Knowledge Quiz: Univariate Visualizations

Suppose you have a large sample from a smooth, roughly normal variable. If you increase the number of bins in a histogram of this variable from a small number (e.g., 5) to a very large number (e.g., 200), what generally happens to the histogram’s appearance?

Its shape reverses, so a right-skewed distribution appears left-skewed.Its appearance stays exactly the same, since only bin width changes, not the underlying data.It becomes progressively smoother, revealing an increasingly clear and simple shape.It becomes more jagged, with bin-to-bin fluctuations caused by having only a few observations per bin.

Wrong: increasing the number of bins does not change which side of the distribution has the longer tail.
Wrong: the number of bins directly changes how the data are grouped, so the plot’s appearance does change.
Wrong: more bins means fewer observations per bin, which introduces more random noise, not less.
Correct: with very many bins, each bin contains very few observations, so bar heights become noisy and jagged.

You want to compare the distributions of a numeric variable across three groups on the same plot. Why might overlaid kernel density plots be preferable to overlaid histograms for this purpose?

Because density plots ignore differences in sample size between the groups being compared.Because density plots don’t require choosing any smoothing parameter, unlike histograms, which require choosing bins.Because density plots draw smooth, continuous curves that can be overlaid without the visual clutter of overlapping bars.Because density plots always reveal more detail about the data than histograms do.

Wrong: density plots are still affected by sample size, just like histograms.
Wrong: density plots still require choosing a smoothing parameter, the bandwidth, which plays a role similar to bin width.
Correct: continuous curves are easier to visually separate and compare than several sets of overlapping bars.
Wrong: whether a density plot reveals more or less detail depends on the chosen bandwidth, just as a histogram depends on bin choice.

In a boxplot, the median line sits much closer to \(Q_1\) than to \(Q_3\), and the upper whisker is noticeably longer than the lower whisker. What does this suggest about the underlying distribution?

The distribution is roughly symmetric.The distribution is right-skewed (positively skewed), with a longer tail toward higher values.The distribution has no variability.The distribution is left-skewed (negatively skewed), with a longer tail toward lower values.

Wrong: a symmetric distribution would have the median roughly centered in the box, with whiskers of similar length.
Correct: a median pulled toward \(Q_1\) combined with a longer upper whisker is the signature of a right-skewed distribution.
Wrong: a distribution with no variability would show a box collapsed to a single line, not this pattern.
Wrong: this pattern would instead show the median pulled toward \(Q_3\) with a longer lower whisker.

Why is a bar plot, rather than a histogram, the appropriate choice for visualizing the distribution of a categorical variable such as ticket class?

Because bar plots and histograms are interchangeable, so either one works equally well for any variable.Because bar plots can only display continuous, numeric data.Because histograms cannot display counts, only proportions.Because a categorical variable has distinct, unordered or ordered levels, not a continuous range that needs to be divided into bins.

Wrong: using the wrong plot type misrepresents the nature of the variable (continuous vs. categorical).
Wrong: bar plots are specifically meant for categorical, not continuous, data.
Wrong: histograms can display raw counts by default, and can be converted to proportions with the probability argument.
Correct: bar plots represent counts per category, with gaps between bars to show that categories are distinct, rather than binning a continuous range.