SPI Mathematics · Statistics

Mean of Raw Data

The mean of a list of numbers is their sum divided by how many numbers there are, so the sum is always the mean times the count.

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Why this matters

Thirteen questions in the bank, ten of them EASY. Most give the mean and ask for a missing value, some ask for the mean of a number pattern such as the first n odd numbers, and the rest ask what happens to the mean when every value is changed in the same way.

Concept 1 of 3: Mean, sum and count

The mean shares the total out equally. If five friends have a mean of 8 sweets each, they have 40 sweets between them, however unevenly they are split. Turn every mean into a sum first and the missing value falls out.

Definition

$\bar{x} = \dfrac{\text{sum of the values}}{\text{number of values}}$ .
Sum $=$ mean $\times$ number of values. This is the step that finds a missing value.
Count every value in the list, unknowns included: 3, x, 5 and y are four values.
With two unknowns and one mean you can find only their sum, such as $x + y$ , not each one.

Mean of n values

\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n} \qquad \sum x = n\bar{x}

$\bar{x}$ the mean
$n$ how many values there are

Worked example

The mean of 4, 9, k, 11 and 6 is 8. Find k.

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The same idea in a practice question:

Practice questionEasy

Example 1 · Statistics · Mean of Raw Data

The mean of 2, 7, 6 and x is 5 and the mean of 18, 1, 6, x and y is 10. What is the value of y?

Dividing by the number of known values

In a list like 5, x, 9, y, 11 there are five values, so the sum is 5 times the mean. Leaving the unknowns out of the count gives a wrong sum.

Looking for x and y separately

One mean gives one equation. With two unknowns it fixes only their sum, so the right option is the one that states

x + y

, not a value of x alone.

Drill 3 more on mean, sum and count

Concept 2 of 3: Mean of a number pattern

When the numbers go up by the same step each time, the mean is simply the middle of the list: the average of the first and last numbers. A few known sums settle the rest.

Definition

Numbers that go up by a fixed step (an A.P.) have mean $= \dfrac{\text{first} + \text{last}}{2}$ .
First n natural numbers 1 to n: mean $\dfrac{n + 1}{2}$ .
First n odd numbers: sum $n^2$ , so mean $n$ .
First n even numbers: sum $n(n + 1)$ , so mean $n + 1$ .
Whole numbers start at 0, natural numbers at 1. The first five whole numbers are 0, 1, 2, 3, 4.
Squares do not go up by a fixed step: add them one by one, or use $1^2 + 2^2 + \dots + n^2 = \frac{n(n + 1)(2n + 1)}{6}$ .

Means of common patterns

\text{mean of } 1, 2, \dots, n = \frac{n + 1}{2} \qquad \text{mean of first } n \text{ odd numbers} = n \qquad \text{mean of first } n \text{ multiples of } k = \frac{k(n + 1)}{2}

Worked example

Find the mean of the first 8 multiples of 5.

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The same idea in a practice question:

Practice questionModerate

Example 2 · Statistics · Mean of Raw Data

The mean of first n odd natural numbers is

\frac{n^2}{81}

, then n =

Starting whole numbers at 1

The first six whole numbers are 0 to 5, not 1 to 6. Starting at 1 gives the answer for natural numbers, and that answer is usually printed as a wrong option.

Taking the sum for the mean

The first n odd numbers add to

n^2

, but their mean is

n^2 \div n = n

. Divide by the count before you compare with the options.

Using first plus last over two for squares

The first and last trick works only when the numbers go up by a fixed step. The squares 1, 4, 9, 16 do not, so add them up instead.

Drill 3 more on mean of a number pattern

Concept 3 of 3: When every value changes the same way

If everyone in a class gets 3 bonus marks, the class average goes up by exactly 3. If every mark is halved, the average is halved. The mean follows whatever you do to every value.

Definition

Add c to every value: the mean goes up by c (not by $nc$ ).
Subtract c from every value: the mean goes down by c.
Multiply every value by k: the mean is multiplied by k. Divide by k: the mean is divided by k.
Do the steps in the order given: divide by x then add y gives $\frac{\bar{x}}{x} + y$ .
Deviations from the mean add to zero: $\sum (x - \bar{x}) = 0$ , for a frequency table too.
Since $\sum x = n\bar{x}$ , the sum less the mean is $n\bar{x} - \bar{x} = (n - 1)\bar{x}$ .

Changing every value

\text{new values } kx + c \;\Rightarrow\; \text{new mean} = k\bar{x} + c \qquad \sum (x - \bar{x}) = 0

Worked example

The mean of a set of numbers is 15. Each number is multiplied by 2 and then 5 is subtracted. Find the new mean.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Statistics · Mean of Raw Data

Mean of a certain number of observations is m. If each observation is divided by x(x

\ne

0) and increased by y, then the mean of new observation is:

Adding n times the change to the mean

Adding 3 to each of n values adds

3n

to the total, but the count is also n, so the mean rises by 3. 'Increases by 3n' is a printed wrong option.

Doing the steps in the wrong order

If the mean is m, dividing every value by x and then adding y gives a new mean of

\frac{m}{x} + y

. Adding first and then dividing gives

\frac{m + y}{x}

, which is a different number.

Drill 4 more on when every value changes the same way

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Mean, sum and count
Mean of n values
$\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n} \qquad \sum x = n\bar{x}$
Mean of a number pattern
Means of common patterns
$\text{mean of } 1, 2, \dots, n = \frac{n + 1}{2} \qquad \text{mean of first } n \text{ odd numbers} = n \qquad \text{mean of first } n \text{ multiples of } k = \frac{k(n + 1)}{2}$
When every value changes the same way
Changing every value
$\text{new values } kx + c \;\Rightarrow\; \text{new mean} = k\bar{x} + c \qquad \sum (x - \bar{x}) = 0$

Watch out for (7)

Dividing by the number of known values
→ Mean, sum and count
Looking for x and y separately
→ Mean, sum and count
Starting whole numbers at 1
→ Mean of a number pattern
Taking the sum for the mean
→ Mean of a number pattern
Using first plus last over two for squares
→ Mean of a number pattern
Adding n times the change to the mean
→ When every value changes the same way
Doing the steps in the wrong order
→ When every value changes the same way

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