Categorical Graphs
A segmented bar chart shows two groups with identical segment proportions. What does this imply about the relationship between the categorical variables?
They are likely independent.
A boxplot shows a very long upper whisker. What does this indicate about the distribution?
Right skewness and/or high variability in the upper half of the data.
A dataset has mean 40 and median 20. What does this imply about the distribution?
It is right-skewed.
A distribution is roughly bell-shaped with mean 50 and SD 10. According to the Empirical Rule, what percentage of data lies between 30 and 70?
About 95%.
A histogram is symmetric but has very long tails. Which measure of spread is most affected by the tails?
Standard deviation or variance.
A pie chart and a bar chart display the same categorical data. Give one situation where the bar chart reveals information the pie chart hides.
Bar charts make small differences between categories easier to compare because they use a common scale.
A cumulative relative frequency plot rises steeply between two values. What does this indicate?
A large proportion of observations fall within that interval.
Why is the median a better measure of center than the mean for a heavily skewed distribution?
The median is resistant to outliers.
A distribution is not bell-shaped. What is the minimum percentage of data guaranteed to lie within 2 standard deviations of the mean?
At least 75% (Chebyshev’s inequality).
A boxplot shows Q1 = 10, median = 20, Q3 = 21. What does this say about the distribution?
Most data is tightly clustered near the median.
A segmented bar graph shows that Group A has a larger proportion in Category X than Group B. What does this tell you about the relationship between the variables?
The variables are associated.
A histogram has a long right tail. What two numerical measures will be pulled in the direction of the tail?
Mean and standard deviation.
If Q1 = 30 and Q3 = 70, what percentage of data lies between 30 and 70?
50%.
Using Chebyshev’s inequality, what proportion of data must lie within 3 standard deviations of the mean?
At least 1 − 1/9 = 89%.
A dataset has mean 60 and SD 5. A value of 75 appears. What does this indicate?
It is 3 SD above the mean — an extreme outlier.
Why is a bar graph preferred over a pie chart when there are more than 6 categories?
Because the slices become too small.
A density histogram has bars of different widths. What must remain constant for it to be valid?
The area of each bar must represent relative frequency.
A dataset has standard deviation 0. What must be true about the data?
All observations are identical.
A dataset is extremely skewed. Can the Empirical Rule still be applied? Why or why not?
No. It only applies to approximately normal distributions.
A comparative boxplot shows Group A with a much larger IQR than Group B. What does this imply?
Group A has more variability in the middle 50% of its data.
Two bar graphs display the same categorical data: one uses raw counts, the other uses relative frequencies. Why might the relative frequency bar graph be more useful when comparing across different sample sizes?
Relative frequency standardizes the data, allowing fair comparison between groups of different sizes.
Two histograms have identical shapes but different scales on the x-axis. What can you conclude?
Different centers or spreads.
A dataset has a very large variance. What does this imply about the standard deviation?
The Standard deviation is also large.
A distribution has mean 100 and SD 20. What is the minimum amount of data between 60 and 140?
At least 75% (k = 2).
A cumulative relative frequency plot shows that 80% of observations are at or below 50. The mean of the dataset is 55. What does this suggest about the shape of the distribution?
Right‑skewed, because most values are below 50 while the mean is pulled higher by larger values.