Statistics Calculator

Descriptive statistics condense a data set into meaningful summary numbers.

• Central tendency measures (mean, median, mode) tell you where the center of the data lies.
• Dispersion measures (range, variance, standard deviation, IQR) tell you how spread out the values are.
• Shape measures (skewness and kurtosis) describe the distribution's symmetry and tail behavior.

Together these statistics give a complete picture of your data without needing to look at every individual value.

When your data represents an entire population, you divide by N to get the population variance and standard deviation. When your data is a sample drawn from a larger population, you divide by N-1 (Bessel's correction) to get an unbiased estimate.

The sample standard deviation is always slightly larger than the population standard deviation for the same data set.

For large N the difference is negligible, but for small samples the correction matters significantly.

• The mean is the sum of all values divided by the count
• the median is the middle value of the sorted data
• the mode is the most frequently occurring value

For the data set {68, 72, 76, 78, 82, 85, 88, 90, 91, 95}, knowing how to calculate the average gives a mean of 825 divided by 10, or 82.5. With an even count of 10 values, the median averages the two middle numbers, (82 + 85) / 2 = 83.5. No value repeats, so this set has no mode.

As Khan Academy explains, these three measures of central tendency answer different questions, so reporting all three gives a fuller picture than any one alone.

Variance is the average of the squared deviations from the mean, and standard deviation is the square root of variance.

You first:

• subtract the mean from each value
• square each result
• sum them

Population variance divides that sum by N; sample variance divides by N-1. For the 10 test scores with mean 82.5, the sample variance is about 78.28 and the sample standard deviation is about 8.85.

Wolfram MathWorld notes that squaring the deviations both removes negative signs and weights larger departures from the mean more heavily, which is why standard deviation is the dominant measure of spread in statistics.

Quartiles split sorted data into four equal parts:

• Q1 is the 25th percentile
• Q2 is the median
• Q3 is the 75th percentile

The interquartile range (IQR) equals Q3 minus Q1 and captures the spread of the middle 50% of the data. For {68, 72, 76, 78, 82, 85, 88, 90, 91, 95}, Q1 is 76, Q3 is 90, so IQR = 90 - 76 = 14.

The five-number summary — minimum, Q1, median, Q3, maximum — is 68, 76, 83.5, 90, 95. The NIST/SEMATECH e-Handbook of Statistical Methods describes this summary as the basis for the box plot.

Skewness measures the asymmetry of a distribution, and kurtosis measures the heaviness of its tails.

A skewness of zero indicates a symmetric distribution; a positive value signals a longer right tail, and a negative value a longer left tail.

Kurtosis compares tail weight to the normal distribution, which has a kurtosis of 3, or an excess kurtosis of 0. Higher kurtosis implies heavier tails and more extreme outliers.

According to Wolfram MathWorld, these third and fourth standardized moments extend the description of a data set beyond center and spread, revealing shape features that the mean and standard deviation cannot capture on their own.

The coefficient of variation (CV) is the standard deviation divided by the mean, expressed as a percentage: CV = (SD / mean) x 100%.

Because it is unitless, the CV lets you compare variability across data sets measured in different units or on different scales. For the 10 test scores, CV is roughly 8.85 / 82.5 x 100%, or about 10.7%, indicating relatively low variability.

Encyclopaedia Britannica describes the CV as a standardized measure of dispersion. A common rule of thumb treats a CV below 15% as low, and one above 30% as high, though ideal thresholds depend on your field and context.

Percentile rank tells you the percentage of values in a data set that fall below a given value, so it builds directly on knowing how to calculate percentages. A simple formula is: percentile rank = (number of values below x / total count) x 100.

If 7 of 10 exam scores are lower than yours, your percentile rank is 70, meaning you scored higher than 70% of the group. Percentiles anchor many standardized reporting systems, from test scores to pediatric growth charts.

Khan Academy emphasizes that percentiles describe relative position rather than absolute value, which is why a raw score becomes far more interpretable once you know the percentile it corresponds to.

Descriptive statistics appear in nearly every quantitative field.

• In manufacturing, Six Sigma programs use standard deviations to quantify defect rates and process capability.
• In finance, standard deviation measures the volatility of returns, while the mean estimates expected return.
• Public health agencies rely on medians and percentiles to summarize outcomes across populations, and educators use them to interpret test performance.

The NIST/SEMATECH e-Handbook of Statistical Methods shows how summary statistics support quality control and experimental analysis. Whether you are grading exams, monitoring a production line, or comparing investment portfolios, these measures turn raw numbers into decisions you can defend.

The empirical rule, or 68-95-99.7 rule, describes how data spread out in a normal distribution:

• Approximately 68% of values fall within one standard deviation of the mean
• about 95% fall within two standard deviations
• and about 99.7% fall within three

For example, if a test has a mean of 82.5 and a standard deviation of about 8.85, roughly 95% of scores would lie between 64.8 and 100.2 if the data were normally distributed.

Encyclopaedia Britannica notes that this rule is the reason standard deviation is so widely used: it converts an abstract spread into concrete probability ranges you can reason about.

• The most frequent error is mixing up sample and population formulas: dividing by N when your data is only a sample understates the true variability, so use N-1 (Bessel's correction) for samples.
• Another mistake is failing to sort the data before finding the median or quartiles, which produces meaningless results.
• People also confuse the mode (most frequent value) with the mean, or report a mode for data where every value is unique.
• Finally, remember that the mean is sensitive to outliers while the median is robust, so a single extreme value can distort your average.

Always confirm which statistic best answers your specific question before drawing conclusions.