Covariance Calculator
Statistics, Data Analysis
Intro: Computes the sample covariance between two variables using the standard (n−1) denominator, showing all intermediate sums.
Worked example
- Compute the sample covariance for X: 2, 4, 5, 7 and Y: 3, 5, 6, 10.
- We use the sample covariance formula $\operatorname{Cov}(X,Y)=\dfrac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})$.
- First find the sample means. For X: $\bar{x}=(2+4+5+7)/4 = 18/4 = 4.5$. For Y: $\bar{y}=(3+5+6+10)/4 = 24/4 = 6$.
- Construct a table with rows for each pair $(x_i, y_i)$ and columns $(x_i-\bar{x})$, $(y_i-\bar{y})$, and their product.
- For i=1: $(2,3)$: $x_1-\bar{x}=2-4.5=-2.5$, $y_1-\bar{y}=3-6=-3$, product $(-2.5)(-3)=7.5$.
- For i=2: $(4,5)$: $x_2-\bar{x}=4-4.5=-0.5$, $y_2-\bar{y}=5-6=-1$, product $(-0.5)(-1)=0.5$.
- For i=3: $(5,6)$: $x_3-\bar{x}=5-4.5=0.5$, $y_3-\bar{y}=6-6=0$, product $(0.5)(0)=0$.
- For i=4: $(7,10)$: $x_4-\bar{x}=7-4.5=2.5$, $y_4-\bar{y}=10-6=4$, product $(2.5)(4)=10$.
- Sum of products: $7.5+0.5+0+10 = 18$.
- Sample size is $n=4$, so $n-1=3$. Covariance is $\operatorname{Cov}(X,Y)=18/3 = 6$.
- Answer: $\boxed{\operatorname{Cov}(X,Y)=6}$, indicating a positive linear association.
FAQs
Is this sample or population covariance?
By default we compute sample covariance with denominator n−1, which is standard in statistics.
Why choose MathGPT?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
How this calculator works
- Type or paste your function (LaTeX like
\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
- Review each step to understand which rule was applied and why.
- Practice with similar problems to lock in the method.