Sample covariance is one of 2 association formulas in the statistics section of this library, and it is used at ap · university level.
Why sample covariance works
When both variables sit above their own means, or both sit below, the two deviations share a sign and their product is positive. When one is high while the other is low, the product is negative. Adding those products measures which pattern dominates, and the n minus 1 matches the sample variance correction.
What each symbol means
Paired values are $(x_i,y_i)$.
Sample covariance: when it holds
$n>1$; covariance depends on measurement units.
When it stops applying
It only detects straight-line co-movement. For x values -2, -1, 0, 1, 2 paired with their squares, the covariance is exactly 0 even though y is completely determined by x, because the rising half and the falling half cancel out.
Sample covariance: a worked example
Positive covariance means larger $x$ values tend to occur with larger $y$ values.
The mistake to avoid
What people do: Comparing covariances from different data sets to judge which relationship is stronger.
Why it goes wrong: Covariance carries the units of both variables multiplied together. Recording the same heights in centimetres instead of metres multiplies the covariance by 100 without changing the relationship one bit.
Do this instead: Standardise before comparing: divide by both standard deviations to get the correlation, which has no units and a fixed scale.
Sample covariance: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. Paired values are $(x_i,y_i)$.
- Check the conditions before substituting. $n>1$; covariance depends on measurement units.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most statistics slips.
Where this formula fits
- Subject
- Statistics formulas — 21 entries in this library
- Topic
- Association
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where sample covariance comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Linear Regression — the lesson behind this formula: fit and interpret a linear relationship honestly.
- Statistics Calculator — check your substitution and the value it produces.
- Study statistics — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 21 statistics formulas — the full grouped reference, or the complete formula library.
Questions about sample covariance
What counts as a large covariance?
There is no answer, because the size depends entirely on the units. Only the sign is interpretable on its own, which is why correlation exists as the standardised version.
What is the covariance of a variable with itself?
It is the variance. Both deviations become the same number, so each product is a square, and the formula collapses into the sample variance formula exactly.
Why divide by n minus 1 here as well?
For the same reason as in the variance: two means were estimated from the same data, so dividing by n would bias the result toward zero. The correction restores an unbiased estimate.
Does zero covariance prove the two variables are independent?
No. The squared example above has zero covariance and a perfect relationship. Zero covariance only rules out a linear trend, not any relationship at all.