Quartiles and percentiles

Where does a value stand?

Quartiles and percentiles describe position: where a value sits relative to the rest of the data.

Quartiles

Quartiles divide ordered data into four equal parts.

Interquartile range = Q3 − Q1: the spread of the middle half.

Percentiles

Percentiles divide data into 100 parts. The 90th percentile is the value below which 90% of the data falls.

• Q1 = 25th percentile
• Median = 50th percentile
• Q3 = 75th percentile

Percentile rank

The percentile rank of a value tells what percent of the data is at or below it. (Definitions vary slightly between textbooks.)

Example: In a class of 30, a student’s score is greater than or equal to 24 of the scores.

Percentile rank ≈ 24/30 × 100 = 80th percentile

This does not mean the student scored 80%. It means they did as well as or better than 80% of the class.

Finding a percentile value

One common method: position = (p/100) × n.

• If it’s not a whole number, round up and use that position.
• If it is, average that value and the next.

Example: 20 ordered values; find the 75th percentile.

0.75 × 20 = 15 → average the 15th and 16th values.

Real-world percentiles

Example: A baby at the 60th percentile for weight weighs more than about 60% of babies the same age.

Z-scores

A z-score shows how many standard deviations a value is from the mean:

z = (x − mean)/σ

For normal data, z-scores convert to percentiles using a table or technology. z = 1 ≈ 84th percentile; z = 2 ≈ 97.7th percentile.

Worked examples

Example 1: Data: 3, 5, 7, 8, 10, 12, 15, 20. Q1, Q3? Q1 = 6, Q3 = 13.5.

Example 2: A score is at the 92nd percentile. What does it mean? It’s at or above about 92% of all scores.

Example 3: Mean 500, σ = 100. z-score of 650? 1.5.

Common mistakes

• Confusing percent correct with percentile.
• Not ordering the data.
• Assuming the 50th percentile is the mean (it’s the median).

Try it yourself

1. Which percentile is Q3?
2. In a group of 50, you scored higher than 40 people. Approximate percentile?
3. Data: 2, 4, 6, 8, 10, 12, 14, 16. Find the median and Q1.
4. z-score for x = 82 with mean 70 and σ = 6?

Answers: 1) 75th 2) about the 80th 3) 9 and 5 4) 2

Quick recap

• Quartiles split data into fourths; percentiles into hundredths.
• Percentile rank shows relative standing, not percent correct.
• z-scores measure distance from the mean in standard deviations.