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: Proportional data expresses each quantity as a fraction or percent of its own total so different-sized groups compare fairly.

Common stuck point: The procedure for proportional data is the easy part; the trap is comparing raw counts across groups of different sizes. Asking "Am I expressing a count as a fraction of its own total so different-sized groups compare fairly?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I expressing a count as a fraction of its own total so different-sized groups compare fairly?

Worked Examples

Example 1

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

Answer

p^=45180=0.25=25%\hat{p} = \frac{45}{180} = 0.25 = 25\%

First step

1
Sample proportion formula: p^=xn\hat{p} = \frac{x}{n}

Full solution

  1. 2
    Substitute: p^=45180=0.25\hat{p} = \frac{45}{180} = 0.25
  2. 3
    Convert to percentage: 0.25×100=25%0.25 \times 100 = 25\%
  3. 4
    Interpret: 25% of sampled students prefer online learning; this estimates the true population proportion
Sample proportion p^=x/n\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 pp.

Example 2

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

Example 3

medium
A school's lunch survey: 8080 of 400400 students chose pasta. Another school reports 2525 of 100100 students chose pasta. Which proportion is higher, and by how many percentage points?

Example 4

medium
A survey reports '60%60\% of voters agree' but only 55 voters were polled. Explain whether this is a meaningful summary.

Example 5

hard
In a population of 10,00010{,}000, 2%2\% have a disease. A test has 90%90\% sensitivity and 90%90\% specificity. Estimate the number of true positives and false positives.

Example 6

hard
A retailer claims '50%50\% off' on a coat originally priced at $240, then adds a 10%10\% tax on the discounted price. Find the final price and the effective discount as a proportion of the original.

Example 7

medium
In year 1, defect rate is 4%4\%; in year 2 it falls to 3%3\%. Compute (a) the absolute change and (b) the relative change.

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 p^\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.

Example 3

easy
Out of 2525 students, 2020 passed. What proportion passed?

Example 4

easy
A pie chart slice is 25%25\%. What is its angle in degrees?

Example 5

easy
City A: 5050 sick of 100100. City B: 3030 sick of 200200. Compare the proportions sick.

Example 6

easy
A breakdown shows 40%,35%,30%40\%, 35\%, 30\% for three categories. What is wrong?

Example 7

easy
Is '100%100\% of patients recovered' impressive if only 11 patient was treated? Why?

Example 8

easy
Convert the proportion 3/83/8 to a percentage.

Example 9

easy
Of 200200 voters, 30%30\% chose candidate X. How many voters is that?

Example 10

easy
Which is a fairer comparison of two differently sized classes' performance: number who passed, or proportion who passed?

Example 11

medium
Class A: 1818 of 2020 passed. Class B: 8080 of 100100 passed. Compute both proportions and state which is higher and by how many percentage points.

Example 12

medium
A survey breakdown: 45%45\% A, 25%25\% B, 20%20\% C, and 'other.' What percent is 'other,' and how many of 400400 respondents chose 'other'?

Example 13

medium
A drug 'reduces risk by 50%50\%.' Baseline risk is 2%2\%. What is the new absolute risk, and the absolute risk reduction?

Example 14

medium
Two pollsters report candidate support: Poll 1 says 52%52\% from 10001000 people; Poll 2 says 60%60\% from 2525 people. Which is more trustworthy and why?

Example 15

medium
A pictograph uses one icon =10= 10 cars. Row A shows 4.54.5 icons, row B shows 33 icons. How many cars each, and what proportion of the total is A?

Example 16

medium
Sales mix: product P is 30%30\% of $2M total revenue. Next year P is 40%40\% of $1.5M total. Did P's revenue rise or fall?

Example 17

medium
A test is 95%95\% accurate. In a population where 1%1\% have a disease, of 10,00010{,}000 people, estimate how many true positives vs false positives (assume 95%95\% sensitivity and 95%95\% specificity).

Example 18

medium
A recipe for 44 servings needs 300300g flour. Scale it proportionally for 66 servings.

Example 19

medium
In a class, the ratio of boys to girls is 3:53:5. If there are 3232 students, how many are girls?

Example 20

challenge
Prove that if two groups have proportions p1p_1 and p2p_2 of a trait, the combined proportion always lies between p1p_1 and p2p_2, and equals their size-weighted average.

Example 21

challenge
A '200%200\% increase' is reported. Starting value is 5050. Find the new value, and explain why '200%200\% increase' differs from '200%200\% of.'

Example 22

challenge
Show that comparing two proportions by their RATIO (relative) vs their DIFFERENCE (absolute) can rank interventions differently, using pp: 0.1→0.050.1\to0.05 vs qq: 0.5→0.40.5\to0.4.

Example 23

easy
In a survey of 250250 commuters, 6060 said they bike to work. What is the sample proportion p^\hat{p} that bike?

Example 24

easy
A pie chart slice represents p^=0.40\hat{p} = 0.40. What is the central angle of that slice in degrees?

Example 25

easy
Town X reports 3030 rainy days out of 100100. Town Y reports 2020 rainy days out of 5050. Which town had the higher proportion of rainy days?

Example 26

easy
A pie chart breakdown shows 35%35\% A, 25%25\% B, 15%15\% C. What percent must be 'other'?

Example 27

medium
Of 1,2001{,}200 ticket holders, 54%54\% are season members. How many ticket holders are NOT season members?

Example 28

medium
A poll of 625625 adults reports p^=0.48\hat{p} = 0.48 favoring a tax measure. Estimate the standard error of p^\hat{p}.

Example 29

medium
In a town, 20%20\% of 2,0002{,}000 households own pets. Of those, 30%30\% own cats. How many households own cats?

Example 30

medium
A vaccine is reported to reduce risk by 40%40\%. Baseline risk is 5%5\%. Find the new absolute risk and the absolute risk reduction in percentage points.

Example 31

medium
In a pictograph, one icon represents 2020 books. Library A shows 3.53.5 icons and Library B shows 55 icons. What proportion of the combined collection is in Library A?

Example 32

medium
A company's revenue mix: Product A was 25%25\% of $8M last year and is 30%30\% of $10M this year. Did Product A's revenue rise or fall, and by how much?

Example 33

hard
In group A, 2020 of 5050 approve a policy (40%40\%); in group B, 3535 of 100100 approve (35%35\%). Compute the pooled approval proportion when the groups are combined.

Example 34

hard
A z-test compares p^1=0.50\hat{p}_1 = 0.50 (n=200) vs p^2=0.40\hat{p}_2 = 0.40 (n=200). Compute the pooled p^\hat{p} and the test statistic zz.

Example 35

hard
A poll reports p^=0.55\hat{p} = 0.55 with a margin of error of ±0.03\pm 0.03 at 95%95\% confidence. State the confidence interval and whether it supports a majority.

Example 36

hard
In a class, 60%60\% are girls. Of the girls, 70%70\% play sports; of the boys, 80%80\% play sports. What proportion of the entire class plays sports?

Example 37

challenge
Two cities each report a 50%50\% approval rate. City A polled 4040 residents, City B polled 4,0004{,}000. Compare their 95%95\% margins of error (approximate z=2z=2).

Example 38

challenge
Simpson's paradox setup: hospital A treats 9090 of 100100 healthy patients and 3030 of 100100 sick patients successfully; hospital B treats 8080 of 100100 healthy and 2020 of 100100 sick. Show that A's overall rate exceeds B's, and confirm A wins within each subgroup as well.
← Back to Proportional Data Formula Explained Practice Mode

Background Knowledge

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

percent as ratio