Full descriptive statistics suite with steps. Enter your data to get mean, median, mode, variance, standard deviation, quartiles, outliers, confidence intervals and z-scores for each value.
Enter Data
Accepts integers, decimals and negative numbers. Non-numeric entries are flagged, not silently dropped. At least 2 valid values are needed.
Summary
Enter data and press Calculate
Full Descriptive Statistics
Statistic
Value
Statistic
Value
ℹ️ Quartiles use linear interpolation at position (n−1)×p, comparable to a commonly used inclusive percentile method. Results may differ slightly from calculators using the median-of-halves method — that's a methodology difference, not an error.
Box Plot Values & Five-Number Summary
Frequency Table & Histogram
Data with Z-Scores & Outlier Detection
#
Value
Deviation (x−μ)
Z-Score
Status
Show Calculation Steps
Show how these results were calculated
Descriptive Statistics Guide
Descriptive statistics summarise the main features of a dataset without drawing conclusions about a larger population.
Measures of Central Tendency
Mean: arithmetic average. Sensitive to outliers. Median: middle value. Robust to outliers. Mode: most frequent value. Can have multiple modes.
Measures of Spread
Range: max − min. Very sensitive to outliers. IQR: Q3 − Q1. The middle 50% of data. Standard deviation: average distance from mean.
Outlier Detection
Outliers are identified using the IQR method: values below Q1−1.5×IQR or above Q3+1.5×IQR are mild outliers; below Q1−3×IQR or above Q3+3×IQR are extreme outliers. Z-scores above |3| also flag potential outliers. Whether an outlier should actually be removed depends on context — a genuine data error is usually safe to correct or exclude, but a real extreme observation is data, not noise, and removing it can quietly bias your results.
Quartile Method
This calculator uses linear interpolation at position (n−1)×p to find Q1 and Q3 — a commonly used inclusive percentile convention. Other tools (including some spreadsheet software) use a median-of-halves method instead, which can give slightly different Q1/Q3 values for the same dataset. Neither is "wrong" — they're different, equally legitimate conventions, so don't assume a mismatch means an error.
Sample vs Population Standard Deviation
Use sample standard deviation (dividing by n−1) when your data is a sample drawn from a larger population you're trying to describe — this is the more common case, and corrects for a systematic bias that would otherwise underestimate spread. Use population standard deviation (dividing by n) only when your dataset genuinely is the entire population of interest, with nothing left out.
⏱️ Last Updated: August 2026 | Reviewed by Mohsin Iqbal
🔑 Key Takeaways
Descriptive statistics summarise data: mean (centre), SD (spread), quartiles (distribution)
Mean is sensitive to outliers; median is resistant — prefer median for skewed data
Skewness > 0: right-skewed (tail to right); < 0: left-skewed (tail to left)
Coefficient of variation (CV) = SD/mean × 100% — undefined when the mean is zero
A 95% confidence interval uses the t-distribution, not a flat 1.96, especially for small samples
Key Descriptive Statistics
Statistic
What It Measures
Formula
Mean (average)
Centre of data
Σxᵢ / n
Median
Middle value (50th percentile)
Middle value when sorted
Mode
Most frequent value(s)
Value appearing most often
Range
Total spread
Max − Min
Sample Variance
Average squared deviation (sample)
Σ(xᵢ−x̄)²/(n−1)
Standard deviation
Typical deviation from mean
√Variance
IQR
Middle 50% spread
Q3 − Q1
95% CI for the mean
Plausible range for the true population mean
x̄ ± t(n−1, 0.975) × SE
Frequently Asked Questions
Mean: sum all values, divide by count — affected by outliers. Median: middle value when sorted — robust to outliers. Mode: most frequent value — can have multiple modes or none. For income data where a few high earners skew the average upward, the median household income is more representative. Australian Bureau of Statistics reports both mean and median income for this reason.
Use the median when data is skewed or contains outliers that would distort the mean. Examples: house prices (a few $10M mansions inflate the mean — median better represents typical buyers), income distributions, time-to-failure data, wait times. Use the mean when data is roughly symmetric and outliers are genuine data points you want included.
An outlier is a data point far from other observations. A common rule: any value more than 1.5 × IQR below Q1 or above Q3 is a mild outlier; more than 3 × IQR is an extreme outlier. Example: IQR=10, Q1=20, Q3=30. Outlier threshold: below 20−15=5 or above 30+15=45.
Sample standard deviation divides by n−1 (Bessel's correction), used when your data is a sample from a larger population. Population standard deviation divides by n, used only when your dataset is the entire population. Sample SD is always slightly larger and is the more common real-world choice.
Because multiple quartile conventions exist. This calculator uses linear interpolation at position (n−1)×p; other tools use a median-of-halves method. Both are legitimate — the small differences you sometimes see are methodology, not an error in either tool.
Negative skewness means the data's tail stretches further to the left (toward smaller values) than the right — the bulk of values sit on the higher side, with a few low outliers pulling the tail out. Positive skew is the mirror image: a long tail to the right.
Excess kurtosis compares a dataset's "tailedness" to a normal distribution (which has excess kurtosis of 0). Positive excess kurtosis means heavier tails and a sharper peak than normal (more extreme values than expected); negative means lighter tails and a flatter peak.
A z-score shows how many standard deviations a value sits from the mean. Z=0 is exactly average; z=+2 is two standard deviations above the mean. Values beyond |z|=2.5–3 are increasingly unusual and often flagged as potential outliers.
Not automatically. If it's a genuine data-entry error, correcting or removing it is usually right. If it's a real, correctly recorded extreme observation, removing it can bias your results — treat outlier detection as a prompt to investigate, not a default deletion rule.
A 95% confidence interval for the mean is a range built from your sample using a procedure that, applied repeatedly to new samples, would contain the true population mean about 95% of the time. It is not a range containing 95% of individual data points, and this calculator computes it using the t-distribution to stay honest for small samples.
CV = SD ÷ mean × 100%, so a mean of zero makes the ratio undefined (division by zero). CV is generally most meaningful for ratio-scale data with a positive, non-zero mean — it's not a sensible comparison for data centred at or crossing zero.
This calculator needs at least 2 valid numbers. Some statistics need more to be meaningful: sample skewness needs more than 2 points, sample kurtosis needs more than 3, and confidence intervals get dramatically wider (and more honest) with very small samples.
Type them directly — e.g. -4.5, 12.75 — separated by commas, spaces, semicolons or new lines. Note that geometric mean is undefined for any dataset containing zero or negative values, and will show "N/A" rather than a misleading number in that case.
Yes — duplicates are completely valid and expected in real data. They directly affect the mode (the most frequent value) and are counted normally in every other statistic.
Minimum, Q1, median, Q3, and maximum — the five values that define a box plot. This calculator also separately reports the lower and upper whiskers, which stop at the most extreme non-outlier values rather than always reaching the true minimum and maximum.
If every value is identical (e.g. 5, 5, 5, 5), standard deviation is exactly 0. Skewness, kurtosis and z-scores are all reported as 0 in that case, rather than the undefined (NaN) result a naive division by zero would otherwise produce.