Covariance Calculator
Online calculator for calculating the covariance of two data series
On this page the covariance of two sorted lists is calculated.
To perform the calculation, enter series of numbers. Then click the 'Calculate' button. The list can be entered unsorted.
Input format
The data can be entered as a series of numbers, separated by semicolons or spaces. You can enter the data as a list (one value per line). Or from a column from Excel spreadsheet by copy & paste
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Covariance is a measure of the linear relationship between two statistical variables.
The covariance can be determined as a sample covariance for a subset, or for the entire set. Different formulas apply for total quantity or sample.
Empirical covariance Formulas
To calculate the covariance of a sample
\(\displaystyle cov(x,y)=\frac{1}{n-1} \left( \sum^n_{i=1} (x_i-\overline{x})(x_i-\overline{y}) \right) \)
Covariance
To calculate the covariance of a total quantity
\(\displaystyle cov(x,y)=\frac{1}{n} \left( \sum^n_{i=1} (x_i-\overline{x})(x_i-\overline{y}) \right) \)
\(n\) Number of data points \(x_i\) Single value of x \(\overline{x}\) Mean of x \(y_i\) Single value of y \(\overline{y}\) Mean of y
Example
In the example we assume that a number of carpenters make a certain number of chairs per day
3 carpenters: 10 chairs
5 carpenters: 16 chairs
7 carpenters: 22 chairs
First, the arithmetic mean is calculated from the number of workers and the number of chairs.
\(\displaystyle 3+4+7=\frac{15}{3}=\color{#44F}{5}\)
\(\displaystyle 10+16+22=\frac{48}{3}=\color{#44F}{16}\)
Calculate covariance:
\(\displaystyle cov(x,y)= ((x_1-\overline{x}) · (y_1-\overline{y})\) \(\displaystyle +(x_2-\overline{x}) · (y_2-\overline{y})\) \(\displaystyle +(x_3-\overline{x}) · (y_3-\overline{y})) \)
\(\displaystyle cov(x,y)= ((3-5) · (10-16)\) \(\displaystyle +(5-5) · (16-16)\) \(\displaystyle +(7-5) · (22-16)) \)
\(\displaystyle = (-2 · -6) +(0 ·0) +(2 · 6) \)
\(\displaystyle = 12 +0 +12 =24 \)
\(\displaystyle = \frac{24}{3}=\color{#44F}{8} \)
In the case of a sample (empirical covariance), divide by \(n-1\) instead of \(n\). In the example above, divide by 2.
More statistics functions
Arithmetic Mean • Contraharmonic Mean • Covariance • Empirical distribution CDF • Deviation • Five-Number Summary • Geometric Mean • Harmonic Mean • Inverse Empirical distribution CDF • Kurtosis • Log Geometric Mean • Lower Quartile • Median • Pooled Standard Deviation • Pooled Variance • Skewness (Statistische Schiefe) • Upper Quartile • Variance
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