Year 11 General Univariate Data Analysis

Summarising Data

20 practice questions 2 video lessons Theory + worked examples

Theory

Mean from a frequency table: $\bar{x} = \frac{\sum f x}{\sum f}$ . For grouped data, use class midpoints in place of $x$ — this gives an estimate of the mean. Median: find position $(n + 1) / 2$ using cumulative frequencies. Standard deviation is the typical distance from the mean; variance is its square. Use sample SD (divide by $n - 1$ ).

This subtopic gathers the formal calculation tools for summary statistics, including mean from a frequency table, mean of grouped data, and the standard deviation.

Mean from a frequency table. If a value $x$ occurs with frequency $f$ , each $x$ contributes $f \cdot x$ to the total. The mean is the grand total of $f x$ divided by the total of $f$ :

$\bar{x} = \frac{\sum f x}{\sum f}$

Median from a frequency table. Find the position $\frac{n + 1}{2}$ where $n = \sum f$ . Use a cumulative frequency column to locate which $x$ -value the median lands on.

Mean of grouped data. Use the midpoint of each class as a representative value:

$midpoint = \frac{lower endpoint + upper endpoint}{2}$

Then apply $\bar{x} = \sum f x / \sum f$ using the midpoints. This gives an estimate, not the exact mean.

Standard deviation $s$ measures the typical distance of values from the mean. The formula is

$s = \sqrt{\frac{\sum (x - \bar{x})^{2}}{n - 1}}$

but in practice you'll use a calculator's statistical functions. Variance is the square of the standard deviation.

The first diagram is a worked frequency table with the $f x$ column, ending in the mean. The second is a decision panel for which summary statistics to use based on the shape of the data.

Each row contributes

f \cdot x

. Sum the

f x

column, divide by

\sum f

Symmetric → mean + SD; skewed or outliers → median + IQR.

The three formulas you need are the mean from frequencies, the class midpoint, and the standard deviation.

Mean from a frequency table

$\bar{x} = \frac{\sum f x}{\sum f}$

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

Class midpoint for grouped data

$midpoint = \frac{lower endpoint + upper endpoint}{2}$

Standard deviation (sample)

$s = \sqrt{\frac{\sum (x - \bar{x})^{2}}{n - 1}}$

s = \sqrt{\frac{\sum (x - \bar{x})^{2}}{n - 1}}

In practice, use your calculator's statistical functions ( $s_{x}$ or $σ_{n - 1}$ for sample standard deviation).

Variance and standard deviation

$variance = s^{2} \Leftrightarrow s = \sqrt{variance}$

Sample vs population. For Year 11 General, almost always use the sample formula (denominator n−1, labelled sx or σn−1). The population formula (denominator n, labelled σx or σn) is only used when you have data on every member of the population.

Mean from a frequency table

Add an $f x$ column: multiply each $x$ by its frequency $f$ .
Total the $f$ column to get $n = \sum f$ and the $f x$ column to get $\sum f x$ .
Divide: $\bar{x} = \frac{\sum f x}{\sum f}$ .

Median from a frequency table

Compute $n = \sum f$ and the median position $\frac{n + 1}{2}$ .
Add a cumulative frequency column to find which $x$ -value contains the median position.
State the median as that $x$ -value.

Mean of grouped data

Compute the midpoint of each class interval.
Multiply each midpoint by the class frequency to get $f x$ .
Apply $\bar{x} = \sum f x / \sum f$ and state the result as an estimate.

Standard deviation by calculator

Enter the data (or the $x$ and $f$ columns from a frequency table) into your calculator's statistics mode.
Read off the sample standard deviation $s_{x}$ (or $σ_{n - 1}$ ).
If asked for variance, square the standard deviation: $variance = s^{2}$ .

EXAMPLE 1 — MEAN FROM FREQUENCY TABLE

For the items-per-order table (

x

1, 2, 3, 4, 5

;

f

5, 12, 18, 10, 5

; total

50

), find the mean.

SOLUTION

Compute the $f x$ column: $5, 24, 54, 40, 25$ . Sum to get $\sum f x = 148$ . Then divide by $\sum f = 50$ .

$\sum f x$	$=$	$148$
$\sum f$	$=$	$50$
$\bar{x}$	$=$	$\frac{148}{50} = 2.96$

Answer: the mean is $2.96$ items per order.

\bar{x} = 2.96

EXAMPLE 2 — MEDIAN FROM FREQUENCY TABLE

20

families' pet counts:

1

pet →

3

;

2 \to 5

;

3 \to 7

;

4 \to 4

;

5 \to 1

. Find the median.

SOLUTION

$n = 20$ (even), so the median is the average of the $10$ th and $11$ th values — equivalently at position $10.5$ .

Cumulative frequencies: after $1$ pet: $3$ ; after $2$ pets: $8$ ; after $3$ pets: $15$ ; after $4$ pets: $19$ ; after $5$ pets: $20$ .

The $10$ th and $11$ th values both fall in the " $3$ pets" group (since cumulative reaches $15$ at $x = 3$ ).

Median

=

3

pets

Answer: the median is $3$ pets.

median = 3

EXAMPLE 3 — GROUPED MEAN

Hours at gym:

0

–

9

→

4

;

10

–

19

→

8

;

20

–

29

→

12

;

30

–

39

→

10

;

40

–

49

→

6

. Estimate the mean.

SOLUTION

Midpoints: $4.5, 14.5, 24.5, 34.5, 44.5$ . Multiply by class frequencies.

$\sum f x$	$=$	$4 (4.5) + 8 (14.5) + 12 (24.5) + 10 (34.5) + 6 (44.5)$
	$=$	$18 + 116 + 294 + 345 + 267$
	$=$	$1040$
$\bar{x}$	$=$	$\frac{1040}{40} = 26 hours$

Answer: the estimated mean is $26$ hours per fortnight (or whatever the unit was).

\bar{x} \approx 26

EXAMPLE 4 — VARIANCE TO STANDARD DEVIATION

A dataset has variance

36

. Find the standard deviation.

SOLUTION

Standard deviation is the positive square root of the variance.

$s$	$=$	$\sqrt{36}$
	$=$	$6$

Answer: the standard deviation is $6$ .

s = 6

Common pitfalls

The denominator is $\sum f$ , not the number of rows. A row with frequency

5

contributes

5

data values, not one. Total your frequencies properly.

Grouped means are estimates. Using midpoints as representative values introduces small errors. State that the grouped-mean answer is approximate (the symbol

\approx

is appropriate).

Sample vs population SD. Use sample SD (divide by

n - 1

, shown as

s_{x}

σ_{n - 1}

on calculators). Population SD divides by

n

and is only used when you really do have data on every member of the population.

Variance is the square of the SD. Don't report variance when asked for SD, or vice versa. If variance

= 36

, SD

= 6

, not

36

Match the summary to the shape. For symmetric data, use mean + SD. For skewed data or data with outliers, use median + IQR. Mean and SD are both pulled by outliers; median and IQR are not.

Frequently asked questions

How do I calculate the mean from a frequency table?

Multiply each value x by its frequency f to get an fx column. Sum the fx column, then divide by the total frequency. The formula is mean equals sigma of fx divided by sigma of f.

How do I find the median from a frequency table?

Find n equals the total frequency. The median is at position (n + 1) divided by 2. Use a cumulative frequency column to find which x-value that position falls in.

How do I estimate the mean of grouped data?

Use the midpoint of each class interval as the representative value. The midpoint is (lower + upper) divided by 2. Then apply mean equals sigma of fx over sigma of f using these midpoints in place of x. The result is an estimate, not the exact mean.

Why is the grouped mean only an estimate?

Grouped data hides the actual values within each interval, so using the midpoint as a stand-in introduces small errors. The estimate is usually close but won't match the true mean from the raw data exactly.

What is the difference between variance and standard deviation?

Standard deviation is the typical distance of values from the mean. Variance is the square of the standard deviation. So if variance equals 36, standard deviation equals the square root of 36, which equals 6.

Should I use the sample or population standard deviation?

Almost always the SAMPLE standard deviation, which divides by n minus 1. On most calculators it appears as s subscript x or sigma subscript n minus 1. Population standard deviation (denoted sigma subscript x or sigma subscript n) is only used when you have data on every member of the population.

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Summarising Data

📖 Theory

Mean from a frequency table

Class midpoint for grouped data

Standard deviation (sample)

Variance and standard deviation

Mean from a frequency table

Median from a frequency table

Mean of grouped data

Standard deviation by calculator

Common pitfalls

Frequently asked questions

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