Descriptive Statistics Calculator

Compute mean, median, mode, standard deviation, quartiles, skewness, and kurtosis from your raw data. Paste or type your numbers separated by spaces, commas, tabs, or newlines.

Quick calculation?

Only need the standard deviation? Use the faster Standard Deviation Calculator. This page computes the full descriptive suite: central tendency, spread, shape, and the five-number summary.

Mean —
Median —
SD (sample) —
IQR —
Enter at least 2 values to begin.

Calculation steps

  1. Enter data to see working.

Summary table

Statistic Value
—

Formulas & definitions

Sample standard deviation

s = √( Σ(xᵢ − x̄)² / (n − 1) )
s
Sample standard deviation
xᵢ
Each observation
x̄
Sample mean
n
Number of observations

Quartiles (linear interpolation)

Position = 1 + (n − 1) × p, where p ∈ {0.25, 0.5, 0.75}
p
Percentile (0.25 for Q1, 0.5 for median, 0.75 for Q3)
n
Number of observations

Quartiles are computed using the linear interpolation method (same as Excel PERCENTILE.INC and NumPy default). Other software may use different conventions (e.g. QUARTILE.EXC, some TI calculators) and may report slightly different Q1 and Q3 values for the same dataset. This is normal and expected.

Sample skewness (Fisher-Pearson)

g₁ = (n / ((n−1)(n−2))) × Σ( (xᵢ − x̄) / s )³
g₁
Adjusted Fisher-Pearson skewness
n
Sample size
s
Sample standard deviation

Sample excess kurtosis

G₂ = (n(n+1) / ((n−1)(n−2)(n−3))) × Σ( (xᵢ − x̄) / s )⁴ − (3(n−1)² / ((n−2)(n−3)))
G₂
Sample excess kurtosis (adjusted for small samples)
n
Sample size
s
Sample standard deviation

Worked example

Suppose you measure 9 response times (in seconds): 12, 15, 14, 18, 16, 19, 14, 17, 15.

  1. n = 9 samples
  2. Σx = 12 + 15 + 14 + 18 + 16 + 19 + 14 + 17 + 15 = 140
  3. x̄ = 140 / 9 ≈ 15.556
  4. Sort: 12, 14, 14, 15, 15, 16, 17, 18, 19. Median = 15 (position 5, middle of 9 values).
  5. Σ(xᵢ − x̄)² ≈ 27.556
  6. s = √(27.556 / 8) ≈ 1.854
  7. Q1 (25th percentile): position = 1 + (9 − 1) × 0.25 = 3. Interpolate between positions 3 and 4 (values 14 and 15): Q1 = 14 + 0.25 × (15 − 14) = 14.25.
  8. Q3 (75th percentile): position = 1 + (9 − 1) × 0.75 = 7. Interpolate between positions 7 and 8 (values 17 and 18): Q3 = 17 + 0.25 × (18 − 17) = 17.25.
  9. IQR = Q3 − Q1 = 17.25 − 14.25 = 3.

The summary table would show n=9, mean≈15.56, median=15, sd≈1.85, Q1=14.25, Q3=17.25, IQR=3, min=12, max=19.

Assumptions & interpretation

When to use

Descriptive statistics summarize the center, spread, and shape of your sample. They make no assumptions about the population or generate a p-value. Use this tool whenever you want to describe your data in a single number or set of numbers.

Skewness interpretation: A value near 0 indicates symmetry; positive values indicate right skew (tail on the right); negative values indicate left skew. Values outside ±1 suggest strong asymmetry.

Kurtosis interpretation: Excess kurtosis compares your data to the normal distribution (which has excess kurtosis = 0). Positive values indicate heavy tails (outliers more common); negative values indicate light tails. Interpretation is most reliable with n ≥ 30.

Quartiles: This calculator uses the linear-interpolation method (PERCENTILE.INC, NumPy default, R quantile type 7). Other methods exist and may yield slightly different values for the same data; this is normal.

Sample vs. population: The standard deviation reported is the sample standard deviation (using n − 1 divisor), appropriate when your data is a sample from a larger population. If your data represents the entire population of interest, use the population SD (divisor n).

Frequently asked questions

What's the difference between sample and population standard deviation?

Sample SD uses divisor n − 1 and estimates the population SD from a sample. Population SD uses divisor n and describes a complete dataset. If you have collected a sample from a larger population, report sample SD. If your data is the entire group of interest, report population SD. This calculator computes sample SD by default because most student datasets are samples.

Why do different calculators report different quartile values?

Quartiles can be computed using several methods, and statisticians disagree on which is "best." This calculator uses linear interpolation (PERCENTILE.INC, NumPy default), which Excel, R, and most statistical software use by default. Some older methods (QUARTILE.EXC, some TI calculators) may report slightly different values. The differences are usually small and decrease as sample size grows. Choose a method and stick with it for consistency within a project.

What is mode, and why might there be no mode?

Mode is the most frequently occurring value. If all values appear equally often, there is no mode. If multiple values tie for highest frequency, all are modes (the data is multimodal). Mode is most useful for categorical data or when you want to know the "typical" value in a non-numeric sense.

What does skewness tell me?

Skewness measures asymmetry. A perfectly symmetric distribution has skewness ≈ 0. Right-skewed data (long tail on the right) has positive skewness; left-skewed data has negative skewness. Large positive or negative skewness (beyond ±1) suggests the data is not bell-shaped. Skewed data may violate assumptions of some statistical tests.

When should I use this tool instead of a t-test or ANOVA?

Use this tool to describe your data (What are the typical values? How spread out are they? Is the distribution symmetric?). Use a t-test or ANOVA to test whether two or more groups differ significantly. Descriptive statistics inform the choice of test; they stand alone when you only want to summarize.

Can I use this calculator for categorical data?

No. Mean, SD, median, and quartiles require numeric data. You can compute mode for categorical data (e.g., "the most common color is blue"), but use this tool only for truly numeric measurements (heights, times, scores, counts).

See also

Paired t-Test

Test whether two paired samples differ significantly.

Open →

Effect Size Calculator

Compute Cohen's d and other effect sizes from group statistics.

Open →

Standard Deviation Calculator

Fast SD and variance calculator for quick reference.

Open →

Sources

  • NIST/SEMATECH e-Handbook of Statistical Methods, sections 1.3.5.11 (skewness and kurtosis formulas) and 1.3.5.5 (quartiles by interpolation).
  • OpenStax Introductory Statistics 2e, chapters 2 (descriptive statistics) and 3 (linear regression, context for skewness).
  • Microsoft Excel PERCENTILE.INC documentation: linear interpolation method for quantiles.
  • R Core Team. The R Project for Statistical Computing. Documentation on quantile() type 7 (linear interpolation) and skewness() / kurtosis() computations.