Updated Descriptive Statistics Tool

Mean & Variance Calculator

Compute mean, variance, standard deviation, full summary statistics and weighted averages for one or two datasets. Compare distributions side-by-side and explore variability with a single, flexible calculator.

Single Dataset Mean & Variance Two-Dataset Comparison Full Summary Statistics Weighted Mean & Variance

Calculate Mean, Variance, Standard Deviation & More

This Mean & Variance Calculator includes four tabs: a basic single-dataset tool, a comparison of two datasets, a full summary statistics tab and a weighted statistics tab. Enter values separated by commas, spaces or line breaks and the calculator will compute sample and population variance, standard deviation, measures of central tendency and dispersion.

Use this tab when you have one list of numeric values and want its mean, variance and standard deviation. The calculator reports both sample and population formulas.

Use this tab to compare the averages and variability of two datasets. The calculator computes means, sample variances, standard deviations and a simple difference-of-means summary.

Use this tab to see a fuller picture of a single dataset, including mean, median, mode, minimum, maximum, range, quartiles and interquartile range.

Use this tab when each value has an associated weight, such as frequencies, probabilities or importance scores. The calculator computes weighted mean and both population-style and count-based weighted variance.

Values and weights lists must have the same length. Weights should be non-negative and not all zero. If weights represent frequencies or counts, the count-based variance uses the sum of weights minus 1 in the denominator.

Mean & Variance Calculator – Complete Guide to Averages and Variability

The Mean & Variance Calculator on MyTimeCalculator is designed to give you a flexible, practical way to explore numerical datasets. It lets you compute basic measures such as mean and variance for one dataset, compare two datasets side-by-side, obtain a full set of summary statistics and work with weighted data when some values count more than others.

For a dataset of \(n\) numeric values \(x_1, x_2, \dots, x_n\), the sample mean is defined as

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i, \]

and the sample variance, which is often used to estimate population variance, is

\[ s^2 = \frac{1}{n - 1} \sum_{i=1}^{n} (x_i - \bar{x})^2. \]

The sample standard deviation is the square root of the sample variance, \(s = \sqrt{s^2}\). A population-style variance (often used when the dataset already represents the whole population) replaces the denominator \(n - 1\) with \(n\).

1. Single Dataset Mean & Variance

The first tab of the calculator focuses on a single dataset. You enter values such as \(x_1, x_2, \dots, x_n\) and the tool returns:

  • the number of data points and their sum,
  • the mean \(\bar{x}\),
  • sample variance and sample standard deviation,
  • population-style variance and standard deviation,
  • minimum, maximum and range (max − min).

This is ideal for quickly summarizing test scores, measurements, financial returns or any other list of numeric values where you want a basic snapshot of average and variability.

2. Comparing Two Datasets

The second tab compares two datasets A and B. For each dataset you obtain the mean, sample variance and standard deviation, then the calculator highlights:

  • the difference between the two means, \(\bar{x}_A - \bar{x}_B\),
  • the ratio of sample variances, \(s_A^2 / s_B^2\),
  • how the spreads compare in terms of standard deviation.

This is helpful when you want to see whether one group tends to have higher or lower values than another and whether one group is more or less variable. It is purely descriptive: the tab focuses on numerical comparison without running a formal hypothesis test.

3. Full Summary Statistics

The third tab provides a richer summary of a single dataset, combining measures of center, spread and distribution shape. Given values \(x_1, x_2, \dots, x_n\), it computes:

  • Mean: the arithmetic average, \(\bar{x}\).
  • Median: the middle value when the data are sorted (or the average of the two middle values when \(n\) is even).
  • Mode: a value that occurs most frequently in the dataset. When there are multiple modes, the calculator reports one of them along with the number of times it appears.
  • Sample variance and standard deviation: based on the usual \(n - 1\) denominator.
  • Minimum, maximum and range: where range equals max minus min.
  • Quartiles and interquartile range (IQR): the first quartile \(Q_1\), median \(Q_2\) and third quartile \(Q_3\), plus the IQR defined as \(Q_3 - Q_1\).

These statistics are often used together to describe the center and spread of a distribution and to identify potential outliers when values fall far beyond the interquartile range.

4. Weighted Mean & Weighted Variance

The fourth tab handles situations where each value \(x_i\) has an associated weight \(w_i\), such as:

  • frequency counts (a value appears \(w_i\) times),
  • probability weights (a value occurs with probability \(w_i\)),
  • importance weights (some observations matter more than others).

The weighted mean is defined by

\[ \mu_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}, \]

provided that the weights are non-negative and not all zero. The calculator also reports two versions of weighted variance:

  • A population-style weighted variance, \[ \sigma_w^2 = \frac{\sum_{i=1}^{n} w_i (x_i - \mu_w)^2}{\sum_{i=1}^{n} w_i}, \] which is natural when the weights reflect probabilities or you treat the weighted data as a full population.
  • A count-based weighted variance, which divides by \(\sum w_i - 1\) when the weights are interpreted as counts, providing an analogue of the usual sample variance.

The corresponding weighted standard deviations are simply the square roots of these variances.

5. How to Use the Mean & Variance Calculator

  1. Choose a tab: decide whether you need a single-dataset summary, a two-dataset comparison, a full set of summary statistics or weighted statistics.
  2. Enter your data: paste values separated by commas, spaces or line breaks. For two datasets or weighted data, make sure each list has the correct length and all entries are numeric.
  3. Run the calculation: click the corresponding button in each tab. The results area will show key statistics along with a short text summary.
  4. Interpret the outputs: use the mean as the central value, the variance and standard deviation to understand variability, and the quartiles and range to see how spread out the data are and whether there are potential outliers.

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Mean & Variance Calculator FAQs

Frequently Asked Questions

Quick answers to common questions about mean, variance, standard deviation, summary statistics and weighted averages.

Population variance divides by \(n\) and describes variability for an entire population. Sample variance divides by \(n - 1\) and is typically used when your data are a sample drawn from a larger population. Dividing by \(n - 1\) instead of \(n\) corrects for bias when using the sample mean as an estimate of the population mean.

Variance measures average squared deviations from the mean and is expressed in squared units. Standard deviation is the square root of variance and returns to the original units of the data, which often makes it easier to interpret. Both describe spread, but standard deviation is usually more intuitive in practical work.

The calculator first sorts the data from smallest to largest. The median is the middle value (or the average of the two middle values when the dataset has an even number of points). Quartiles split the sorted data into four parts; \(Q_1\) is a median of the lower half, \(Q_2\) is the overall median and \(Q_3\) is a median of the upper half. The exact algorithm matches common textbook conventions used for descriptive statistics.

Weighted formulas are appropriate when some values occur more often, represent larger groups or have different importance. For example, if each value represents an average for a group of different size, weighting by group size gives a mean and variance that reflect the combined population rather than treating each group equally regardless of size.

Yes. You can enter the same values used in an exercise or spreadsheet and compare the mean, variance and other statistics with your own results. This is helpful for verifying manual calculations. For written assignments, you should still show the underlying formulas and steps, as the calculator provides final numbers rather than detailed working.

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