Investigating the Association Between a Numerical and a Categorical Variable (VCE SSCE General Mathematics): Revision Notes

Worked Example: Using a parallel dot plot

The parallel dot plots shown above display the number of sit-ups performed by $15$ people before and after completing a gym program.

Here, the numerical variable number of sit-ups is the response variable, and the categorical variable gym program (with values 'before' and 'after') is the explanatory variable.

Step 1: Locate the median number of sit-ups performed before and after the gym program.

Before: $M = 26$ sit-ups

After: $M = 32$ sit-ups

Step 2: Determine the IQR of sit-ups performed before and after the gym program.

Before: $IQR = 6$ sit-ups

After: $IQR = 10$ sit-ups

Step 3: Compare the values and draw a conclusion.

There are reasonable differences in both the median and IQR of the number of sit-ups performed before and after the gym program. This provides evidence that the number of sit-ups performed is associated with completing the gym program.

Report:

The median number of sit-ups performed after attending the gym program ($M = 32$) is higher than the median number performed before attending ($M = 26$). The variability in the number of sit-ups has also increased from $IQR = 6$ to $IQR = 10$. Thus we can conclude that the number of sit-ups performed is associated with completing the gym program.

Worked Example: Using a back-to-back stem plot

Consider a back-to-back stem plot displaying the distribution of life expectancy (in years) for the same $13$ countries in $1970$ and $2010$.

Here, the numerical variable life expectancy is the response variable, and year is the explanatory variable. Although year is numerical, because it only takes two values ($1970$ and $2010$), we treat it as a categorical variable.

Step 1: Determine the median life expectancies for $1970$ and $2010$.

$1970$: $M = 67$ years

$2010$: $M = 76$ years

Step 2: Determine the quartiles and IQR for each year.

$1970$: $IQR = 12.5$ years

$2010$: $IQR = 8$ years

Step 3: Compare the values and draw a conclusion.

The differences in median and IQR between $1970$ and $2010$ are sufficient to conclude that the distribution of life expectancy has changed over this time period.

Report:

There is an association between year and life expectancy. The median life expectancy has increased between $1970$ and $2010$, from $M = 67$ years to $M = 76$ years. Over the same period, life expectancy has also become less variable ($IQR$ in $1970 = 12.5$ years; $IQR$ in $2010 = 8$ years).

Worked Example: Comparing pulse rates by gender

The parallel boxplots below compare resting pulse rates (in beats per minute) for a group of $70$ male students and $90$ female students.

Here, the numerical variable resting pulse rate is the response variable, and the categorical variable gender is the explanatory variable.

Step 1: Compare the medians.

Female median: approximately $72$ beats per minute

Male median: approximately $65$ beats per minute

The median for females is higher than that for males.

Step 2: Compare the spread.

Female IQR: approximately $15$ beats per minute

Male IQR: approximately $10$ beats per minute

The IQR for females is greater than that for males.

Step 3: Compare the shape.

Both distributions are approximately symmetric.

Step 4: Locate any outliers.

There are two outliers for males: one at $40$ beats per minute and one at $120$ beats per minute.

Step 5: Write the report.

Report:

There is an association between resting pulse rate and gender. On average, the resting pulse rate for males is lower (median: male $= 65$, female $= 72$) and less variable than that for females (IQR: male $= 10$, female $= 15$). The distributions of resting pulse rates for both male and female students were approximately symmetric. One male was found to have an extremely low pulse rate of $40$, while another had an extremely high pulse rate of $120$.

Worked Example: Comparing salaries across age groups

The parallel boxplots below compare salary distributions for workers in a certain industry across four different age groups: $20$–$29$ years, $30$–$39$ years, $40$–$49$ years, and $50$–$65$ years.

Here, the numerical variable salary is the response variable, and the categorical variable age group is the explanatory variable.

Step 1: Compare the medians.

The median salary increases with age:

• $20$–$29$ years: $64,000
• $30$–$39$ years: approximately $66,000
• $40$–$49$ years: approximately $68,000
• $50$–$65$ years: $72,000

Step 2: Compare the IQRs.

The IQR also increases with age:

• $20$–$29$ years: approximately $12,000
• $50$–$65$ years: approximately $20,000

Step 3: Compare the shapes.

The shape of the distribution changes with age group:

• $20$–$29$ years: symmetric
• Progressively more positively skewed as age increases

Step 4: Locate the outliers.

• $20$–$29$ years: no outliers
• $30$–$39$ years: no outliers
• $40$–$49$ years: one outlier at $110,000
• $50$–$65$ years: three outliers at $119,000, $126,000, and $140,000

Step 5: Write the report.

Report:

In this industry there is an association between salary and age group. The median salaries increase across the age groups, from $64,000 for $20$–$29$ year-olds to $72,000 for $50$–$65$ year-olds. The salaries also become more variable, with the IQR increasing from around $12,000 for $20$–$29$-year-olds to around $20,000 for $50$–$65$-year-olds. The shape of the distribution of salaries changes with age group, from symmetric for $20$–$29$-year-olds to progressively more positively skewed as age increases. There are no outliers in the $20$–$29$ and $30$–$39$ age groups. Outliers begin to appear at $110,000 for the $40$–$49$ age group, and at $119,000, $126,000, and $140,000 for the $50$–$65$ age group.