Statistics Calculator
Paste a set of numbers to get a full descriptive-statistics summary — mean, median, mode, standard deviation, quartiles, outliers — plus a distribution histogram.
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Disclaimer: This free tool is provided “as is,” without warranties of any kind, and is for general informational purposes only — not professional, legal, financial, medical, tax, or engineering advice. Results may contain errors; verify anything important independently and use at your own risk. We accept no liability for any loss or damage arising from its use. See our Terms of Use for details.
Step-by-Step Guide
Paste or type your numbers separated by commas, spaces, tabs, or newlines. Select whether the data represents a sample or a population — this affects how variance and standard deviation are calculated. The full descriptive statistics grid and a histogram update immediately.
Sample vs. population
The distinction matters for variance and standard deviation. Population variance divides by n (the total count). Sample variance divides by n−1 (Bessel's correction), which corrects for the bias introduced when estimating a population parameter from a subset. Use population when your data is the complete set; use sample when your data is a random subset of a larger group and you want to estimate the population's variability.
[12, 7, 3, 14, 7, 9, 21, 7, 10, 4] (10 values): count=10, sum=94, mean=9.4, median=(7+9)/2=8.0, mode=7 (appears 3 times), min=3, max=21, range=18. Sample std dev = √(variance with n−1) ≈ 5.52. Q1=6.5, Q2=8, Q3=11, IQR=4.5. Outliers (beyond 1.5×IQR from Q1/Q3): none in this set. Histogram shows clustering around 7–10.
Who it's for
Students, analysts, researchers, and anyone summarizing data.
Core Features
- Count, sum, mean, median, mode, min, max, and range.
- Sample and population variance and standard deviation.
- Quartiles, IQR, and 1.5·IQR outlier detection.
- A quick bar histogram of the distribution.
🛡️ No tracking — your inputs, keys, and details never leave this client sandbox.
What is the difference between mean, median, and mode?
Mean is the arithmetic average — sum all values and divide by count. Median is the middle value when the data is sorted — it is resistant to outliers, unlike the mean. Mode is the most frequently occurring value — there can be multiple modes or no mode if all values appear equally. For symmetric distributions (like a normal distribution), these three measures are equal or close; for skewed distributions, they diverge — which one is 'best' depends on what you want to communicate.
What is standard deviation and how do I interpret it?
Standard deviation (σ or s) measures how spread out the values are around the mean. A small standard deviation means values cluster tightly around the mean; a large one means they are widely spread. For a normally distributed dataset, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. This is the '68-95-99.7 rule' or the empirical rule.
What are quartiles and IQR?
Quartiles divide the sorted data into four equal parts. Q1 (25th percentile) is the median of the lower half; Q2 (50th percentile) is the overall median; Q3 (75th percentile) is the median of the upper half. The interquartile range (IQR = Q3 − Q1) contains the middle 50% of the data. IQR is used to detect outliers: values more than 1.5×IQR below Q1 or above Q3 are considered outliers (the Tukey fence rule).
What is the histogram showing?
The histogram divides the data range into equal-width bins and shows how many values fall into each bin. It visualizes the distribution shape: symmetric (bell-curve), right-skewed (tail to the right), left-skewed (tail to the left), bimodal (two peaks), or uniform. The distribution shape helps you choose the right summary statistics — for heavily skewed data, the median is a better central tendency measure than the mean.
The statistic that gets reported most often is often the least informative
Most people assume that the mean (average) is the primary summary of a dataset. In practice, the mean is the most easily manipulated and most frequently misleading summary statistic available. A dataset of salaries at a company where most employees earn $50,000 and the CEO earns $5,000,000 will show a mean salary of perhaps $150,000 — a number that describes nobody's actual pay. The median — the middle value — would be far closer to the typical employee's experience. Understanding which measure to use, and when to be skeptical of a reported average, is one of the most practically useful things statistics teaches.
How experts interpret summary statistics differently than beginners
Beginners compute the mean and stop. Analysts look at mean vs. median first: a large gap between them signals a skewed distribution or outliers. They look at the standard deviation relative to the mean (the coefficient of variation) to judge whether the spread is meaningful. They check the histogram shape before drawing any conclusion — a bimodal distribution (two peaks) may represent two distinct populations mixed together, and summarizing it with a single mean is actively misleading. The IQR and outlier flags tell you whether extreme values are likely errors or genuine data points that warrant investigation.
Variance and standard deviation
Variance is the average squared deviation from the mean: σ² = Σ(x − μ)² / n for a population, or s² = Σ(x − x̄)² / (n−1) for a sample. Squaring the deviations makes them all positive and gives larger weight to points far from the mean. Standard deviation is the square root of variance, putting it back in the same units as the original data. Bessel's correction (dividing by n−1 rather than n for a sample) compensates for the fact that the sample mean is itself estimated from the data, which causes the deviations to be slightly understated.
The normal distribution and why it matters
Many natural phenomena follow an approximately normal (Gaussian) distribution — a symmetric bell-shaped curve defined by its mean and standard deviation. Test scores, measurement errors, adult heights, and many biological measurements tend to be approximately normal. The central limit theorem states that the mean of any sufficiently large sample, regardless of the underlying distribution, will be approximately normally distributed. This makes the standard deviation a universal measure of spread: you can compare the variability of datasets with different means by comparing their standard deviations, and you can use the 68-95-99.7 rule to reason about probabilities.