Frequency tables

Statistics and probability, grade 10 (4 units) Lesson 2 of 20Learning

Frequency tables

From a list to a frequency table

A frequency table sums up a long list: one row holds the values of the variable, and under each value its frequency – how many times it appears in the list.

20 students were asked how many books they read in the summer holidays. The answers, as a list:
2, 0, 3, 1, 2, 4, 2, 1, 0, 2, 3, 2, 1, 5, 2, 3, 1, 2, 4, 0

Number of books	0	1	2	3	4	5	Total
Frequency	3	4	7	3	2	1	20

Each value is counted once, and crossed off in the list so as not to count it twice.
The frequencies add up to the number of data: $\displaystyle {3+4+7+3+2+1}={20}$ .
In the table you see at once that the commonest answer is 2 books; in the list you have to search.

Relative frequency

The relative frequency of a value is its frequency divided by the number of all the data – what part of the data it is. It is written as a fraction, a decimal or a percent.

\displaystyle \frac{f}{n}

$\displaystyle f$ – the value’s frequency, $\displaystyle n$ – the number of all the data.

Number of books	0	1	2	3	4	5	Total
Frequency	3	4	7	3	2	1	20
Relative frequency	0.15	0.2	0.35	0.15	0.1	0.05	1
In percent	15%	20%	35%	15%	10%	5%	100%

For example, 7 of the 20 students read 2 books: $\displaystyle {\frac{7}{20}}={0.35}={35\%}$ . The relative frequencies always add up to 1, that is 100%.

Cumulative frequency

The cumulative frequency of a value is the number of data less than or equal to it. It is found by adding up the frequencies from the left, starting at the least value.

Number of books	0	1	2	3	4	5
Frequency	3	4	7	3	2	1
Cumulative frequency	3	7	14	17	19	20

14 students read 2 books or fewer: $\displaystyle {3+4+7}={14}$ .
The cumulative frequency of the last value is the number of all the data, 20.
The difference between two neighbouring cumulative frequencies gives back a frequency: $\displaystyle {17-14}={3}$ students read exactly 3 books.

A table of classes

When a continuous quantitative variable has many different values, a table with one value a column is long, with a frequency of 1 or 2 in each column. Then the values are split into classes of equal width, and the data in each class are counted.

The heights of a class’s 14 students, in cm:
148, 153, 155, 158, 162, 163, 163, 167, 169, 174, 177, 178, 185, 188

Height (cm)	140–150	150–160	160–170	170–180	180–190
Frequency	1	3	5	3	2

The class 160–170 is “from 160 up to less than 170”: a student 170 cm tall is counted in the class 170–180.
All the classes are 10 cm wide.
Sometimes an end class has no bound: “180 and over” or “less than 150”. This is an open class – its bound is not known.

What each table keeps and what it loses

Each kind of table answers a different question well, and every move between tables keeps something and loses something.

A list – all the data themselves, but the overall picture is hard to see.
A frequency table – how many times each value came up; the order in which the data were collected is lost.
A table of classes – the overall picture; the exact values inside each class are lost.
Relative frequency – lets you compare groups of different sizes; the number of data is lost, unless it is written.
Cumulative frequency – answers “how many data up to a given value” at once.

An example: in class 10A, 7 of 20 students read 2 books; in class 10B, 8 of 32. Class 10B has more such students, but their part is smaller: $\displaystyle {\frac{7}{20}}={35\%}$ against $\displaystyle {\frac{8}{32}}={25\%}$ .

Exercise 1

25 students were asked how many brothers and sisters they have. The answers:
1, 2, 0, 1, 3, 2, 1, 1, 4, 2, 0, 1, 2, 3, 1, 2, 1, 0, 2, 1, 3, 1, 2, 1, 5

Build a frequency table of the data.

What is the relative frequency, in percent, of the students with exactly one brother or sister?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power