Statistics Calculators
From basic descriptive statistics to confidence intervals and sample size calculations — our free statistics calculators cover the core tools used in research, data analysis, and academic coursework.
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Descriptive Statistics
Mean, median, mode, standard deviation, and variance
Inference & Probability
Confidence intervals, sample size, and probability
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Statistics Glossary — Key Terms Explained
- Standard Deviation
- A measure of how spread out values are around the mean. Low SD = values clustered near the mean. High SD = values spread widely. Formula: σ = √[Σ(x − μ)² ÷ N] for population; s = √[Σ(x − x̄)² ÷ (n−1)] for sample.
- Confidence Interval
- A range of values within which the true population parameter falls with a given probability. A 95% CI means that if we repeated the study 100 times, 95 of the confidence intervals would contain the true value.
- p-value
- The probability of observing a result at least as extreme as what was observed, assuming the null hypothesis is true. p < 0.05 is the conventional threshold for statistical significance. Does NOT measure the probability that the null hypothesis is true.
- Statistical Significance
- A result is statistically significant if the probability of observing it by chance alone is below a threshold (typically 5%). Not the same as practical significance — a result can be statistically significant but too small to matter in practice.
- Normal Distribution
- A symmetric bell-shaped probability distribution where 68% of values fall within 1 standard deviation of the mean, 95% within 2 SD, and 99.7% within 3 SD. Many natural phenomena follow approximately normal distributions.
Frequently Asked Questions
What is the difference between standard deviation and variance?
Variance is the average of squared differences from the mean. Standard deviation is the square root of variance — expressed in the same units as the original data, making it more interpretable. If heights have a mean of 170cm and variance of 100 cm², the standard deviation is 10cm (more useful: 'heights vary by about ±10cm').
What sample size do I need for a survey?
For a 95% confidence interval with ±5% margin of error, you need approximately 385 responses for a large population. For ±3% margin: ~1,067 responses. For ±1% margin: ~9,604 responses. For smaller populations, fewer are needed. The sample size calculator gives exact figures for any combination of confidence level and margin of error.
What does a 95% confidence interval mean?
If you ran the same survey 100 times, approximately 95 of the resulting confidence intervals would contain the true population value. It does NOT mean 'there's a 95% chance the true value is in this specific interval' — that's a common misinterpretation. The true value either is or isn't in the interval; the 95% refers to the method's reliability.
What is the difference between mean and median?
Mean: sum of all values ÷ count. Affected by outliers. Median: the middle value when sorted. Not affected by outliers. For symmetric distributions, they're equal. For skewed distributions (e.g., incomes), median is more representative. Median UK household income is more meaningful than mean income, which is pulled up by very high earners.