 100 and SD 10 has a CV of 10%. A dataset with mean 10 and SD 10 has a CV of 100% — far more variable relative to its scale.
Quickly compute descriptive statistics
The Descriptive Statistics Calculator lets you paste in any list of numbers and instantly see count, mean, median, mode, min, max, range, variance, and standard deviation — with a toggle for sample vs population variance.