Mean, Median, Mode & More
Last reviewed: May 2026
A statistics calculator computes essential descriptive statistics: central tendency (mean, median, mode), spread (range, variance, standard deviation), and distribution properties. These transform raw data into interpretable summaries — what is typical, how much variation exists, and whether data is symmetric or skewed.1
| Measure | Best For | Weakness |
|---|---|---|
| Mean | Symmetric data | Sensitive to outliers |
| Median | Skewed data | Ignores actual values |
| Mode | Categorical data | May not be unique |
| Std Dev | Spread around mean | Assumes ~normal distribution |
| IQR | Skewed spread | Ignores tails |
Statistics transforms raw data into actionable insight through three categories of measures. Central tendency (mean, median, mode) identifies where the data clusters. Dispersion (range, variance, standard deviation, IQR) describes how spread out the values are. Shape (skewness, kurtosis) characterizes the distribution's symmetry and tail behavior. Together, these measures paint a complete picture: a dataset with a mean of 100, standard deviation of 5, and zero skewness describes a tight, symmetric cluster. A dataset with the same mean but a standard deviation of 30 and positive skewness describes a wide, right-trailing distribution. This calculator computes all key measures instantly, helping you understand your data's story at a glance.
| Measure | What It Tells You | Formula/Method | Sensitive to Outliers? |
|---|---|---|---|
| Mean | Center of the data (balance point) | Sum ÷ count | Yes — highly |
| Median | Middle value (50th percentile) | Middle of sorted values | No — robust |
| Mode | Most frequent value | Highest frequency count | No |
| Range | Total spread | Max − min | Yes — extremely |
| Variance | Average squared deviation from mean | Σ(x−μ)² ÷ n (or n−1) | Yes — highly |
| Standard deviation | Typical distance from mean | √variance | Yes — highly |
| IQR | Spread of middle 50% | Q3 − Q1 | No — robust |
| Skewness | Asymmetry direction | Third standardized moment | Somewhat |
Robust measures (median, IQR) are preferred for exploratory data analysis because they faithfully represent the bulk of the data regardless of extreme values. Sensitive measures (mean, standard deviation) are used in inferential statistics because their mathematical properties enable hypothesis testing and confidence interval construction — but only when the underlying assumptions (typically normality) are reasonably met.
Standard deviation quantifies variability in the same units as the data, making it directly interpretable. A class scoring a mean of 80 with a standard deviation of 5 has most students between 75 and 85 — a tightly clustered, consistent performance. The same mean with a standard deviation of 15 means students range from 65 to 95 — wide variation suggesting diverse preparation levels. In manufacturing, a production line with mean weight 500g and standard deviation 2g produces very consistent output, while a standard deviation of 10g may indicate process control issues. The coefficient of variation (CV = standard deviation ÷ mean × 100%) allows comparison of variability across datasets with different scales — a CV of 10% indicates the same relative consistency whether the data measures milligrams or tons.
A critical distinction: population parameters describe the entire group of interest; sample statistics estimate those parameters from a subset. When you measure every item (all students in a class, all products in a batch), you have population data and divide by n for variance. When you measure a sample (100 voters from a state, 30 products from a shipment), you divide by n−1 for variance (Bessel's correction), which compensates for the tendency of samples to underestimate population variability. This calculator lets you select either approach. In practice, nearly all real-world data is sample data — you rarely measure every individual in a population. The n−1 correction matters most for small samples (under 30) and becomes negligible for large ones.
Percentiles divide data into 100 equal parts, indicating the percentage of values falling below a given point. The 75th percentile means 75% of observations are lower. Quartiles are specific percentiles: Q1 (25th percentile), Q2/median (50th), and Q3 (75th). The interquartile range (IQR = Q3 − Q1) measures the spread of the middle 50% of data and is the basis for boxplot construction and outlier detection. Values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR are flagged as outliers. For income data with Q1 = $35,000 and Q3 = $85,000 (IQR = $50,000), outliers begin above $85,000 + $75,000 = $160,000 and below $35,000 − $75,000 = −$40,000 (only the upper bound matters here). See our Standard Deviation Calculator for focused variability analysis.
| Distribution | Shape | Real-World Examples | Key Property |
|---|---|---|---|
| Normal (Gaussian) | Symmetric bell curve | Heights, test scores, measurement error | 68-95-99.7 rule |
| Right-skewed | Long right tail | Income, wealth, city population | Mean > median |
| Left-skewed | Long left tail | Age at retirement, exam scores (easy test) | Mean < median |
| Uniform | Flat / equal probability | Dice rolls, random number generators | All values equally likely |
| Bimodal | Two peaks | Mixed populations, exam with two groups | Two distinct clusters |
Identifying your data's distribution shape is the first step in choosing the right analytical approach. Normal distributions support z-tests, t-tests, and ANOVA directly. Skewed distributions may need transformation (logarithmic, square root) or non-parametric methods. Bimodal distributions suggest the data contains two distinct subpopulations that should be analyzed separately.
Statistical significance answers the question: "Could this result have occurred by chance?" The p-value is the probability of observing a result as extreme as yours if the null hypothesis (typically "no effect") were true. A p-value of 0.03 means there is a 3% probability of seeing this result by chance — conventionally considered "statistically significant" at the standard α = 0.05 threshold. However, statistical significance does not equal practical importance. A study of 10,000 people might find a statistically significant 0.5-point blood pressure difference between two drugs (p = 0.001) that is too small to be clinically meaningful. Always consider effect size (how big the difference is) alongside significance (how confident you are that the difference is real).
Correlation measures the strength and direction of a linear relationship between two variables, ranging from −1.0 (perfect negative) through 0 (no linear relationship) to +1.0 (perfect positive). Ice cream sales and drowning deaths are positively correlated (both increase in summer), but ice cream does not cause drowning — temperature is the confounding variable driving both. Establishing causation requires controlled experiments where one variable is manipulated while others are held constant, or sophisticated observational study designs that account for confounders. The correlation coefficient r captures only linear relationships — two variables can have a perfect non-linear relationship (like a parabola) with r near zero. Always plot your data before interpreting correlation statistics to verify the relationship type.
Start every analysis by plotting the data — histograms for single variables, scatterplots for relationships. Plots reveal outliers, skewness, clusters, and patterns that no single summary statistic captures. Report both central tendency and dispersion: "average salary is $65,000" is incomplete without "with a standard deviation of $20,000" or "ranging from $35,000 to $150,000." Use the median for skewed data and the mean for symmetric data. Check for outliers before computing mean-based statistics. Compare groups using both visual comparisons (side-by-side boxplots) and statistical tests (t-tests for means, Mann-Whitney for medians). Round final results appropriately — reporting a mean of 78.23456 implies false precision if your measurements are only accurate to whole numbers.
The reliability of any statistical analysis depends on how the sample was collected and its size. Random sampling — where every member of the population has an equal chance of selection — is the gold standard because it minimizes systematic bias. Convenience sampling (surveying whoever is easiest to reach) and voluntary response sampling (letting people opt in) produce biased results that may not represent the population. For sample size, the general rule is: larger samples produce more precise estimates. Doubling the sample size reduces the margin of error by a factor of √2 (approximately 30%). To achieve a ±3% margin of error at 95% confidence for a proportion near 50%, you need approximately 1,067 respondents. For ±1%, you need roughly 9,604 respondents. Most national polls use 1,000–2,000 respondents, yielding margins of ±2–3% — sufficient for most purposes. Use our Percentage Calculator for quick proportion calculations in survey analysis.
The five-number summary — minimum, Q1, median, Q3, maximum — provides a comprehensive snapshot of any distribution in just five values. It captures center (median), spread (IQR = Q3 − Q1), range (max − min), and skewness (comparing distances from median to Q1 versus Q3). Boxplots visualize this summary: the box spans Q1 to Q3 with a line at the median, whiskers extend to the farthest non-outlier points, and outliers appear as individual dots. Side-by-side boxplots are one of the most effective tools for comparing groups — you can instantly see whether two distributions overlap, which has more variability, and where outliers fall. For datasets with 20 or more values, the five-number summary paired with a boxplot communicates more useful information than any single average or variability measure alone.
Effect size measures the magnitude of a difference or relationship independent of sample size. Cohen's d compares group means: d = (mean₁ − mean₂) / pooled standard deviation. Values of 0.2 are considered small, 0.5 medium, and 0.8 large. A drug that lowers cholesterol by 2 mg/dL (d = 0.1) may reach statistical significance with 10,000 participants but has negligible clinical impact. A drug lowering cholesterol by 30 mg/dL (d = 0.8) represents a meaningful health improvement. For correlations, r of 0.1 is small, 0.3 is medium, and 0.5 is large. Reporting effect sizes alongside p-values provides the complete picture: statistical significance tells you whether the effect exists, effect size tells you whether it matters.
→ Check for outliers first. They affect mean and SD disproportionately.
→ Use median for skewed data. Income, home prices, and wait times are almost always skewed.
→ Report both mean and SD. Mean alone is incomplete — SD shows how reliable it is.
→ Visualize your data. Numbers without context mislead. Plot first, then calculate.
See also: Standard Deviation · Z-Score · Percentage · Scientific Calculator