Variance and standard deviation | CAIE 9709 Probability & Statistics 1 Notes

Probability & Statistics 1

Mathematics9709

The same mean can have different spread

A distribution concentrated near its mean has small variance. Moving probability farther away increases variance, even if the mean stays unchanged.

Keep E(X) = 2. Move the probability mass.

0.5

01234

E(X) = 2

Var(X) = 1

The probability mass is one unit from the mean.

The mean can stay fixed while the variance changes. Variance increases when more probability lies farther from the mean.

Variance is never negative.
Variance is zero only when the variable always has one value.
Variance has squared units. Standard deviation returns to the original units.

Find the mean of the squares, then subtract

\operatorname{Var}(X)=\sum x^2p-[E(X)]^2

Symbols

$x²$ = square of a possible value (squared context unit)
$E(X)$ = expectation found first (same as X)
$Var(X)$ = variance of X (squared context unit)

Calculate $E(X)=\sum xp$ and keep full accuracy.
Calculate $\sum x^2p$ . Square the values, not the probabilities.
Subtract $[E(X)]^2$ once, at the end.

Worked example

Continue the formula-defined distribution

Use the values $-2,-1,0,2,3$ with probabilities $6/28,3/28,2/28,6/28,11/28$ . (9709/51/O/N/24 Q2)

The weighted-value numerator is $-12-3+0+12+33=30$ .

E(X)=\frac{30}{28}=\frac{15}{14}

After squaring the values, the weighted numerator is $24+3+0+24+99=150$ .

\sum x^2p=\frac{150}{28}=\frac{75}{14}

\operatorname{Var}(X)=\frac{75}{14}-\left(\frac{15}{14}\right)^2

Answer

\operatorname{Var}(X)=\frac{825}{196}=4.21\text{ (3 s.f.)}

Common mistake

Variance is not

\sum x^2p

alone, and it is not

\sum xp^2

. Keep the two weighted sums separate.

Use STAT mode for speed, then show the method

For a completed distribution table, one-variable STAT mode is usually the fastest and safest calculation method.

Enter the possible values as the data list.
Enter the matching probabilities as frequency weights. If your calculator requires whole-number frequencies, multiply all probabilities by the same common denominator first.
Read the population mean $\bar x$ for $E(X)$ and the population standard deviation $\sigma_x$ .
Square $\sigma_x$ to obtain the variance. Do not use the sample value $s_x$ .

Worked example

Turn probabilities into whole-number weights

Probabilities $1/12,3/12,5/12,3/12$ can be entered as frequencies 1, 3, 5 and 3. Multiplying every weight by 12 does not change the weighted mean or population variance.

Examiner note

Let the calculator do the arithmetic, but write

E(X)=\sum xp

and

\operatorname{Var}(X)=\sum x^2p-[E(X)]^2

with substituted values. Current mark schemes award method marks for those steps, not for an unexplained calculator result.

Your turnIndependent check [4 marks]

$X=0,1,2$ has probabilities 0.25, 0.50 and 0.25. Find $E(X)$ , $\operatorname{Var}(X)$ and the standard deviation.

Show worked answer

Use frequency weights 1, 2 and 1, or calculate $E(X)=1$ and $\sum x^2p=1.5$ directly.

Answer

E(X)=1

Answer

\operatorname{Var}(X)=0.5

Answer

\operatorname{SD}(X)=0.707\text{ (3 s.f.)}

The same mean can have different spread

A distribution concentrated near its mean has small variance. Moving probability farther away increases variance, even if the mean stays unchanged.

Keep E(X) = 2. Move the probability mass.

0.5

01234

E(X) = 2

Var(X) = 1

The probability mass is one unit from the mean.

The mean can stay fixed while the variance changes. Variance increases when more probability lies farther from the mean.

Variance is never negative.
Variance is zero only when the variable always has one value.
Variance has squared units. Standard deviation returns to the original units.

Find the mean of the squares, then subtract

\operatorname{Var}(X)=\sum x^2p-[E(X)]^2

Symbols

$x²$ = square of a possible value (squared context unit)
$E(X)$ = expectation found first (same as X)
$Var(X)$ = variance of X (squared context unit)

Calculate $E(X)=\sum xp$ and keep full accuracy.
Calculate $\sum x^2p$ . Square the values, not the probabilities.
Subtract $[E(X)]^2$ once, at the end.

Worked example

Continue the formula-defined distribution

Use the values $-2,-1,0,2,3$ with probabilities $6/28,3/28,2/28,6/28,11/28$ . (9709/51/O/N/24 Q2)

The weighted-value numerator is $-12-3+0+12+33=30$ .

E(X)=\frac{30}{28}=\frac{15}{14}

After squaring the values, the weighted numerator is $24+3+0+24+99=150$ .

\sum x^2p=\frac{150}{28}=\frac{75}{14}

\operatorname{Var}(X)=\frac{75}{14}-\left(\frac{15}{14}\right)^2

Answer

\operatorname{Var}(X)=\frac{825}{196}=4.21\text{ (3 s.f.)}

Common mistake

Variance is not

\sum x^2p

alone, and it is not

\sum xp^2

. Keep the two weighted sums separate.

Use STAT mode for speed, then show the method

For a completed distribution table, one-variable STAT mode is usually the fastest and safest calculation method.

Enter the possible values as the data list.
Enter the matching probabilities as frequency weights. If your calculator requires whole-number frequencies, multiply all probabilities by the same common denominator first.
Read the population mean $\bar x$ for $E(X)$ and the population standard deviation $\sigma_x$ .
Square $\sigma_x$ to obtain the variance. Do not use the sample value $s_x$ .

Worked example

Turn probabilities into whole-number weights

Probabilities $1/12,3/12,5/12,3/12$ can be entered as frequencies 1, 3, 5 and 3. Multiplying every weight by 12 does not change the weighted mean or population variance.

Examiner note

Let the calculator do the arithmetic, but write

E(X)=\sum xp

and

\operatorname{Var}(X)=\sum x^2p-[E(X)]^2

with substituted values. Current mark schemes award method marks for those steps, not for an unexplained calculator result.

Your turnIndependent check [4 marks]

$X=0,1,2$ has probabilities 0.25, 0.50 and 0.25. Find $E(X)$ , $\operatorname{Var}(X)$ and the standard deviation.

Show worked answer

Use frequency weights 1, 2 and 1, or calculate $E(X)=1$ and $\sum x^2p=1.5$ directly.

Answer

E(X)=1

Answer

\operatorname{Var}(X)=0.5

Answer

\operatorname{SD}(X)=0.707\text{ (3 s.f.)}