A frequency table communicates the frequency of each result from a set of data. This is often represented as a column table with the far-left column describing the result and any columns to the right recording frequencies of different result types.

Frequency tables can help us find the least or most common results among categorical data. They can also allow us to calculate what fraction of the data a certain result represents.

When working with numerical data, frequency tables can also help us to answer other questions that we might have about how the data are distributed.

For example, the following list of colours were recorded in the frequency table below:

\text{White, Black, White, Black, Black, Blue, Blue, White, Red, White,}\\ \text{ White, Blue, Orange, Blue, White, White, Orange, Red, Blue, Red}

Car colour	Frequency
\text{Red}	3
\text{Black}	3
\text{White}	7
\text{Orange}	2
\text{Blue}	5

We can find the total number of colours by adding up the frequencies to get 3+3+7+2+5=20. The least common colour is orange, since it had the lowest frequency of 2. So the fraction of colours that are orange is \dfrac{2}{20}=\dfrac{1}{10}.

We can find the mode, mean, median and range from a frequency table. These will be the same as the mode, mean, median and range from a list of data but we can use the frequency table to make it quicker.

The cumulative frequency is the sum of the frequencies of the score and each of the scores below it. The cumulative frequency of the first row will be the frequency of that row. For each subsequent row, add the frequency to the cumulative frequency of the row before it.

Examples

Example 1

Consider the following data set.

Score	Frequency	Cumulative frequency
2	3	3
3	5	8
4	3	11
5	4	15
6	8	23
7	2	25

How many scores are there in total?

Worked Solution

Create a strategy

We can find the total number of scores in the last entry of the cumulative frequency column.

Apply the idea

This value is the same as adding all the entries in the frequency column. \text{Number of scores}=25

Find the median score.

Worked Solution

Create a strategy

The median is the middle score when the data has been ordered.

Apply the idea

Since there are 25 scores in total, the median will be the 13th score.

Looking at the cumulative frequency table, there are 11 scores less than or equal to 4 and 15 scores less than or equal to 5. This means that the 13th score is 5. \text{Median}=5

Example 2

We want to find the mean of the following data set.

\text{Score }(x)	\text{Frequency }(f)	xf
2	7	14
3	2	6
4	8	32
5	5	25
6	4	24
7	7	49

How many scores are there in the data set?

Worked Solution

Create a strategy

We can find the total number of scores in the data set by adding all the values in the frequency (f) column.

Apply the idea

\displaystyle \text{Number of scores}	\displaystyle =	\displaystyle 7+2+8+5+4+7	Add the frequencies
	\displaystyle =	\displaystyle 33	Evaluate

What is the total sum of all the scores in the data set?

Worked Solution

Create a strategy

We can find the total sum of all the scores in the data set by adding all the values in the xf column.

Apply the idea

\displaystyle \text{Sum of scores}	\displaystyle =	\displaystyle 14+6+32+25+24+49	Add the xf values
	\displaystyle =	\displaystyle 150	Evaluate

Find the mean for this data set.

Worked Solution

Create a strategy

To find the mean using a frequency table, divide the total of the xf column (the sum of the scores) by the total of the f column (the number of scores).

Apply the idea

\displaystyle \text{Mean}	\displaystyle =	\displaystyle \dfrac{150}{33}	Divide the sum by the number of scores
	\displaystyle =	\displaystyle 4.5	Evaluate

Idea summary

We can use the frequency table to find the mean, mode, median, and range of a data set.

The cumulative frequency is the sum of the frequencies of the score and each of the scores below it. Adding a cumulative frequency column to a frequency table is helpful for finding the median.

Adding an xf column to a frequency table is helpful for finding the mean.

Grouped frequency tables

When the data are more spread out, sometimes it doesn't make sense to record the frequency for each separate result and instead we group results together to get a grouped frequency table.

A grouped frequency table combines multiple results into a single group. We can find the frequency of a group by adding all the frequencies of the results contained in that group.

The modal class in a grouped frequency table is the group that has the greatest frequency. If there are multiple groups that share the greatest frequency then there will be more than one modal class.

The drawback of a grouped frequency table is that the data becomes less precise, since we have grouped multiple data points together rather than looking at them individually.

Examples

Example 3

Complete the frequency table for the data set below.

77,\,54,\,53,\,56,\,73,\,55,\,94,\,95,\,76,\,52,\,72,46,\,85, \\61,\,48,\,90,\,64,\,70,\,40,\,52,\,57,\,88,\,59,\,95,\,61

Class	Frequency	Cumulative frequency
40-49
50-59
60-69
70-79
80-89
90-99

Worked Solution

Create a strategy

Divide the data into the classes.

To find the cumulative frequency, add the frequency of each row to the cumulative frequency of the previous row.

Apply the idea

The group 40 - 49 includes the scores: 40,\,46,\,48.

The group 50 - 59 includes the scores: 52,\,52,\,53,\,54,\,55,\,57,\,59.

The group 60 - 69 includes the scores: 61,\,61,\,64.

The group 70 - 79 includes the scores: 70,\,72,\,73,\,76,\,77.

The group 80 - 89 includes the scores: 85,\,88.

The group 90 - 99 includes the scores: 90,\,94,\,95,\,95.

So we get the following frequency column:

Class	Frequency	Cumulative frequency
40-49	3
50-59	8
60-69	3
70-79	5
80-89	2
90-99	4

For the first row the cumulative frequency is equal to the frequency.

Then by adding the frequency of each row to the cumulative frequency of the previous row we can complete the cumulative frequency column:

Class	Frequency	Cumulative frequency
40-49	3	3
50-59	8	3+8=11
60-69	3	11+3=14
70-79	5	14+5=19
80-89	2	19+2=21
90-99	4	21+4=25

Idea summary

A grouped frequency table combines multiple results into a single group. We can find the frequency of a group by adding all the frequencies of the results contained in that group.

Summarise data from a grouped frequency table

When finding the mean and median of grouped data we want to first find the class centre of each group. The class centre is the mean of the highest and lowest possible scores in the group.

Examples

Example 4

We want to estimate the median for this data set.

\text{Class}	\text{Frequency}	\text{Cumulative frequency}
21 - 25	4	4
26 - 30	3	7
31 - 35	3	10
36 - 40	2	12
41 - 45	5	17
46 - 50	8	25

How many scores are there in total?

Worked Solution

Create a strategy

Use the last value in the cumulative frequency column.

Apply the idea

The last value in the cumulative frequency column is 25 so there are 25 scores.

Estimate the median.

Worked Solution

Create a strategy

Use the cumulative frequency column and the class centre of the appropriate group.

Apply the idea

Since there are 25 scores, the middle score will be the \dfrac{25+1}{2}=13th score.

The 13th to 17th scores are all in the class 41-45. So to estimate the median, we need to find the class centre of this group.

\displaystyle \text{Class centre}	\displaystyle =	\displaystyle \dfrac{41+45}{2}	Find the average
	\displaystyle =	\displaystyle 43	Evaluate

So the median is 43.

Example 5

Estimate the mean for this data set. Round your answer to one decimal place.

\text{Class}	\text{Class centre } (x)	\text{Frequency } (f)	xf
6 - 10	8	2	16
11 - 15	13	1	13
16 - 20	18	9	162
21 - 25	23	8	184
26 - 30	28	7	196
31 - 35	33	5	165

Worked Solution

Create a strategy

Divide the sum of the xf column by the sum of the f column.

Apply the idea

\displaystyle \text{Sum of } xf	\displaystyle =	\displaystyle 16+13+162+184+196+165	Add the values in the xf column
	\displaystyle =	\displaystyle 736	Evaluate
\displaystyle \text{Sum of } f	\displaystyle =	\displaystyle 2+1+9+8+7+5	Add the frequencies
	\displaystyle =	\displaystyle 32	Evaluate
\displaystyle \text{Mean}	\displaystyle =	\displaystyle \dfrac{736}{32}	Divide the sum of xf by the sum of f
	\displaystyle =	\displaystyle 23.0	Evaluate

Idea summary

When estimating the mean and median of grouped data we use the class centre of each group.

The class centre is the mean of the highest and lowest possible scores in the group.

10.02 Frequency tables

Ideas

Frequency tables

Examples

Example 1

Example 2

Grouped frequency tables

Examples

Example 3

Summarise data from a grouped frequency table

Examples

Example 4

Example 5

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