Statistics
Statistics turns raw data into summaries, predictions and decisions. The tools here cover the core operations: describing a data set, standardizing values, working with the normal distribution, building confidence intervals and fitting a regression line.
Most statistical questions reduce to two flavours: describing a data set you already have, and inferring something about a wider population from a sample of it. Descriptive tools answer the first; z-scores, the normal distribution and confidence intervals answer the second. Regression bridges the two.
Descriptive statistics
Mean, median, mode, range and standard deviation summarize a data set in a handful of numbers. The descriptive calculator reports population and sample versions side by side.
Z-scores and the normal distribution
A z-score standardizes a value to standard-deviation units; the normal distribution calculator turns z-scores into probabilities through Φ(z).
Confidence intervals
A confidence interval gives a range that likely contains the true population mean. The width depends on the sample standard deviation, the sample size and the chosen confidence level.
Linear regression
Least-squares regression fits a straight line to paired data, minimizing the sum of squared vertical distances. The correlation r and r² report how well the line fits.
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Frequently asked questions
What is the difference between population and sample standard deviation?
The population version divides by n; the sample version divides by n − 1, which corrects bias when estimating from a sample.
What does a 95% confidence interval mean?
If the sampling were repeated many times, about 95% of the intervals built this way would contain the true mean.
When is a correlation r considered strong?
Loosely, |r| above about 0.7 is considered strong, between 0.3 and 0.7 moderate, and below 0.3 weak.