Several observations versus one value multiplied

By the end of this lesson

By the end you can tell the difference between "the total of $n$ separate items" and " $n$ times one item", find the mean and variance of each, and find the mean and variance of totals made of several different parts.

Key to the symbols

Symbol	Meaning
$X$	one observation, such as the mass of one bag of flour
$\mu$ , $\sigma^2$	the mean and variance of $X$
$X_1, X_2, \dots, X_n$	$n$ separate observations of $X$ (the first bag, the second bag, ...)
$n$	how many observations there are
$nX$	one observation multiplied by $n$

The small numbers below the $X$ (called subscripts) just label which bag is which. Each $X_i$ has the same distribution as $X$ , and they are independent of each other.

Two bags, or one bag doubled?

A bag of flour has mass $X$ kg with mean 5 and variance 0.04. You buy two bags.

The total of two bags is $X_1 + X_2$ . These are two different bags, each with its own random error. Using the two-variable rule with $a = b = 1$ :

$\text{Var}(X_1 + X_2) = 0.04 + 0.04 = 0.08$

One bag weighed and doubled is $2X$ . This is a single random mass multiplied by 2. Using the one-variable rule:

$\text{Var}(2X) = 2^2 \times 0.04 = 0.16$

Both have mean $10$ kg. But $2X$ is twice as variable as $X_1 + X_2$ .

Why separate items vary less

With two separate bags, one is often a little heavy while the other is a little light, so their errors partly cancel. With $2X$ there is only one error, and doubling the bag doubles that error too.

You can see it with a tiny example. Let $X$ be $0$ or $2$ , each with probability $\frac{1}{2}$ , so $\text{Var}(X) = 1$ .

$2X$ is $0$ or $4$ , each with probability $\frac{1}{2}$ . It is always 2 away from its mean of 2, so $\text{Var}(2X) = 4$ .
$X_1 + X_2$ is $0$ , $2$ , $2$ or $4$ , each with probability $\frac{1}{4}$ . Its squared distances from 2 are $4, 0, 0, 4$ , so $\text{Var}(X_1 + X_2) = \frac{8}{4} = 2$ .

So $\text{Var}(X_1 + X_2) = 2 = 1 + 1$ , but $\text{Var}(2X) = 4 = 2^2 \times 1$ .

Definition

Totals of n observations

If $X_1, X_2, \dots, X_n$ are independent observations, each with mean $\mu$ and variance $\sigma^2$ :

$E(X_1 + X_2 + \cdots + X_n) = n\mu \qquad \text{Var}(X_1 + X_2 + \cdots + X_n) = n\sigma^2$

For one observation multiplied by $n$ :

$E(nX) = n\mu \qquad \text{Var}(nX) = n^2\sigma^2$

How to tell which one a question means:

"the total mass of 5 bags", "the sum of 3 readings" → separate items → $n\sigma^2$ ;
"the cost of a bag at £2 per kg", "3 times the length", "a recipe that uses double the amount" → one value scaled → $n^2\sigma^2$ (or $a^2\sigma^2$ for a multiplier $a$ ).

Worked example: Example 1: people in a lift

The mass of an adult is a random variable with mean 75 kg and standard deviation 10 kg. Four adults, chosen independently, get into a lift. Find the mean and standard deviation of their total mass.

Have a go first?

Check 1

CalculateQuestion 1 · Recall

Multiple choiceQuestion 2 · Apply

Answer the 2 questions above to continue.