AP®StatisticsCollege BoardStudy GuidesUnit 1: Exploring One-Variable Data and Collecting DataSummary StatisticsTables & Relative Frequency

Tables & Relative Frequency (College Board AP® Statistics): Study Guide

Syllabus Edition

First teaching 2026

First exams 2027

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on 21 May 2026

Frequency tables

How are frequency tables used for ungrouped data?

Frequency tables can be used for ungrouped data when you have lots of the same values within a data set
- They can be used to collect and present data easily
If a particular value has a frequency of 3 this means that there are three of that value in the data set
For example, the number of pets owned by a group of individuals
- could be presented as a list, e.g. 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3 , 3, 3, 3, 3
- or alternatively, in a frequency table

Number of pets	0	1	2	3
Frequency	3	5	4	6

How are measures of center calculated from frequency tables with ungrouped data?

The mode is the data value that has the highest frequency
The median is the middle value of the data when put in order
- Use cumulative frequencies (running totals) to find the median
  - The median is the data value that is halfway through the total frequency
The mean can be calculated by
- multiplying each value $x_{i}$ by its frequency $f_{i}$
- summing these together to get $\sum_{} f_{i} x_{i}$
- then dividing by the total frequency $n = \sum_{} f_{i}$ = Σf_i
- The formula, $\bar{x} = \frac{1}{n} \sum_{i = 1}^{n} f_{i} x_{i}$ , is not given in the exam
Your calculator can calculate these statistical measures by inputting the values and their frequencies into your calculator and calculating one-variable statistics

How are measures of variability calculated from frequency tables with ungrouped data?

The range is the largest value of the data minus the smallest value of the data
- It is not the largest frequency minus the smallest frequency
The interquartile range, IQR, is the third quartile minus the first quartile, $IQR = Q 3 - Q 1$
- The quartiles can be found by listing out the data and calculating by hand
  - or by using that Q1 is the data value that is a quarter of the way through the total frequency, etc.
- or by inputting the values and their frequencies into your calculator and calculating one-variable statistics
The standard deviation and variance can also be calculated by hand using the formulas after listing out the values
- or by inputting the values and their frequencies into your calculator and calculating one-variable statistics

Worked Example

The table shows data for the shoe sizes of students in class 11A.

Shoe size	Frequency
6	1
6.5	1
7	3
7.5	2
8	4
9	6
10	11
11	2
12	1

(a) Find the mean shoe size for the class, giving your answer to 3 significant figures.

(b) Find the median shoe size.

Answer:

(a)

It helps to label shoe size (x) and frequency (f)
Add an extra column and calculate the values of 'shoe size × frequency', (xf)
Find the total frequency and total xf value

Shoe size (x)	Frequency (f)	xf
6	1	6 × 1 = 6
6.5	1	6.5 × 1 = 6.5
7	3	7 × 3 = 21
7.5	2	7.5 × 2 = 15
8	4	8 × 4 = 32
9	6	9 × 6 = 54
10	11	10 × 11 = 110
11	2	11 × 2 = 22
12	1	12 × 1 = 12
	Total = 31	Total = 278.5

Use the formula that the mean is the total of the xf column divided by the total frequency

Mean $= \frac{278.5}{31} = 8.983 870 . . .$

Give your final answer to 3 significant figures

The mean shoe size is 8.98 (3 s.f.)

(b)

The median is the ${(\frac{n + 1}{2})}^{t h}$ value where $n$ is the total frequency

$\frac{n + 1}{2} = \frac{31 + 1}{2} = \frac{32}{2} = 16$

The median is the 16^th value
There are 1 + 1 + 3 + 2 + 4 = 11 values in the first five rows of the table
There are 11 + 6 = 17 values in the first six rows of the table
Therefore the 16^th value must be in the sixth row

The median shoe size is 9

(c)

The first quartile is the median of the lower half
Therefore, it is the midpoint of the 8^th and 9^th values

8^th and 9^th values are both 8

First quartile is 8

The third quartile is the median of the upper half
Therefore, it is the midpoint of the 23^rd and 24^th values

23^rd and 24^th values are both 11

Third quartile is 11

Find the difference between the quartiles

11 - 8 = 3

The interquartile range of shoe sizes is 3

How are frequency tables used for grouped data?

Frequency tables can be used for grouped data when you have lots of values within the same interval
- Class intervals will be written using inequalities and without gaps
  - $10 \leq x < 20$ and $20 \leq x < 30$
If a particular class interval has a frequency of 3 this means that there are three data items in that class interval
- You do not know the exact data values when you are given grouped data
For example, the heights of students within a class
- could be presented in a grouped frequency table

Height, $h$	Frequency
$150 \leq h < 155$	1
$155 \leq h < 160$	3
$160 \leq h < 165$	5
$165 \leq h < 170$	4
$170 \leq h < 175$	2

Relative frequency tables

What is a relative frequency table for ungrouped data?

A relative frequency table gives the proportion of data items falling into each category
- This may be presented as a decimal, fraction or percentage

If a particular value has a relative frequency of 0.4 this means that 0.4 or 40% of the items in the data set have that value
- We do not know the exact number of data items in each group
- However, we can calculate this if we are also told the total frequency

For example, the ungrouped relative frequency table below shows the number of siblings that a particular group of individuals have
- 30% of the group have no siblings
- 50% of the group have 1 sibling
- 15% of the group have 2 siblings
- 5% of the group have 3 siblings

Number of siblings	0	1	2	3
Relative frequency	0.3	0.5	0.15	0.05

What is a relative frequency table for grouped data?

The same idea of relative frequency can be applied to grouped data
- If a particular class interval has a relative frequency of $\frac{7}{10}$ this means that $\frac{7}{10}$ or 70% of the items in the data set have a value within that class interval
  - We do not know the exact number of data items in each group
  - However, we can calculate this if we are also told the total frequency

For example, the grouped relative frequency table below shows the number of weights of a sample of cats
- 8% of the group weigh between 1 kg and 2 kg
- 42% of the group weigh between 2 kg and 3 kg
- 31% of the group weigh between 3 kg and 4 kg
- 19% of the group weigh between 4 kg and 5 kg

Weight of cat, $w$ (kg)	$1 < w \leq 2$	$2 < w \leq 3$	$3 < w \leq 4$	$4 < w \leq 5$
Relative frequency	$\frac{8}{100}$	$\frac{42}{100}$	$\frac{31}{100}$	$\frac{19}{100}$

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