Proportional Data Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Proportional Data.

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

Concept Recap

Proportional data expresses quantities as fractions or percentages of a whole, enabling fair comparison across groups of different sizes.

Raw counts can mislead when groups differ in size — saying "100 people in City A vs. 100 in City B have a disease" ignores that City A may be ten times larger.

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: Always ask: proportional to what? A proportion only makes sense relative to a clearly defined whole — and the choice of denominator changes the story dramatically.

Common stuck point: Percentages can mislead with small samples—50\% of 2 is just 1.

Worked Examples

Example 1

easy
In a survey, 45 out of 180 students prefer online learning. Calculate the sample proportion \hat{p} and interpret it.

Solution

  1. 1
    Sample proportion formula: \hat{p} = \frac{x}{n}
  2. 2
    Substitute: \hat{p} = \frac{45}{180} = 0.25
  3. 3
    Convert to percentage: 0.25 \times 100 = 25\%
  4. 4
    Interpret: 25% of sampled students prefer online learning; this estimates the true population proportion

Answer

\hat{p} = \frac{45}{180} = 0.25 = 25\%
Sample proportion \hat{p} = x/n is the fundamental estimate of a population probability from data. It is always between 0 and 1. The sample proportion is an unbiased estimator of the true population proportion p.

Example 2

medium
A poll finds \hat{p} = 0.52 supporting a candidate from n=400 voters. Calculate the standard error of \hat{p} and construct an approximate 95% confidence interval.

Practice Problems

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

Example 1

easy
In a class of 25 students, 15 passed the test. Calculate \hat{p} and find the number expected to pass in a class of 100 using this estimate.

Example 2

hard
Compare two proportions: Group A: 30/120 support policy. Group B: 45/150 support policy. Test if the proportions differ using a z-test for difference in proportions.
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Background Knowledge

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

percent as ratio