AFCAT Numerical Ability · Statistics

The Mean, and How Far the Data Spread

Find the mean of a list or a frequency table, use how the mean moves when the data change, join the means of groups, and measure spread with the standard deviation.

All Statistics notes

Why this matters

The mean is the total divided by the count, and nearly every question here is solved by working with the total rather than the mean. Marks are lost by averaging two group means without weighting them by size, and by forgetting that a wrongly copied value changes the total but not the count.

Concept 1 of 4: Mean of a list and of a frequency table

The mean shares the total out equally: add the values and divide by how many there are. In a frequency table each value counts as many times as its frequency, so multiply each value by its frequency before adding.

Definition

A list: $\bar x = \dfrac{\sum x}{n}$ .
A frequency table: $\bar x = \dfrac{\sum f x}{\sum f}$ . Divide by the total frequency, not by the number of rows.
Grouped data: use the mid-point of each class as its value.
A missing frequency: the total frequency gives one equation and the mean gives another. Solve the two together.
Evenly spaced values such as $x, x + 3, x + 6$ have their mean at the middle value.
Values $y$ and $\tfrac 1y$ : work with their sum, using $y^2 + \tfrac{1}{y^2} = \left(y + \tfrac 1y\right)^2 - 2$ and $y^3 + \tfrac{1}{y^3} = \left(y + \tfrac 1y\right)^3 - 3\left(y + \tfrac 1y\right)$ .

Mean of a frequency table

\bar x = \dfrac{\sum f x}{\sum f}

Worked example

The values 1, 2, 3 and 4 occur with frequencies 3, 5, 8 and 4. Find the mean.

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Example 1 · Statistics · Mean and Standard Deviation

X	0	1	2	3	4
Frequency	4	$f$	9	$g$	4

The table above gives the frequency distribution of a discrete variable

X

with two missing frequencies. If the total frequency is 25 and the arithmetic mean of

X

is 2, then what is the value of the missing frequency

f

Divide by the total frequency

For the values 1 to 4 with frequencies 3, 5, 8 and 4, divide

\sum fx

by 20, the total frequency. Dividing by 4, the number of different values, gives a mean five times too big.

A class counts at its mid-point

The class 10-20 enters the sum as 15. Using 10 or 20 moves every class the same way and shifts the mean by half a class width.

Drill 3 more on mean of a list and of a frequency table

Concept 2 of 4: How the mean moves when the data change

Because the mean is the total over the count, any change to the data is a change to the total. Add the same number to every value and the mean moves by that number. Correct one wrongly copied value and the total moves by the error while the count stays the same.

Definition

Adding $k$ to every value adds $k$ to the mean. Multiplying every value by $k$ multiplies the mean by $k$ .
A value $b$ copied as $a$ : the correct total is $n\bar x - a + b$ . Divide by the same $n$ .
The deviations from the mean add to zero: $\sum f(x - \bar x) = 0$ .
Bounds: if a new value can be no lower than 0, the new mean of $n + 1$ values is at least the old total divided by $n + 1$ .

Correcting a wrong value

\bar x_{\text{correct}} = \dfrac{n \bar x - a + b}{n}

Worked example

The mean of 40 observations is 35. One value, 62, was copied as 26. Find the correct mean.

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The same idea in a practice question:

Practice questionEasy

Example 2 · Statistics · Mean and Standard Deviation

Average score of 50 students in a class is 44. Later on it was found that the score 23 was incorrectly recorded as 73. The correct average score is

A wrong value does not change the count

Fixing a miscopied value changes the total only. Divide the corrected total by the same number of observations, not by one fewer.

Adding and multiplying act differently

Adding 3 to every value adds 3 to the mean; multiplying every value by 3 triples it. Applying the adding rule to a multiplication, or the other way round, is the usual slip.

Drill 3 more on how the mean moves when the data change

Concept 3 of 4: Combined mean of groups

Two groups' means cannot simply be averaged unless the groups are the same size. Turn each mean back into a total, add the totals, and divide by the number in all the groups together.

Definition

$\bar x = \dfrac{n_1 \bar x_1 + n_2 \bar x_2 + \dots}{n_1 + n_2 + \dots}$ .
A missing group mean: the total of everyone, less the totals of the known groups, divided by the size of the missing group.
The combined mean lies between the group means, nearer the mean of the larger group.

Combined mean

\bar x = \dfrac{n_1 \bar x_1 + n_2 \bar x_2}{n_1 + n_2}

Worked example

A class has 24 boys averaging 62 marks and 16 girls averaging 72 marks. Find the class average.

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The same idea in a practice question:

Practice questionEasy

Example 3 · Statistics · Mean and Standard Deviation

In a class of 50 students, 10 have failed and their average marks are 28. The total marks obtained by the entire class are 2800. The average marks of those who have passed are

Do not average the averages

A group of 10 with mean 20 and a group of 30 with mean 40 combine to

\tfrac{200 + 1200}{40} = 35

, not 30. The larger group pulls the mean its way.

Drill 3 more on combined mean of groups

Concept 4 of 4: Spread: range, variance and standard deviation

Two sets can share a mean and still spread very differently. The range looks only at the two ends; the standard deviation uses how far every value sits from the mean.

Definition

Range is the highest value minus the lowest.
Variance is the mean of the squared deviations, $\sigma^2 = \dfrac{\sum (x - \bar x)^2}{n}$ . The standard deviation $\sigma$ is its square root.
Shortcut: $\sigma^2 = \dfrac{\sum x^2}{n} - \bar x^2$ .
Adding $k$ to every value leaves $\sigma$ unchanged. Multiplying every value by $k$ multiplies $\sigma$ by $|k|$ .

Standard deviation

\sigma = \sqrt{\dfrac{\sum (x - \bar x)^2}{n}}

Worked example

Find the standard deviation of 2, 4, 4, 4, 5, 5, 7 and 9.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 4 · Statistics · Mean and Standard Deviation

The standard deviation of 7, 9, 11, 13, 15 is

Adding a constant does not change the spread

Adding 10 to every value moves the mean by 10, but every distance from the mean stays the same, so

\sigma

does not change.

Take the square root at the end

A variance of 20 is not the standard deviation.

\sigma = \sqrt{20} \approx 4.47

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 (4)

Mean of a list and of a frequency table
Mean of a frequency table
$\bar x = \dfrac{\sum f x}{\sum f}$
How the mean moves when the data change
Correcting a wrong value
$\bar x_{\text{correct}} = \dfrac{n \bar x - a + b}{n}$
Combined mean of groups
Combined mean
$\bar x = \dfrac{n_1 \bar x_1 + n_2 \bar x_2}{n_1 + n_2}$
Spread: range, variance and standard deviation
Standard deviation
$\sigma = \sqrt{\dfrac{\sum (x - \bar x)^2}{n}}$

Watch out for (7)

Divide by the total frequency
→ Mean of a list and of a frequency table
A class counts at its mid-point
→ Mean of a list and of a frequency table
A wrong value does not change the count
→ How the mean moves when the data change
Adding and multiplying act differently
→ How the mean moves when the data change
Do not average the averages
→ Combined mean of groups
Adding a constant does not change the spread
→ Spread: range, variance and standard deviation
Take the square root at the end
→ Spread: range, variance and standard deviation

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