Statistics Guide: Mean, Median, Mode, Variance & More

Statistics is the science of collecting, organizing, analyzing, and interpreting numerical data. Descriptive statistics — the focus of this guide — summarize what a dataset looks like: its center, its spread, its shape, and its extremes.

Whether you are interpreting test scores, analyzing business data, evaluating experimental results, or simply making sense of numbers in the news, understanding the core statistical measures gives you the tools to draw accurate conclusions. This guide covers every key descriptive statistic with formulas, worked examples, and guidance on when to use each measure.

Central tendency describes where the "center" of a dataset falls. Three measures answer this in different ways.

Mean (arithmetic average): sum all values and divide by the count.

Formula: μ = (Σx) ÷ n

• Dataset: 4, 7, 13, 2, 9
• Sum: 4 + 7 + 13 + 2 + 9 = 35
• Count: 5
• Mean: 35 ÷ 5 = 7

The mean uses every data point, making it sensitive to outliers. One very large or very small value pulls the mean away from the typical.

Median: the middle value when data is sorted in order. If there is an even count of values, the median is the average of the two middle values.

• Dataset: 4, 7, 13, 2, 9 → Sorted: 2, 4, 7, 9, 13
• Middle value (position 3 of 5): 7

• Even dataset: 3, 7, 9, 15 → Middle two: 7 and 9
• Median: (7 + 9) ÷ 2 = 8

The median is unaffected by outliers. For skewed distributions (income, house prices), the median is the more representative measure.

Mode: the value that appears most frequently. A dataset can have one mode, multiple modes, or no mode (if all values appear equally).

• Dataset: 3, 5, 5, 7, 8, 5, 2 → Mode: 5 (appears 3 times)
• Dataset: 1, 2, 3, 4 → No mode (all values appear once)
• Dataset: 2, 2, 5, 5, 9 → Bimodal: 2 and 5

The mode is the only measure of central tendency applicable to categorical data (e.g., most common color, most popular product).

Spread (also called dispersion or variability) describes how much the values in a dataset differ from each other. Two datasets can have identical means but very different spreads.

Range: the simplest measure — the difference between the maximum and minimum values.

• Dataset: 12, 7, 25, 4, 19 → Range: 25 − 4 = 21

Range is easy to compute but sensitive to outliers — one extreme value changes it dramatically.

Variance: the average of the squared deviations from the mean. Squaring eliminates negatives and emphasizes larger deviations.

Population variance formula: σ² = Σ(x − μ)² ÷ N

Sample variance formula: s² = Σ(x − x̄)² ÷ (n − 1)

• Dataset: 2, 4, 4, 4, 5, 5, 7, 9 — Mean = 5
• Deviations: −3, −1, −1, −1, 0, 0, 2, 4
• Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16
• Sum of squared deviations: 32
• Population variance: 32 ÷ 8 = 4
• Sample variance: 32 ÷ 7 ≈ 4.57

Variance is in squared units (e.g., dollars squared), which is not intuitively interpretable. Standard deviation fixes this.

Standard deviation: the square root of variance. It returns the measure of spread to the original units of the data.

• Population standard deviation (σ): √4 = 2
• Sample standard deviation (s): √4.57 ≈ 2.14

Interpretation: on average, data points in the example dataset are about 2 units away from the mean of 5.

Quartiles divide a sorted dataset into four equal parts. They describe the distribution more fully than a single center measure.

• Q1 (first quartile / 25th percentile) — 25% of values fall below this point
• Q2 (second quartile / median / 50th percentile) — 50% of values fall below
• Q3 (third quartile / 75th percentile) — 75% of values fall below

Interquartile range (IQR): IQR = Q3 − Q1

The IQR represents the spread of the middle 50% of data and is resistant to outliers, making it more robust than range.

Example: Dataset: 1, 3, 5, 7, 8, 10, 12, 14, 15, 20

• Q1 = 5, Q2 (Median) = 9, Q3 = 14
• IQR = 14 − 5 = 9

Outlier detection using the IQR method (Tukey's fences):

• Lower fence: Q1 − 1.5 × IQR
• Upper fence: Q3 + 1.5 × IQR
• Values outside these fences are flagged as potential outliers

Using the example above: Lower fence = 5 − 1.5(9) = −8.5. Upper fence = 14 + 1.5(9) = 27.5. The value 20 is within range; no outliers in this dataset.

Quartile data is visualized using a box plot (box-and-whisker plot), which shows Q1, median, Q3, and the whiskers extending to the fences.

The normal distribution is a symmetric, bell-shaped probability distribution described entirely by its mean (μ) and standard deviation (σ). It is the most important distribution in statistics because many natural phenomena approximate it, and because of the central limit theorem (sample means tend toward normal distributions regardless of the underlying population shape).

The empirical rule (68-95-99.7 rule):

• 68% of data falls within 1 standard deviation of the mean (μ ± σ)
• 95% of data falls within 2 standard deviations (μ ± 2σ)
• 99.7% of data falls within 3 standard deviations (μ ± 3σ)

Example: Adult male heights in the US are approximately normally distributed with mean 70 inches (5'10") and standard deviation 3 inches.

• 68% of men are between 67 and 73 inches tall
• 95% are between 64 and 76 inches
• 99.7% are between 61 and 79 inches

Z-scores standardize values to a common scale: Z = (x − μ) ÷ σ. A Z-score tells you how many standard deviations a value is from the mean. Z = 0 is the mean; Z = 2 is 2 standard deviations above; Z = −1.5 is 1.5 standard deviations below.

Not all data is normally distributed. Skewed distributions (income, asset prices, city populations) require different interpretive approaches. Always visualize data before assuming normality.

A critical distinction in statistics:

Population — the complete set of all individuals or observations you are studying. Population parameters (like μ for mean, σ for standard deviation) are fixed, true values.

Sample — a subset of the population used to estimate population parameters. Sample statistics (x̄, s) are estimates that carry uncertainty.

This distinction affects formulas:

• Population variance divides by N (all data points)
• Sample variance divides by n − 1 (Bessel's correction) to reduce underestimation bias

When to use population vs. sample statistics:

• Use population statistics when you have the complete dataset — all students in a class, all products in a batch, all transactions in a month.
• Use sample statistics when your data is a subset drawn to estimate characteristics of a larger group — a poll of 1,000 voters representing millions, or a quality sample from a production line.

Most real-world statistics work with samples. The Statistics Calculator computes both population and sample versions of variance and standard deviation.

Statistical literacy requires knowing which measure to use for which situation:

When the mean is misleading: income distributions are right-skewed. US median household income (~$75,000) is substantially below the mean (~$100,000+) because a small number of very high earners pull the mean up. The median better represents the typical household.

When standard deviation misses the picture: for bimodal or heavily skewed distributions, standard deviation does not describe the shape well. A dataset of [1, 1, 1, 100, 100, 100] has a mean of ~50 and a large standard deviation — but no values near the mean at all.

Comparing spread across groups: use the coefficient of variation (CV = σ ÷ μ × 100%) when comparing spread across datasets with different units or different scales. A CV of 10% means the spread is 10% of the mean, regardless of whether the units are dollars or kilograms.

Checking for normality: before applying techniques that assume a normal distribution, visualize the data with a histogram or Q-Q plot. Look for symmetric bell shape, the absence of heavy tails, and whether the mean and median are close.

Sample size matters: small samples produce unstable statistics. Standard deviation estimated from 5 data points is unreliable. As sample size grows, sample statistics converge on population parameters. A rule of thumb for basic analysis: n ≥ 30 provides reasonably stable estimates.

These tools calculate descriptive statistics for any dataset:

• Statistics Calculator — enter any dataset to get mean, median, mode, variance, standard deviation, quartiles, IQR, range, and frequency distribution
• Random Number Generator — generate random datasets to practice with or use in simulations
• Scientific Calculator — evaluate statistical formulas manually with logarithm and exponent support

Browse all Math Calculators including proportion, fraction, Pythagorean theorem, and scientific notation tools.