Distribution (Intuition) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Distribution (Intuition).

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A distribution describes how data values are spread out across their range β€” which values occur, how often, and whether the data is symmetric or skewed.

If you took many measurements, where would most values fall? What's the shape?

Read the full concept explanation β†’

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 captures both center and spreadβ€”the full picture of data.

Common stuck point: A distribution is a whole shape, not a single number β€” summarizing it with only the mean loses information about spread, skewness, and outliers.

Sense of Study hint: Draw a histogram or dot plot of your data. Describe three things: the center, the spread, and the shape (symmetric, skewed, or lumpy).

Worked Examples

Example 1

easy
Describe the distribution of heights of adult men (approximately normally distributed with mean 70 inches and SD 3 inches). What percentage of men are between 64 and 76 inches tall?

Solution

  1. 1
    Normal distribution: symmetric, bell-shaped, centered at \mu = 70 inches
  2. 2
    64 inches is \frac{64-70}{3} = -2 SD below the mean; 76 inches is \frac{76-70}{3} = +2 SD above the mean
  3. 3
    Empirical rule: approximately 95% of data falls within \mu \pm 2\sigma
  4. 4
    Therefore approximately 95% of men are between 64 and 76 inches tall

Answer

Approximately 95% of men have heights between 64 and 76 inches.
The normal distribution is described by its mean (center) and standard deviation (spread). The 68-95-99.7 rule: 68% within Β±1Οƒ, 95% within Β±2Οƒ, 99.7% within Β±3Οƒ. This makes normal distributions highly predictable.

Example 2

medium
Compare three distributions: (A) uniform (equal probability for all outcomes), (B) right-skewed (most values small, few very large), (C) bimodal (two peaks). Give a real-world example of each.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A distribution has mean 50 and median 65. Is this distribution symmetric, left-skewed, or right-skewed? Explain.

Example 2

hard
The Central Limit Theorem says that sample means follow a normal distribution for large n, regardless of the original population's shape. Explain why this is remarkable and give an example using a right-skewed population.
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Background Knowledge

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

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