Mathematics · Statistics and probability

Sample Covariance

Sample covariance describes whether two variables tend to lie together above or below their means.

AdvancedUniversity/collegeBachelor foundations

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Formula

s_{x y} = \frac{1}{n - 1} i = 1 \sum n (x_{i} - \overset{x}{ˉ}) (y_{i} - \overset{y}{ˉ})

Product of the units of x and y.

Variables & units

Variables & units
Symbol	Meaning	Unit
$xᵢ,yᵢ$	Paired observations	units of x and y
x̄,ȳ	Sample means	units of x and y
$n$	Number of observation pairs, n≥2	dimensionless
$sₓᵧ$	Sample covariance	unit of x · y

LaTeX

s_{xy}=\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar x)(y_i-\bar y)

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Derivation & background

Exam blueprint

Validity range

Derivation steps

Sample covariance describes whether two variables tend to lie together above or below their means.

1Identify the quantities in the equation and check its assumptions: Product of the units of x and y.
2Set up the model for “Sample Covariance” and assign each value to a symbol.
3Work through the example and verify it with an independent relation or limiting-case check: For x=(1,2,3), y=(2,4,6), means are 2 and 4; deviation products are 2,0,2. Thus s_xᵧ=4/2=2. Check: doubling the units of y doubles the covariance.

Rearrangements

Covariance from centered data

s_{xy}=\frac{\sum x_i y_i-n\bar x\bar y}{n-1}

Keep paired observations correctly matched.

Task variant

Rework the example for “Sample Covariance” and explain why the check is valid.

For x=(1,2,3), y=(2,4,6), means are 2 and 4; deviation products are 2,0,2.
Thus s_xᵧ=4/2=2.

Check: doubling the units of y doubles the covariance.

Common mistakes

Ignoring an assumption of the formula: A positive sign means deviations often share the same sign; a negative sign indicates opposing deviations

Check units and model conditions before substitution; then verify the result in the original relation or a limiting case.

Exam context

Joint variation, portfolios, multivariate statistics, and calculation of correlation.

Formula cluster

Statistics and probability

Related relations in the formula sheet: stichprobenvarianz, pearson-korrelationskoeffizient, lineare-regression.

s²=Σ(x_i−x̄)²/(n−1)

Sample Variance

r=s_xᵧ/(s_xsᵧ)

Pearson Correlation Coefficient

ŷ = a + b·x

Linear Regression (Line of Best Fit)

Worked example

For x=(1,2,3), y=(2,4,6), means are 2 and 4; deviation products are 2,0,2. Thus s_xᵧ=4/2=2. Check: doubling the units of y doubles the covariance.

Applications

Joint variation, portfolios, multivariate statistics, and calculation of correlation.

How do you calculate with Sample Covariance?

Here is how to work through a typical Sample Covariance (sₓᵧ=Σ(xᵢ−x̄)(yᵢ−ȳ)/(n−1)) task step by step:

1
Task
Rework the example for “Sample Covariance” and explain why the check is valid.
Solution path
1. For x=(1,2,3), y=(2,4,6), means are 2 and 4; deviation products are 2,0,2.
2. Thus s_xᵧ=4/2=2.
Check: doubling the units of y doubles the covariance.

Frequently asked questions about Sample Covariance

What does the “Sample Covariance” relation describe?+

Sample covariance describes whether two variables tend to lie together above or below their means. The notation s_xᵧ=Σ(x_i−x̄)(y_i−ȳ)/(n−1) shows which quantities are connected. Here, x_i,y_i: Paired observations (units of x and y); x̄,ȳ: Sample means (units of x and y); n: Number of observation pairs, n≥2 (dimensionless). Product of the units of x and y. An application depends on whether the supplied quantities match the definitions and reference directions. Common uses include Joint variation, portfolios, multivariate statistics, and calculation of correlation.. Read the variables and their dimensions before substituting numbers; this often reveals whether the result should be a length, rate, probability, or dimensionless statistic.

How can I solve a “Sample Covariance” problem transparently?+

Start with a diagram or a list of known quantities, assign each one a symbol, and convert units to a consistent form. Then choose the appropriate equation and substitute values only afterward. Example: For x=(1,2,3), y=(2,4,6), means are 2 and 4; deviation products are 2,0,2. Thus s_xᵧ=4/2=2. Check: doubling the units of y doubles the covariance. Verification is part of the calculation: A positive sign means deviations often share the same sign; a negative sign indicates opposing deviations Also compare the magnitude and unit with the context. Round only at the end, since early rounding can affect multi-step calculations, especially when small differences or sums are involved.

What assumptions and limitations apply to “Sample Covariance”?+

A positive sign means deviations often share the same sign; a negative sign indicates opposing deviations. Its magnitude depends on measurement units, so it is not directly comparable across scales. Correlation normalizes it by both standard deviations. The n−1 denominator gives an unbiased estimate under independent, identically distributed sampling. These are not minor footnotes: when assumptions fail, the formula can produce a precise but inappropriate result. Before calculating, identify which values are observed, estimated, or modeled, and whether special cases change the equation. Joint variation, portfolios, multivariate statistics, and calculation of correlation. For numerical methods, also check step size and stability; for statistics, sampling design and distribution; for calculus, differentiability, boundary, and orientation. Use another method when these conditions do not hold.

How is “Sample Covariance” related to Sample Variance?+

Both relations belong to the same topic, but answer different questions. “Sample Covariance”: Sample covariance describes whether two variables tend to lie together above or below their means. The companion relation “Sample Variance” The corrected sample variance measures spread around the sample mean and estimates the population variance. Choose based on the quantity sought and the model: Joint variation, portfolios, multivariate statistics, and calculation of correlation. The worked example For x=(1,2,3), y=(2,4,6), means are 2 and 4; deviation products are 2,0,2. Thus s_xᵧ=4/2=2. Check: doubling the units of y doubles the covariance. shows which quantities actually enter this relation. Compare both formulas using the same data and identify the companion’s extra assumption. This clarifies which relation fits.

Which common reasoning error can undermine a “Sample Covariance” calculation?+

A statistic alone proves neither causation nor a hypothesis. Check sampling, independence, denominator and distribution assumptions before interpreting a test or interval. The symbols must also match their definitions exactly: x_i,y_i: Paired observations (units of x and y); x̄,ȳ: Sample means (units of x and y); n: Number of observation pairs, n≥2 (dimensionless). Do not calculate mechanically; write the assumption beside the setup, then check dimensions and sign. Also note: A positive sign means deviations often share the same sign; a negative sign indicates opposing deviations The example For x=(1,2,3), y=(2,4,6), means are 2 and 4; deviation products are 2,0,2. Thus s_xᵧ=4/2=2. Check: doubling the units of y doubles the covariance. lets you follow these checks directly. If the assumptions do not fit, choose another relation from the chapter instead of forcing values into an unsuitable formula.

Quanta exam set

Curated exam set for "Sample Covariance":

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Which formula describes Sample Covariance?

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Question (front)

How do you rearrange Sample Covariance for Covariance from centered data?

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Question (front)

Which common mistake happens with Sample Covariance?

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Subject references and further reading

This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.

[1]MIT OpenCourseWare, 18.05, Reading 7b: Covariance and Correlation

Related formulas

f(x)=\frac{\exp[-\frac12(x-\mu)^T\Sigma^{-1}(x-\mu)]}{(2\pi)^{d/2}|\Sigma|^{1/2}}

Multivariate Normal Distribution with Covariance Matrix

More formulas: Probability and statistics

P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}

Bayes' Theorem

f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Normal Distribution (Gaussian Bell Curve)

P(X = k) = \binom{n}{k} \cdot p^{k} \cdot (1-p)^{n-k}

Binomial Distribution

E(X) = \mu = \sum_{i} x_{i} \cdot p_{i}

Expected Value

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Symbol

Meaning

Unit

xᵢ,yᵢ

Paired observations

units of x and y

x̄,ȳ

Sample means

units of x and y

n

Number of observation pairs, n≥2

dimensionless

sₓᵧ

Sample covariance

unit of x · y

Derivation & background

Exam blueprint

Validity range

Derivation steps

Sample covariance describes whether two variables tend to lie together above or below their means.

1Identify the quantities in the equation and check its assumptions: Product of the units of x and y.
2Set up the model for “Sample Covariance” and assign each value to a symbol.
3Work through the example and verify it with an independent relation or limiting-case check: For x=(1,2,3), y=(2,4,6), means are 2 and 4; deviation products are 2,0,2. Thus s_xᵧ=4/2=2. Check: doubling the units of y doubles the covariance.

How do you calculate with Sample Covariance?

Here is how to work through a typical Sample Covariance (sₓᵧ=Σ(xᵢ−x̄)(yᵢ−ȳ)/(n−1)) task step by step:

Task

Rework the example for “Sample Covariance” and explain why the check is valid.

Solution path

For x=(1,2,3), y=(2,4,6), means are 2 and 4; deviation products are 2,0,2.
Thus s_xᵧ=4/2=2.

Check: doubling the units of y doubles the covariance.

Frequently asked questions about Sample Covariance

What does the “Sample Covariance” relation describe?+

How can I solve a “Sample Covariance” problem transparently?+

What assumptions and limitations apply to “Sample Covariance”?+

How is “Sample Covariance” related to Sample Variance?+

Which common reasoning error can undermine a “Sample Covariance” calculation?+

Retain Sample Covariance for exams

Create a curated FSRS exam set for sₓᵧ=Σ(xᵢ−x̄)(yᵢ−ȳ)/(n−1): formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential