AkCalculators

Descriptive Statistics: What they are and how to compute them (mean, median, mode, variance, std dev)

Descriptive statistics summarize and describe the features of a dataset. The most common descriptive statistics are measures of central tendency — such as the mean, median and mode — and measures of spread — such as range, variance, and standard deviation. These measures give you a quick sense of where the data is centered and how much variability there is. This tutorial explains each measure, shows how to compute them step-by-step, and gives practical tips for using the Statistics Calculator effectively.

Mean (average)

The arithmetic mean is the sum of values divided by the count. If your dataset is x₁, x₂, …, xₙ then:

mean = (x₁ + x₂ + ... + xₙ) / n

Example: for values 2, 4, 6, the mean is (2+4+6)/3 = 4. The mean is sensitive to extreme values (outliers).

Median

The median is the middle value when the data are sorted. If n is odd, it is the middle element. If n is even, it is the average of the two middle elements. The median is robust to outliers and often a better measure of central tendency for skewed data.

Mode

The mode is the most frequently occurring value(s). A dataset can be unimodal (one mode), multimodal (multiple modes), or have no mode when all values are unique. Mode is particularly useful for categorical data or discrete numeric values.

Range

Range is a simple measure of spread: range = max − min. It gives a quick feel for the spread but is affected by outliers.

Variance and standard deviation

Variance measures average squared deviation from the mean. For a population of N values:

population variance σ² = (1/N) Σ (xi − μ)²

For a sample (estimating a population variance), use the unbiased estimator that divides by N−1:

sample variance s² = (1/(N−1)) Σ (xi − x̄)²

Standard deviation is the square root of variance and has the same units as the data:

σ = sqrt(σ²),  s = sqrt(s²)

The difference between population and sample formulas matters when drawing inferences. For descriptive summaries of a full population, use the population formula. For an estimate from a sample, use the sample version to correct bias.

Step-by-step computation (example)

Suppose we have the numbers: 3, 7, 7, 2, 9.

1. Sort for median: 2, 3, 7, 7, 9 → median = 7 (middle value).
2. Mean: sum = 28, n = 5 → mean = 28 / 5 = 5.6.
3. Mode: 7 appears twice → mode = 7.
4. Range: max − min = 9 − 2 = 7.
5. Variance (population): compute squared differences: (3.6² + 1.6² + 1.4² + 1.4² + 3.4²) sum = ... / 5 → population variance.

The Statistics Calculator automates these steps and shows them when you enable step-by-step output.

Practical tips

• Input cleanliness: paste raw numbers from spreadsheets into the comma mode. Remove headers or non-numeric text first.
• Small datasets: mode and median remain useful; watch sample vs population variance.
• Outliers: beware of extreme values that skew the mean — consider median or trimmed means.
• Missing data: exclude missing values. This calculator ignores non-numeric tokens and reports them.

When to use which statistic

Use the mean when the distribution is roughly symmetric and you care about total sum. Use median when the distribution is skewed or contains outliers. Use mode for categorical or discrete variables. Use variance and standard deviation to quantify spread; if you need interpretability in original units prefer standard deviation.

Reporting results

When publishing results, always report which formula you used (population vs sample) and report the sample size. For reproducibility include the raw data or a CSV export, which this calculator provides.

Digit precision

The display precision controls how many decimal places to show. Internally calculations use JavaScript numeric precision; for extremely high-precision needs use specialized numeric libraries.

Conclusion

Descriptive statistics give a compact summary of your data. This Statistics Calculator lets you paste or enter data, toggle input modes, view step-by-step calculations, and export results. It’s ideal for classroom use, quick analyses, and double-checking computations before reporting or model-building.