Variance, standard deviation and linear variance of a discrete random variable

To calculate variance by subtracting the squared mean from the expected square, which two quantities do you need?

Vary the multiplier in the displayed transformation. The exact probabilities are two ninths, five ninths and two ninths; their table decimals are rounded. Use the exact fractions, not the rounded entries, to calculate moments. This table tracks original outcomes, not a sorted distribution of distinct Y values. At a equal to zero all rows map to 5 and their probabilities combine to 1. Compare a multiplier and its negative: the means differ when a is nonzero, but their variances and standard deviations agree

Why is the standard deviation non-negative for a negative multiplier?

Variance and standard deviation

Variance measures the spread of a discrete random variable in squared units. Standard deviation is the principal square root of variance, so it has the same units as the variable

1. 1Square before weighting Square each value before multiplying it by its probability \(\displaystyle (-3)^2\times\frac{1}{4}=\frac{9}{4}\)
2. 2Find the two moments Calculate E(X^2) and E(X) as separate probability-weighted sums \(\displaystyle \operatorname{E}(X^2)=(-2)^2\left(\frac{1}{4}\right)+4^2\left(\frac{1}{4}\right)=5\)
3. 3Subtract the squared mean Subtract the square of E(X) from E(X^2) \(\displaystyle \operatorname{Var}(X)=5-\left(\frac{1}{2}\right)^2=\frac{19}{4}\)
4. 4Take the principal root Use the non-negative square root of the variance for standard deviation \(\displaystyle \operatorname{SD}(X)=\sqrt{\frac{19}{4}}=\frac{\sqrt{19}}{2}\)

Complete distribution and mean

\(\displaystyle \begin{array}{c|ccc}x&-2&0&4\\\Pr(X=x)&1/4&1/2&1/4\end{array},\quad\operatorname{E}(X)=\frac12\)

For X = -2, 0 and 4 with probabilities 1/4, 1/2 and 1/4, calculate E(X^2)

Complete distribution and mean

\(\displaystyle \begin{array}{c|ccc}x&-2&0&4\\\Pr(X=x)&1/4&1/2&1/4\end{array},\quad\operatorname{E}(X)=\frac12\)

Using E(X^2) = 5 and E(X) = 1/2, calculate Var(X)

Squaring after taking the expectation. One of these lines is right. Which one?

Find two different weighted sums

A score model has values −2,1,4 with probabilities 2/9,5/9,2/9. Calculate each squared-value contribution. Squaring happens before the probability weight is applied; it is not the square of a weighted contribution

Distribution and requested quantity

\(\displaystyle (-2)^2\left(\frac29\right)\)

Find the squared-value contribution for X=−2.

Distribution and requested quantity

\(\displaystyle 1^2\left(\frac59\right)\)

Find the squared-value contribution for X=1.

Distribution and requested quantity

\(\displaystyle 4^2\left(\frac29\right)\)

Find the squared-value contribution for X=4.

Distribution and requested quantity

\(\displaystyle \frac89+\frac59+\frac{32}{9}\)

Add the squared-value contributions to find E(X²)

Distribution and requested quantity

\(\displaystyle (-2)\left(\frac29\right)+1\left(\frac59\right)+4\left(\frac29\right)\)

Calculate E(X) from the original values, not their squares

Distribution and requested quantity

\(\displaystyle \operatorname{E}(X^2)=5,\quad\operatorname{E}(X)=1\)

Calculate Var(X) from the two moments

Check variance using squared distances from the mean

The mean is 1. The deviations are −3,0,3, so their weighted squares are 2,0,2. Their sum agrees with the moment formula. Squaring prevents signed deviations from cancelling

Distribution and requested quantity

\(\displaystyle \operatorname{Var}(X)=4\)

Find the standard deviation of this score model

If X is measured in points, which units belong to its variance and standard deviation?

Square the multiplier for variance

Define Y=−(3/2)X+4 for the same model. The constant moves every value equally and does not change distances from the transformed mean. The negative multiplier reverses the order of values but does not make the spread negative

Distribution and requested quantity

\(\displaystyle \operatorname{Var}(X)=4,\quad Y=-\frac32X+4\)

Find Var(Y) for Y=−(3/2)X+4.

Distribution and requested quantity

\(\displaystyle \operatorname{SD}(X)=2,\quad Y=-\frac32X+4\)

Find SD(Y), retaining the non-negative convention

Check the transformed deviations directly

The original values −2,1,4 map to 7,5/2,−2. The transformed mean is 5/2. Its deviations are 9/2,0,−9/2; weighting their squares gives 9. This checks the variance rule without treating the constant as extra spread

Keep variance exact before taking a rounded root

The preceding expected-value model used scores −3,2,7 with probabilities 1/6,1/2,1/3. Its mean is 17/6 and expected square is 119/6. Calculate variance exactly before rounding standard deviation

Distribution and requested quantity

\(\displaystyle \operatorname{E}(X^2)=\frac{119}{6},\quad\operatorname{E}(X)=\frac{17}{6}\)

Find the exact variance of this second model

Exact variance

\(\displaystyle \operatorname{Var}(X)=\frac{425}{36}\)

Give the standard deviation correct to two decimal places

Check the rounding interval without rounding the variance early

The squared half-unit boundaries enclose the exact variance, so its principal root rounds to 3.44 at two decimal places. The exact standard deviation remains 5√17/6; the decimal is an approximation

Do not apply a variance-additivity rule without its premise

Return to the first model with Var(X)=4 and let Z=2X. Then X+Z=3X, so its variance is 9Var(X), not Var(X)+Var(Z). Expectation additivity did not need independence; a variance-additivity statement needs an appropriate covariance condition

Distribution and requested quantity

\(\displaystyle \operatorname{Var}(X)=4,\quad X+Z=3X\)

Find Var(X+Z) when Z=2X

Write a complete variance and standard-deviation calculation, ungraded

A new score model measured in points has the distribution shown below. Construct a column of each value times its probability and a column of each squared value times its probability, calculate the mean and expected square, and check variance using weighted squared deviations. Give the exact standard deviation of the original variable, then the variance and standard deviation after multiplying it by negative two and adding one; include units and explain the effect of the negative multiplier. The whole calculation is self-reviewed rather than graded by the scalar checks

Compare both variance methods and the transformed spread

For values −4, 2 and 8 with respective probabilities one quarter, one half and one quarter, the mean is 2 and the expected square is 22, giving variance 18. The weighted squared deviations are 9,0,9, which also total 18. The negative multiplier reverses the order of values; the constant has no effect on spread

Check the order of operations and the requested quantity

Square each value before weighting for E(X²), then subtract the square of the mean. Use the principal root for standard deviation, a² for transformed variance and |a| for transformed standard deviation. Keep exact intermediate quantities until the requested final rounding