Complete descriptive statistics, linear regression, and Pearson correlation. Enter a dataset to instantly get mean, median, mode, variance, SD, skewness, kurtosis, percentiles, regression line, and R².
Descriptive statistics summarise and describe the features of a dataset. They answer: "What does my data look like?" Tools include mean, median, mode, range, variance, standard deviation, skewness, and kurtosis.
Inferential statistics use sample data to make conclusions about a larger population. Tools include confidence intervals, hypothesis tests, regression analysis, and ANOVA.
This calculator focuses on descriptive statistics and two key inferential techniques — linear regression and Pearson correlation — which are foundational in both UK A-level and US AP Statistics curricula.
The arithmetic mean (x̄) is the sum of all values divided by n: x̄ = Σx / n. It is sensitive to outliers — a single extreme value can shift the mean significantly.
The median is the middle value when data is sorted. For even n, it is the average of the two middle values. The median is robust to outliers — one extreme value does not change it (unless there are very few values).
The mode is the most frequently occurring value. A dataset can have no mode, one mode (unimodal), or multiple modes (bimodal/multimodal). The mode is the only average that can be used for categorical (non-numeric) data.
The variance (s²) measures the average squared deviation from the mean: s² = Σ(xᵢ − x̄)² / (n−1) for a sample. The standard deviation (s) is the square root of the variance — in the same units as the data, making it more interpretable.
Population variance uses n in the denominator; sample variance uses n−1 (Bessel's correction) to get an unbiased estimate. This calculator uses sample statistics (n−1).
Skewness measures the asymmetry of the distribution. Positive skew = tail to the right (most data is left). Negative skew = tail to the left. A normal distribution has skewness = 0.
Kurtosis measures the "tailedness" or "peakedness" relative to a normal distribution. Excess kurtosis = 0 for a normal distribution; positive = heavy tails (leptokurtic); negative = light tails (platykurtic). Financial returns often have positive excess kurtosis (fat tails).
Simple linear regression models the relationship between two variables as a straight line: ŷ = mx + b, where m is the slope and b is the y-intercept.
An R² of 0.85 means that 85% of the variation in Y is explained by the linear relationship with X. The remaining 15% is unexplained (residual).
The Pearson correlation coefficient r measures the strength and direction of a linear relationship between two variables. It ranges from −1 (perfect negative) to +1 (perfect positive), with 0 indicating no linear relationship.
| |r| value | Strength | Interpretation |
|---|---|---|
| 0.00 – 0.19 | Negligible | No meaningful linear relationship |
| 0.20 – 0.39 | Weak | Small but detectable relationship |
| 0.40 – 0.59 | Moderate | Meaningful relationship |
| 0.60 – 0.79 | Strong | Substantial linear relationship |
| 0.80 – 1.00 | Very Strong | Near-perfect linear relationship |
A classic warning in statistics: correlation does not imply causation. Two variables can be strongly correlated because they share a common cause (confounding), because the relationship is coincidental (spurious), or because one truly causes the other. Only carefully designed experiments with random assignment can establish causality.
Famous examples of spurious correlations: the number of Nicolas Cage films released per year correlates with swimming pool drowning rates. Ice cream sales correlate with shark attacks — both caused by hot weather (the confounder).
In the UK, A-level Statistics (within Mathematics or as a standalone) covers descriptive statistics, probability, binomial and normal distributions, hypothesis testing, and regression. The AQA, Edexcel, and OCR specifications all require students to interpret regression lines and correlation coefficients.
In the US, AP Statistics is one of the most popular Advanced Placement courses, covering exploratory data analysis, sampling, probability, statistical inference, and regression. The AP exam requires students to interpret context-specific results, not just compute them.
| Feature | Excel | SPSS | R | This Calculator |
|---|---|---|---|---|
| Descriptive stats | Yes (Analysis ToolPak) | Yes | Yes (summary()) | Yes |
| Linear regression | Yes (LINEST) | Yes | Yes (lm()) | Yes |
| Pearson r | Yes (CORREL) | Yes | Yes (cor()) | Yes |
| Cost | Paid | Paid | Free | Free |
The mean is the arithmetic average (sum ÷ n). The median is the middle value. For symmetric distributions they are similar. For skewed distributions (like income), the median is more representative because it is not pulled by extreme values.
Standard deviation measures how spread out data is around the mean. A low SD means values cluster near the mean; a high SD means values are more dispersed. In a normal distribution, 68% of data falls within ±1 SD, 95% within ±2 SD.
R² (coefficient of determination) measures the proportion of variance in Y explained by the linear model. R²=0.85 means 85% of Y's variation is explained by X. R²=1.00 is a perfect fit; R²=0 means no linear relationship.
Pearson's r measures the strength and direction of the linear relationship between two variables. r=+1 is perfect positive correlation, r=-1 is perfect negative, r=0 is no linear correlation. It is sensitive to outliers.
Skewness measures asymmetry. Positive skewness: right tail is longer (e.g., income distributions). Negative skewness: left tail is longer. Rule of thumb: |skewness| > 1 indicates substantially skewed data.
Kurtosis measures "tailedness." Excess kurtosis > 0 means heavier tails (more outliers) than a normal distribution. This matters in finance — heavy-tailed return distributions mean extreme events (crashes) are more likely than normal distribution models predict.
The P-th percentile is the value below which P% of data falls. Sort the data; P25 = Q1 (25th percentile), P50 = median, P75 = Q3. The IQR (interquartile range) = Q3 − Q1, used to detect outliers (values beyond Q3 + 1.5×IQR).
No. Two variables can correlate for many reasons: common cause (confounding), reverse causation, coincidence, or direct causation. Only randomised controlled experiments can establish causation. Always consider the plausibility of the mechanism before inferring causation.
Results are for educational purposes. Sample statistics (n−1 denominator) are used. For research publications, use validated statistical software and consult a statistician to verify assumptions.