Distribution Shape Examples: 42 Problems with Answers
Start with the recap, study the fully worked examples, then use the practice problems to
check your understanding of Distribution Shape.
This page combines explanation, solved examples, and follow-up practice so you can move
from recognition to confident problem-solving in Statistics.
Concept Recap
Distribution shape describes the overall pattern of how data values are spread when displayed in a histogram or dot plot. Common shapes include symmetric (bell curve), skewed left, skewed right, uniform (all values equally common), and bimodal (two peaks).
If you make a histogram, what shape emerges? A bell curve? A slope leaning one way? Two peaks? The shape tells you about what's typical and what's unusual in your data.
Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:Distribution Shape asks how a value or feature behaves inside the full distribution.
Common stuck point:Students often know a procedure related to distribution shape but skip the recognition step: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary? That leads to a calculation or graph that looks reasonable but answers a different question.
Sense of Study hint:Ask: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary?
Common Mistakes to Watch For
Before you work through the examples, skim the mistake guide so you know which shortcuts and
sign errors to avoid.
Data: 10,12,15,15,17,18,20. Compare mean and median to identify the shape.
Example 3
medium
A histogram of household incomes in a city shows a peak on the left with a long tail extending to the right. Describe the shape and state whether the mean or median is likely larger.
Example 4
medium
Classify each distribution shape: (a) Test scores cluster around 75 with equal tails. (b) Marathon finish times have a peak at 4 hours with a long tail for slower runners. (c) Ages at a family reunion show peaks at 10 and 40.
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
A histogram is a mirror image about its center peak. What shape is it?A histogram that is a mirror image about its center peak — what shape is this?
Example 2
easy
A distribution has a long tail stretching to the RIGHT (toward large values). What is its skew?This distribution has a long tail stretching to the RIGHT — what is its skew?
Example 3
easy
A distribution has a long tail stretching to the LEFT. What is its skew?This distribution has a long tail stretching to the LEFT — what is its skew?
Example 4
easy
A histogram where every bar is the same height is called what shape?A histogram where every bar is the same height — what shape is this called?
Example 5
easy
A histogram has TWO distinct peaks. What is this shape called?This histogram has TWO distinct peaks — what is this shape called?
Example 6
easy
For a roughly symmetric, single-peaked distribution, the mean and median are approximately what?
Example 7
easy
Home prices in a city: most are moderate, a few mansions are very expensive. What skew is expected?
Example 8
easy
Which shape best describes adult human heights in a large population?
Example 9
medium
A data set has mean 50 and median 42. What shape is implied?
Example 10
medium
A data set has mean 30 and median 38. What shape is implied?
Example 11
medium
For a right-skewed distribution, order mean, median, and mode from smallest to largest.For this right-skewed distribution, order mode, median, and mean from smallest to largest.
Example 12
medium
For a LEFT-skewed distribution, order mean, median, and mode from smallest to largest.For this left-skewed distribution, order mean, median, and mode from smallest to largest.
Example 13
medium
Test scores where almost everyone scored very high and a few scored low. What skew?Almost everyone scored high; a few scored low — what skew does this create?
Example 14
medium
Which measure of center is more appropriate to report for a strongly right-skewed income data set, and why?
Example 15
medium
A dot plot of exam retakes shows a tall cluster at 0 retakes and a long thin tail toward 5+ retakes. Classify the shape.
Example 16
medium
A bimodal histogram of student heights mixes two grade levels. What does the shape suggest about the data?Heights from two mixed grade levels — two peaks. What does this bimodal shape suggest?
Example 17
medium
A uniform distribution over equally likely values has what skewness and mean-median relationship?
Example 18
challenge
A distribution is perfectly symmetric and bimodal (two equal peaks). Where do its mean and median fall relative to the two peaks?A perfectly symmetric bimodal distribution — where do mean and median fall relative to the two peaks?
Example 19
challenge
A right-skewed distribution has its values all multiplied by −1. What is the new shape?
Example 20
challenge
Distribution A is symmetric; distribution B is right-skewed. Both have median 50. Which has the larger mean, and why?
Example 21
easy
Salaries of workers at a large company range from $20,000 to $1,000,000 with most under $80,000. What skew is expected?
Example 22
easy
A fair die is rolled 6000 times and outcomes counted. What shape does the histogram of outcomes have?
Example 23
easy
A dot plot shows clusters around 5 and 25 with very few values in between. What shape is this?
Example 24
easy
Time-to-complete data for a video game level: most players finish quickly, a few take very long. What shape?
Example 25
medium
A test with mean 75 and median 82 is what shape?
Example 26
medium
Histogram shows: 5 values at 1, 20 at 2, 5 at 3. What shape?
Example 27
medium
A histogram of ages at a children's birthday party shows two peaks: one around age 7 (kids) and one around age 35 (parents). What shape?
Example 28
medium
Choose the best center to summarize a strongly skewed distribution.
Example 29
medium
A distribution has mean 50 and median 50. Is the distribution necessarily symmetric?
Example 30
hard
A box plot has a much longer whisker on the LEFT than on the right and a median close to the right side of the box. What is the likely skew?
Example 31
hard
For a right-skewed distribution, what is the typical relationship Q3−median vs. median−Q1?
Example 32
hard
A data set has mean 20, median 20, and one cluster between 5 and 35 with no extreme tail. Is the most likely shape uniform or symmetric unimodal?
Example 33
hard
What shape do daily counts of customers entering a small shop, where most days have 10-30 but a few special-event days reach 100+, typically take?
Example 34
medium
Shoe sizes for a school of 400 students. What is the most likely shape?
Example 35
medium
A histogram of digits 0-9 for the last digit of phone numbers (random) is expected to be what shape?
Example 36
challenge
A distribution has THREE distinct modes. What is the term for this shape, and what does it commonly suggest about the data?
Example 37
medium
A histogram of exam scores is left-skewed. What does this tell us about the relationship between the mean and median?
Example 38
medium
A distribution has most values near the high end, with a tail stretching toward smaller values. What is the shape, and how do the mean and median compare?