Question

Categorical Data. In Exercises 23 and 24, use the given categorical data to construct the relative frequency distribution.

Clinical Trial When XELJANZ (tofacitinib) was administered as part of a clinical trial for this rheumatoid arthritis treatment, 1336 subjects were given 5 mg doses of the drug, and here are the numbers of adverse reactions: 57 had headaches, 21 had hypertension, 60 had upper respiratory tract infections, 51 had nasopharyngitis, and 53 had diarrhoea. Does any one of these adverse reactions appear to be much more common than the others? (Hint: Find the relative frequencies using only the adverse reactions, not the total number of treated subjects.)

Short answer

The relative frequency distribution is constructed as follows:

Adverse reactions	Relative Frequency
Headache	23.55%
Hypertension	8.68%
Upper respiratory tract infection	24.79%
Nasopharyngitis	21.07%
Diarrhea	21.9%

There is no adverse reaction that occurs much frequently than others.

Step-by-step solution

Given information

The frequency distribution of the adverse reactions of a drug as a part of a treatment is provided.

Define relative frequency

The relative frequency of a class interval in a grouped frequency distribution is obtained using the given formula:

$\text{Relative}\text{frequency}\text{of}\text{a}\text{class}\text{interval}=\frac{\text{Frequency}\text{of}\text{a}\text{class}\text{interval}}{\text{Sum}\text{of}\text{all}\text{frequencies}}\times 100\%$

Calculate the relative frequencies

The table below shows the frequencies of different adverse reactions:

Adverse reactions	Frequency
Headache	57
Hypertension	21
Upper respiratory tract infection	60
Nasopharyngitis	51
Diarrhea	53

The sum of all frequencies is shown below:

$$\begin{gathered} \text{Sum}\text{of}\text{all}\text{frequencies}=57+21+60+51+53 \\ =242 \end{gathered}$$

Using the formula mentioned above, the relative frequencies for the different categories are calculated as shown below:

$$\begin{gathered} \text{Relative}{\text{freq}}_1=\frac{57}{242}\times 100\% \\ =23.55\% \\ \text{Relative}{\text{freq}}_2=\frac{21}{242}\times 100\% \\ =8.68\% \end{gathered}$$

$$\begin{gathered} \text{Relative}{\text{freq}}_3=\frac{60}{242}\times 100\% \\ =24.79\% \\ \text{Relative}{\text{freq}}_4=\frac{51}{242}\times 100\% \\ =21.07\% \end{gathered}$$

$$\begin{gathered} \text{Relative}{\text{freq}}_5=\frac{53}{242}\times 100\% \\ =21.9\% \end{gathered}$$

Adverse reactions	Relative Frequency
Headache	23.55%
Hypertension	8.68%
Upper respiratory tract infection	24.79%
Nasopharyngitis	21.07%
Diarrhea	21.9%
Total	99.99%

Analysis

It can be observed that the highest relative frequency,equal to 224.79%, is not very high as compared to the other frequencies.

Therefore, there is not one particular adverse reaction that appears to occur much more frequently than others.