## 2.2 Frequency, relative frequency and cumulative tables for ordinal data

Consider another variable from the survey data set called Smoke, which indicates how much a student smokes. This categorical variable can take the following values (or 'levels'):

• Heavy
• Regularly
• Occasionally
• Never

The levels presented above are in an arbitrary order. However, it would make sense to present them from least regular to most regular as follows:

• Never
• Occasionally
• Regularly
• Heavy

and thus, we can consider Smoke to be an ordinal categorical variable. The frequency and relative frequency tables are both presented below:

Table 2.3: Frequency and Relative Frequency Tables for the Smoke survey data.
Smoke Frequency
Never 189
Occasionally 19
Regularly 17
Heavy 11
Smoke Relative Frequency (%)
Never 80.08
Occasionally 8.05
Regularly 7.20
Heavy 4.66

Since the Smoke variable is ordinal, it can be useful to convert our frequency and relative frequency tables into cumulative tables, so that as the categories go up, the values are added together. Such tables are called cumulative frequency tables and cumulative relative frequency tables respectively. Putting it all together, we could present the four types of tables together in one table as below:

Table 2.4: Complete table for Smoke survey data.
Smoke Frequency Cumulative Frequency Relative Frequency (%) Cumulative Relative Frequency (%)
Never 189 189 80.08 80.08
Occasionally 19 208 8.05 88.14
Regularly 17 225 7.20 95.34
Heavy 11 236 4.66 100.00

Use Table 2.4 to answer the following questions (please give answers to 2 decimal places where relevant):

1. How many students smoke occasionally?
2. How many students do not smoke heavily?
3. How many students smoke at least occasionally?
4. What percentage of students smoke regularly? %
5. What percentage of students do not smoke regularly? %
6. What percentage of students smoke at least occasionally? %
1. 19
2. 225
3. 47 (The answer can be calculated as follows: $$19 + 17 + 11$$)
4. 7.2
5. 92.8
6. 19.92 (or 19.91). The answer can be calculated as follows: $$100 - 80.08$$ OR $$8.05 + 7.2 + 4.66$$