What each summary answers

The arithmetic mean is the sum divided by the number of values. It uses every observation and is often what “average” means, but it can be pulled strongly by an extreme value. The median is the middle value after sorting, or the mean of the two middle values when the count is even. It is more resistant to extremes and can better describe a typical household price, waiting time or income in a skewed distribution.

Minimum and maximum identify the observed endpoints, while their difference is the range. The range is easy to understand but depends only on two observations. Standard deviation uses every value to describe spread around the mean. None of these summaries shows the full shape, clusters, gaps or data-quality problems, so retain the original dataset and inspect a plot where the decision matters.

Population and sample standard deviation

Population standard deviation divides the sum of squared deviations by N, the number of values. Use it when the entries are the entire population you intend to describe, such as every transaction in a closed batch. Sample standard deviation divides by N − 1. That Bessel correction makes the variance estimate less biased when a sample is used to infer variability in a larger population.

The two values converge as the sample grows but can differ materially for a small dataset. With only one observation, population standard deviation is zero because there is no within-list variation, while sample standard deviation is undefined because N − 1 is zero. The page says that at least two values are needed rather than inventing a sample result.

How the calculation is performed

After validating every token, the script sums the values and calculates the mean. It subtracts that mean from each observation, squares each deviation and adds the squares. Squaring prevents positive and negative deviations from cancelling. The appropriate variance divides that sum by N or N − 1, and standard deviation is the square root of variance, returning the result to the original unit.

A stable professional statistics system may use specialised algorithms for extremely large values or streaming datasets. This calculator is intended for ordinary lists that fit comfortably in the browser. If values are near the largest representable number or differ only in very low-order digits, floating-point cancellation can affect the result. Use audited statistical software for high-volume or high-stakes analysis.

Preparing a clean dataset

Separate values with spaces, commas or semicolons. Decimal points follow South African English notation in this module. Do not use a comma as a thousands separator because it is treated as a value boundary; enter 12500 rather than 12,500. Units should not be typed into the list. Convert every observation to the same unit first and record that unit in your analysis.

The parser rejects invalid text instead of dropping it. Silent omission would change the count, mean and spread without warning. Missing data needs an explicit decision: investigate the source, distinguish genuinely absent values from zero, and document any exclusion or imputation. A blank response is not automatically the numerical value zero.

Worked interpretation of the default list

The values 12, 15, 15, 18, 20 and 22 sum to 102, so the mean is 17. The sorted middle pair is 15 and 18, giving a median of 16.5. The mean sits slightly above the median because the larger observations contribute more to the balance point. The observed range runs from 12 to 22.

The sample standard deviation is larger than the population standard deviation because the same squared-deviation total is divided by 5 instead of 6. Reporting “standard deviation” without saying which definition was used can make two correct calculations appear inconsistent. Label the choice, the count and the unit beside the number.

Outliers and skewed distributions

An outlier can be a data-entry error, an unusual but real event, or evidence that the population contains different groups. Do not remove it merely because it raises standard deviation. Check the source, define an exclusion rule before looking for a preferred answer, and consider reporting median and percentiles alongside mean. A trimmed or winsorised statistic is a different method and should be named explicitly.

For non-negative data with a long upper tail, mean plus or minus one standard deviation is not automatically a meaningful interval. The familiar empirical-rule percentages depend on an approximately normal distribution. Plot the values or use distribution-aware methods before translating standard deviation into probabilities.

From description to inference

These outputs describe the entered numbers. They do not calculate a confidence interval, test a hypothesis, correct sampling bias or establish causation. A representative sample depends on how units were selected, non-response, measurement quality and study design. Increasing the number of decimal places cannot fix a biased sample.

South African surveys may need weights, strata and clusters to represent a population correctly. A simple unweighted mean of respondent records can differ from an official weighted estimate. For public reporting, academic work or regulated decisions, follow the statistical method specified by the data owner and retain reproducible cleaning and analysis steps.

Grouped observations and frequency tables

If a value occurs several times, it must appear that many times in the pasted list for this unweighted calculator. A frequency table can be expanded manually for a short dataset, but a large table is better handled by software that accepts value-frequency pairs. Using each distinct value once would discard the frequencies and usually change the mean, median and standard deviation.

Class intervals create a further approximation. Replacing every observation in an interval with its midpoint can estimate grouped statistics, but the exact within-class values have been lost. Label the result as grouped-data analysis and retain the class definitions.

Comparing variation across units

Standard deviation uses the same unit as the observations. Converting metres to centimetres multiplies both the mean and standard deviation by 100. Comparing raw SD between datasets with very different means or units can therefore be unhelpful. A coefficient of variation divides SD by a non-zero mean, but it has its own assumptions and is not displayed here. Never compare spread by silently mixing currencies, years or measurement scales.