Exercises: Interpret Shape, Center, and Spread in Context
Work through each section in order. For interpretation problems, write in
complete sentences about the real-world quantity, not just numbers. Remember:
skew points toward the TAIL, the median and IQR resist outliers while the mean
and standard deviation do not, and "lower" or "higher" is good or bad only
relative to what is being measured.
Warm-Up: Describing and Comparing Distributions
These problems review description vocabulary you already know.
A student is asked to describe the distribution of daily commute times for 40 workers. Which response is a complete description of the distribution?
'Commute times are right-skewed: a typical commute is about 25 minutes, and the middle half of commutes span about 15 minutes.'
mean , median , IQR
The longest commute is 70 minutes.
Most workers drive to work each day.
A histogram of household incomes has most values bunched on the left with a long tail stretching toward the high incomes on the right. How is this distribution described?
Right-skewed, because the long tail points toward the high (right) values.
Left-skewed, because most of the data (the peak) sits on the left.
Symmetric, because it has one peak.
There is not enough information to name the shape.
A data set has nine values clustered near 50 and one extreme value of 500. Adding the extreme value strongly changes one center but barely changes the other. Which center changes a lot: the ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ? Which center barely changes: the ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ? (Answer "mean" or "median" in each blank.)
Fluency Practice
Read each plot or summary and answer in context.
A distribution of test scores is roughly symmetric with one peak. Which center and which spread should you report to describe a typical score and its variability? (Answer "mean" or "median" for the center, and "standard deviation" or "IQR" for the spread.)
The waiting times at a clinic are right-skewed, with a long tail toward the very long waits. A report says "the distribution is left-skewed because most patients wait only a short time." What is wrong with this claim?
Skew is named for the tail, not the peak. The long tail is toward the long waits (the right), so the distribution is RIGHT-skewed.
Nothing is wrong; most of the data is on the left, so it is left-skewed.
The distribution is actually symmetric because it has a single peak.
You cannot name skew from waiting-time data.
The box plots below show customer wait times at two coffee shops. Which statement correctly interprets BOTH the center and the spread difference?
Store A is usually faster (lower median), but its waits are less predictable (larger IQR); Store B is a bit slower on average but more consistent (smaller IQR).
Store A is better in every way because its median is lower.
The two stores are the same because both boxes reach up to about 8 minutes.
Store B is faster because its box is narrower.
Two classes take the same exam. Class X has a higher mean score than Class Y. A student concludes "a higher mean is always better, so Class X did better." On a DIFFERENT measure — the number of errors on a typing test — Class X also has the higher mean. What is the correct reasoning?
Whether higher is better depends on the context: a higher exam score is good, but a higher error count is bad. You must check what is being measured first.
A higher mean is always better, so Class X is better on both measures.
A higher mean is always worse, so Class Y is better on both measures.
You can never compare two groups using their means.
A data set of nine ordinary values gains one extreme outlier. Which pair of statistics changes the MOST because of the outlier?
The mean and the standard deviation.
The median and the IQR.
All four statistics change by the same amount.
None of the statistics change.
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