# Covariance Calculator URL: https://speedycalculator.com/calculators/statistics/covariance/ Last updated: 2026-10-30 > The covariance calculator finds the sample and population covariance of two paired data sets, along with both means, the sum of cross-products, both standard deviations and the correlation coefficient that standardizes the covariance. ## Inputs - Data set X - Data set Y (same order as X) ## Results - Sample covariance - Population covariance - Sum of cross-products Σ(x − x̄)(y − ȳ) - Mean of X - Mean of Y - Standard deviation of X (sample) - Standard deviation of Y (sample) - Correlation coefficient r - Direction - Number of pairs ## Formula ``` Sample cov(X, Y) = Σ(xᵢ − x̄)(yᵢ − ȳ) ÷ (n − 1) · Population cov = Σ(xᵢ − μx)(yᵢ − μy) ÷ N · r = cov ÷ (sx · sy) ``` ## Explanation Covariance measures whether two variables tend to move in the same direction. For each pair of values the calculator multiplies the deviation of x from its mean by the deviation of y from its mean. When both are above or both are below their means the product is positive; when one is above and the other below it is negative. The average of these products is the covariance. A positive covariance means the variables rise together, a negative one means one tends to fall as the other rises, and a value near zero means there is no linear tendency either way. Investors use covariance between asset returns to build diversified portfolios: assets with low or negative covariance smooth each other's swings. Covariance is also the building block of regression slopes, principal component analysis and the variance of a sum, Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y). The size of a covariance depends on the units – measuring heights in centimetres instead of metres multiplies it by 100 – so it is hard to read on its own. Dividing it by both standard deviations gives the unit-free correlation coefficient r, which the calculator shows too. ### How to use the Covariance Calculator - **Data set X** - **Data set Y (same order as X)** Results update as you type, and everything is calculated in your browser. ### How it works Enter the two data sets in the same order. The calculator finds both means, multiplies the paired deviations, adds them up and divides by n − 1 for a sample or by N for a whole population. ### Worked example Take X = 2.1, 2.5, 3.6, 4.0 and Y = 8, 10, 12, 14. The means are 3.05 and 11. The deviations of X are −0.95, −0.55, 0.55 and 0.95, and of Y −3, −1, 1 and 3, so the cross-products are 2.85, 0.55, 0.55 and 2.85, adding to 6.8. The sample covariance is 6.8 ÷ 3 = 2.2667 and the population covariance 6.8 ÷ 4 = 1.7. With sx = 0.8963 and sy = 2.582, the correlation is 2.2667 ÷ (0.8963 × 2.582) = 0.9795 – a very strong positive relationship. ### Covariance and portfolio risk For two assets held in proportions w₁ and w₂, the variance of the portfolio return is w₁²σ₁² + w₂²σ₂² + 2w₁w₂·Cov(R₁, R₂). A negative covariance makes the last term negative, which is why mixing assets that do not move together lowers risk without necessarily lowering the expected return. ### Inputs explained FieldWhat to enterData set XA list of numbers separated by commas, spaces or new linesData set Y (same order as X)A list of numbers separated by commas, spaces or new lines ### What each result means - **Sample covariance** = cov([Data set X], [Data set Y (same order as X)]) - **Population covariance** = pcov([Data set X], [Data set Y (same order as X)]) - **Sum of cross-products Σ(x − x̄)(y − ȳ)** = cov([Data set X], [Data set Y (same order as X)])×(count([Data set X]) - 1) - **Mean of X** = mean([Data set X]) - **Mean of Y** = mean([Data set Y (same order as X)]) - **Standard deviation of X (sample)** = stdev([Data set X]) - **Standard deviation of Y (sample)** = stdev([Data set Y (same order as X)]) - **Correlation coefficient r** = corr([Data set X], [Data set Y (same order as X)]) - **Number of pairs** = count([Data set X]) ## Worked example Inputs — Data set X: 2.1, 2.5, 3.6, 4.0; Data set Y (same order as X): 8, 10, 12, 14. Results — **Sample covariance = 2.266667**; Population covariance = 1.7; Sum of cross-products Σ(x − x̄)(y − ȳ) = 6.8; Mean of X = 3.05; Mean of Y = 11; Standard deviation of X (sample) = 0.896289; Standard deviation of Y (sample) = 2.581989; Correlation coefficient r = 0.979457; Direction = Positive – X and Y tend to rise together; Number of pairs = 4. ## Key terms - **Covariance**: The average product of paired deviations from the means. - **Cross-product**: (xᵢ − x̄)(yᵢ − ȳ) for one pair of observations. - **Covariance matrix**: A table of the variances (diagonal) and covariances (off-diagonal) of several variables. ## FAQ **What does a positive covariance mean?** That above-average values of X tend to go with above-average values of Y. Its size alone does not say how strong the link is – use the correlation coefficient for that. **What is the difference between sample and population covariance?** The sample version divides by n − 1 and is an unbiased estimate when your data are a sample; the population version divides by N. With 4 pairs and Σ = 6.8 they are 2.267 and 1.7. **Can covariance be negative?** Yes. Negative covariance means the variables move in opposite directions, like the price of a good and the quantity bought. **How is covariance related to variance?** The covariance of a variable with itself is its variance: Cov(X, X) = Var(X). **How is covariance related to correlation?** r = Cov(X, Y) ÷ (sx · sy). Correlation is covariance rescaled to lie between −1 and +1. **Why do the data sets need the same length?** Covariance pairs each x with one y, so every observation must have both values, in the same order. **How is covariance used in finance?** The covariance of two stocks' returns enters the portfolio variance; combining assets with low covariance lowers the overall risk. ## Sources - [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/) - [OpenStax – Introductory Statistics 2e](https://openstax.org/details/books/introductory-statistics-2e) - [Penn State STAT 500 – Applied Statistics](https://online.stat.psu.edu/stat500/)