How to make a frequency table

When we want to describe the frequency of our observations, that is, how often each observation occurs, we use a frequency table. It gives a good and easy overview of our data. The typical hurdle isn't the counting, it's keeping the four columns apart: which one is a count, which one is a running total, and which ones are percentages.

When do I use this?

Whenever you've collected a data set and want to get an overview of it. In statistics almost everything reads off the frequency table: the bar chart is one of its columns drawn as bars, the cumulative frequency graph is another column drawn as a curve, and the mode and median can be found in it.

We'll use the shoe sizes of a class of 20:

$$\{38, 38, 38, 38, 39, 39, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 43, 45, 46\}$$

The procedure

We build the table one column at a time, from left to right, because each column uses the one before it.

1. Write down the observations

We start by writing our observations up: one row for each different value that occurs. For the shoe sizes that is 38, 39, 40, 41, 42, 43, 45 and 46. There is no 44 in the class, so there is no row for 44.

2. Count the frequency

Then we look at how many times each observation occurs in the data set. That number is the frequency: it shows how many there are of our observation. Size 38 occurs 4 times, size 39 occurs 5 times, and so on.

3. Add up the cumulative frequency

Cumulative frequency means that we add the frequencies together as we go down the table. After the first row it's 4. After the second row it's $4 + 5 = 9$. After the third row it's $9 + 3 = 12$. The last row must end on 20, the total number of observations, which is a nice check that nothing has been missed.

4. Work out the relative frequency

In the relative frequency we write how big a part the frequency of that observation makes up of the total. It can be given as a percentage, a fraction or a decimal. Size 38 is 4 of the 20 observations:

$$\frac{4}{20} = 20\%$$

Size 39 is $\tfrac{5}{20} = 25\%$. Size 43 is $\tfrac{1}{20} = 5\%$. If you want to see why a fraction and a percentage are the same number, that's in what is a fraction? and what is a percentage?.

5. Add up the cumulative relative frequency

In the cumulative relative frequency we just add the relative frequencies together down the column, in the same way as in step 3: $20\%$, then $20\% + 25\% = 45\%$, then $45\% + 15\% = 60\%$, and so on until the last row reaches $100\%$.

The finished table

Observation	Frequency	Cumulative frequency	Relative frequency	Cumulative relative frequency
38	4	4	$\tfrac{4}{20}$, 20%	$\tfrac{4}{20}$, 20%
39	5	9	$\tfrac{5}{20}$, 25%	$\tfrac{9}{20}$, 45%
40	3	12	$\tfrac{3}{20}$, 15%	$\tfrac{12}{20}$, 60%
41	3	15	$\tfrac{3}{20}$, 15%	$\tfrac{15}{20}$, 75%
42	2	17	$\tfrac{2}{20}$, 10%	$\tfrac{17}{20}$, 85%
43	1	18	$\tfrac{1}{20}$, 5%	$\tfrac{18}{20}$, 90%
45	1	19	$\tfrac{1}{20}$, 5%	$\tfrac{19}{20}$, 95%
46	1	20	$\tfrac{1}{20}$, 5%	$\tfrac{20}{20}$, 100%
Total	20	20	100%	100%

Reading down the third column, you can see that we're adding the different observations together. Reading down the last column, you can see the percentages being added down the rows.

Grouped data

When we make a frequency table for a grouped data set, it's exactly the same thing we do. We just have to remember that the frequency is for the whole interval, and not for the individual observations. If the row is "heights from 150 cm up to and including 155 cm", the frequency is how many people fall anywhere in that interval.

Worked example: reading the table

Once the table exists, questions answer themselves:

• How many pupils have size 41? Frequency of 41: $3$.
• How many pupils have size 40 or smaller? Cumulative frequency at 40: $12$.
• What share of the class has size 42? Relative frequency: $\tfrac{2}{20} = 10\%$.
• What share has size 41 or smaller? Cumulative relative frequency at 41: $75\%$.

That last kind of reading is the one the cumulative frequency graph is built from.

Common mistakes

• "Frequency and relative frequency are the same thing." The frequency is a count (5 pupils). The relative frequency is that count as a part of the whole ($\tfrac{5}{20} = 25\%$).
• "The cumulative frequency is just the frequency written again." It's the running total. It grows down the table and ends on the total number of observations.
• "In grouped data I count each observation separately." The frequency is for the whole interval, not the individual observations inside it.
• "The last row doesn't matter." It's your check. The cumulative frequency must end on the total, and the cumulative relative frequency must end on 100%.

Related

The frequency table is the starting point in statistics. From here you can draw a bar chart of the frequency column, draw a cumulative frequency graph of the last column, or read off the mean, median, mode and range.