Statistics Calculator

How to use this tool

1. Enter your numbers. Paste complete finite numbers separated by commas, spaces, or new lines; malformed tokens are ignored rather than partially parsed.
2. Choose sample or population. Choose Sample for a subset of a larger population and Population for the full dataset.
3. Review results. The calculator shows descriptive statistics, sample shape estimates, and potential Tukey outliers immediately, and marks undefined results as N/A.

Standard deviation explained

Standard deviation measures how spread out the values in a dataset are from the mean. A low standard deviation means values cluster close to the mean; a high standard deviation means they are spread out widely.

Use sample standard deviation (÷ n−1) when your data is a sample drawn from a larger population. Use population standard deviation (÷ n) when you have data for the entire population.

The standard error of the mean estimates how much a sample mean varies from sample to sample. This calculator uses the selected standard deviation divided by the square root of the count: SEM = SD ÷ √n. It follows the current Sample or Population setting; a one-value sample has no defined sample standard error, while a one-value population has SEM 0.

Sample vs population statistics

Quartiles and IQR

Quartiles divide your sorted data into four equal parts. Q1 is the 25th percentile, Q3 is the 75th percentile, and the interquartile range (IQR = Q3 − Q1) measures the spread of the middle 50% of the data. This calculator uses the median-of-halves method: for an odd number of values, it excludes the overall median from both halves before finding Q1 and Q3. Other software may use a different percentile interpolation rule, so quartiles can differ slightly for the same list. Its potential-outlier flags use Tukey's 1.5 × IQR fences, Q1 − 1.5×IQR and Q3 + 1.5×IQR. The fences are a review heuristic; flagged values remain in the data and are never deleted.

Measures of central tendency

• Mean: the sum of all values divided by the count. Best for symmetric distributions without outliers. Sensitive to extreme values - a single outlier can pull the mean far from the center of the data.
• Median: the middle value when sorted. Robust to outliers and better represents "typical" in skewed distributions (e.g., income data, where a few very high earners distort the mean).
• Mode: the most frequently occurring value. Most useful for categorical data or when identifying the most common outcome.

Skewness and kurtosis

Skewness measures the asymmetry of the distribution. A positive skew (right skew) has a long tail to the right and the mean is pulled above the median - income and wealth distributions are classic examples. A negative skew (left skew) has the tail to the left.

The Shape results use the adjusted Fisher–Pearson sample statistics: skewness G1 applies the small-sample correction to the third standardized moment, and excess kurtosis G2 applies the correction to the fourth moment and subtracts the normal-distribution baseline. G1 is available from 3 values, and G2 from 4; both are undefined when all values are identical. These estimates can vary widely in small samples. A normal distribution has excess kurtosis 0; positive values indicate relatively heavier tails and negative values relatively lighter tails. Shape results stay sample-based even when Population spread is selected.

To make the convention explicit, let S₂ = Σ(x − x̄)², S₃ = Σ(x − x̄)³, and S₄ = Σ(x − x̄)⁴. The adjusted skewness is G1 = n√(n−1)S₃ ÷ ((n−2)S₂^3/2), and adjusted excess kurtosis is G2 = n(n+1)(n−1)S₄ ÷ ((n−2)(n−3)S₂²) − 3(n−1)² ÷ ((n−2)(n−3)).

Z-score

A Z-score expresses how many standard deviations a value is from the mean:

z = (x − μ) / σ

A Z-score of +2 means the value is 2 standard deviations above the mean. In a normal distribution, about 95% of values fall within ±2 standard deviations (Z-score between −2 and +2). Z-scores enable comparison of values from different datasets with different units and scales. This page summarizes a dataset and does not calculate individual Z-scores; use the related Z-Score Calculator to standardize one observation.

Outlier detection

The calculator lists the original observations outside the Tukey 1.5×IQR fences as potential outliers. The rule is resistant to extreme values because it uses the middle half of the data, but it is still a heuristic: unusual values may be valid and should be checked in context. The list does not remove observations or change the count. Z-score thresholds are another common screening method, but this calculator does not apply them because it does not compute a Z-score for each observation.

Input and one-value samples

The calculator includes only complete finite number tokens separated by commas, spaces, or new lines. Other tokens are ignored as whole tokens: for example, 1abc is not read as the value 1. This prevents an accidental numeric prefix from changing a dataset without notice.

A population containing one known value has variance and standard deviation of zero. A one-value sample is different: its sample variance divides by n−1, which is zero when n is one. The tool therefore keeps the available descriptive statistics but displays sample variance, standard deviation, and standard error as N/A until at least two observations are provided. Skewness and kurtosis also require their stated minimum counts and nonzero variance.