The frequency (f) of a particular observation is the number of times the observation occurs in the data. The distribution of a variable is the pattern of frequencies of the observation. Frequency distributions are portrayed as frequency tables, histograms, or polygons.
Frequency distributions can show either the actual number of observations falling in each range or the percentage of observations. In the latter instance, the distribution is called a relative frequency distribution.
Frequency distribution tables can be used for both categorical and numeric variables. Continuous variables should only be used with class intervals, which will be explained shortly.
A survey was taken on Maple Avenue. In each of 20 homes, people were asked how many cars were registered to their households. The results were recorded as follows:
1, 2, 1, 0, 3, 4, 0, 1, 1, 1, 2, 2, 3, 2, 3, 2, 1, 4, 0, 0
Use the following steps to present this data in a frequency distribution table.
Your frequency distribution table for this exercise should look like this:
|Number of cars (x)||Tally||Frequency (f)|
By looking at this frequency distribution table quickly, we can see that out of 20 households surveyed, 4 households had no cars, 6 households had 1 car, etc.
A cumulative frequency distribution table is a more detailed table. It looks almost the same as a frequency distribution table but it has added columns that give the cumulative frequency and the cumulative percentage of the results, as well.
At a recent chess tournament, all 10 of the participants had to fill out a form that gave their names, address and age. The ages of the participants were recorded as follows:
36, 48, 54, 92, 57, 63, 66, 76, 66, 80
Use the following steps to present these data in a cumulative frequency distribution table.
1 + 2 = 3
10.0. (1 ÷ 10) X 100 = 10.0
10.0. (1 ÷ 10) X 100 = 10.0
The cumulative frequency distribution table should look like this:
|Lower Value||Upper Value||Frequency (f)||Cumulative frequency||Percentage||Cumulative percentage|
If a variable takes a large number of values, then it is easier to present and handle the data by grouping the values into class intervals. Continuous variables are more likely to be presented in class intervals, while discrete variables can be grouped into class intervals or not.
To illustrate, suppose we set out age ranges for a study of young people, while allowing for the possibility that some older people may also fall into the scope of our study.
The frequency of a class interval is the number of observations that occur in a particular predefined interval. So, for example, if 20 people aged 5 to 9 appear in our study's data, the frequency for the 5–9 interval is 20.
The endpoints of a class interval are the lowest and highest values that a variable can take. So, the intervals in our study are 0 to 4 years, 5 to 9 years, 10 to 14 years, 15 to 19 years, 20 to 24 years, and 25 years and over. The endpoints of the first interval are 0 and 4 if the variable is discrete, and 0 and 4.999 if the variable is continuous. The endpoints of the other class intervals would be determined in the same way.
Class interval width is the difference between the lower endpoint of an interval and the lower endpoint of the next interval. Thus, if our study's continuous intervals are 0 to 4, 5 to 9, etc., the width of the first five intervals is 5, and the last interval is open, since no higher endpoint is assigned to it. The intervals could also be written as 0 to less than 5, 5 to less than 10, 10 to less than 15, 15 to less than 20, 20 to less than 25, and 25 and over.
In summary, follow these basic rules when constructing a frequency distribution table for a data set that contains a large number of observations:
In deciding on the width of the class intervals, you will have to find a compromise between having intervals short enough so that not all of the observations fall in the same interval, but long enough so that you do not end up with only one observation per interval.
It is also important to make sure that the class intervals are mutually exclusive.
Thirty AA batteries were tested to determine how long they would last. The results, to the nearest minute, were recorded as follows:
423, 369, 387, 411, 393, 394, 371, 377, 389, 409, 392, 408, 431, 401, 363, 391, 405, 382, 400, 381, 399, 415, 428, 422, 396, 372, 410, 419, 386, 390
Use the steps in Example 1 and the above rules to help you construct a frequency distribution table.
The lowest value is 363 and the highest is 431.
Using the given data and a class interval of 10, the interval for the first class is 360 to 369 and includes 363 (the lowest value). Remember, there should always be enough class intervals so that the highest value is included.
The completed frequency distribution table should look like this:
|Battery life, minutes (x)||Tally||Frequency (f)|
An analyst studying these data might want to know not only how long batteries last, but also what proportion of the batteries falls into each class interval of battery life.
This relative frequency of a particular observation or class interval is found by dividing the frequency (f) by the number of observations (n): that is, (f ÷ n). Thus:
Relative frequency = frequency ÷ number of observations
The percentage frequency is found by multiplying each relative frequency value by 100. Thus:
Percentage frequency = relative frequency X 100 = f ÷ n X 100
Use the data from Example 3 to make a table giving the relative frequency and percentage frequency of each interval of battery life.
Here is what that table looks like:
|Battery life, minutes (x)||Frequency (f)||Relative frequency||Percent frequency|
An analyst of these data could now say that:
Keep in mind that these analytical statements have assumed that a representative sample was drawn. In the real world, an analyst would also refer to an estimate of variability (see section titled Measures of spread) to complete the analysis. For our purpose, however, it is enough to know that frequency distribution tables can provide important information about the population from which a sample was drawn.