Distribution Shape Examples in Statistics

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 full concept explanation →

How to Use These Examples

  • 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.

Mean vs Median Mistakes

Worked Examples

Example 1

medium
Data: 1,1,2,2,2,3,81, 1, 2, 2, 2, 3, 8. Sketch reasoning: is this symmetric, left-skewed, or right-skewed?

Answer

right-skewed\text{right-skewed}

First step

1
Median =2= 2; mean =19/7≈2.71= 19/7 \approx 2.71.

See the full worked solution + why-it-works coaching

SetupKey insightWhy it worksCommon pitfallConnection

Unlock answer keys One Family plan — every worked solution, all subjects

Example 2

hard
Data: 10,12,15,15,17,18,2010, 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?

Example 2

easy
A distribution has a long tail stretching to the RIGHT (toward large values). What is its skew?

Example 3

easy
A 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?

Example 5

easy
A 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 5050 and median 4242. What shape is implied?

Example 10

medium
A data set has mean 3030 and median 3838. What shape is implied?

Example 11

medium
For a right-skewed distribution, order mean, median, and mode from smallest to largest.

Example 12

medium
For a 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?

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 00 retakes and a long thin tail toward 5+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?

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?

Example 19

challenge
A right-skewed distribution has its values all multiplied by −1-1. What is the new shape?

Example 20

challenge
Distribution AA is symmetric; distribution BB is right-skewed. Both have median 5050. 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 60006000 times and outcomes counted. What shape does the histogram of outcomes have?

Example 23

easy
A dot plot shows clusters around 55 and 2525 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 7575 and median 8282 is what shape?

Example 26

medium
Histogram shows: 55 values at 11, 2020 at 22, 55 at 33. What shape?

Example 27

medium
A histogram of ages at a children's birthday party shows two peaks: one around age 77 (kids) and one around age 3535 (parents). What shape?

Example 28

medium
Choose the best center to summarize a strongly skewed distribution.

Example 29

medium
A distribution has mean 5050 and median 5050. 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−medianQ_3 - \text{median} vs. median−Q1\text{median} - Q_1?

Example 32

hard
A data set has mean 2020, median 2020, and one cluster between 55 and 3535 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 1010-3030 but a few special-event days reach 100100+, typically take?

Example 34

medium
Shoe sizes for a school of 400400 students. What is the most likely shape?

Example 35

medium
A histogram of digits 00-99 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?
← Back to Distribution Shape Common Mistakes Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

stat histogrambar graph