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: The shape of a distribution — symmetric, skewed left, skewed right, or bimodal — determines which summary statistics are most appropriate and meaningful.

Common stuck point: Students confuse the direction of skew: a right-skewed distribution has its long tail pointing right, meaning a few unusually large values pull the mean up.

Sense of Study hint: First, create a histogram or dot plot of your data. Then look at the overall shape: is it roughly symmetric, does it lean to one side, or does it have multiple peaks? Finally, describe the shape using standard terms (symmetric, left-skewed, right-skewed, uniform, or bimodal) and note any outliers.

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

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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.

Solution

  1. 1
    Step 1: A peak on the left with a long right tail is a right-skewed (positively skewed) distribution.
  2. 2
    Step 2: In right-skewed distributions, the few high values pull the mean to the right.
  3. 3
    Step 3: Therefore, the mean is larger than the median.

Answer

Right-skewed distribution. Mean > Median.
Distribution shape affects the relationship between mean and median. In right-skewed data, extreme high values inflate the mean, making it larger than the median.

Example 2

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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

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A histogram of exam scores is left-skewed. What does this tell us about the relationship between the mean and median?

Example 2

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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.

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