Quartiles Math Example 5

Follow the full solution, then compare it with the other examples linked below.

Example 5

hard
A student scores at the 75th percentile on a standardized test with scores {45,52,60,67,72,78,83,89,95,98}\{45, 52, 60, 67, 72, 78, 83, 89, 95, 98\}. Confirm that Q3Q_3 equals the 75th percentile score and find what percent of students scored below Q1Q_1.

Solution

  1. 1
    Find Q2Q_2: n=10n=10, median =72+782=75= \frac{72+78}{2} = 75
  2. 2
    Upper half: {78,83,89,95,98}\{78, 83, 89, 95, 98\}; Q3=89Q_3 = 89 — this is the 75th percentile
  3. 3
    Lower half: {45,52,60,67,72}\{45, 52, 60, 67, 72\}; Q1=60Q_1 = 60 — this is the 25th percentile
  4. 4
    Percent below Q1Q_1: by definition, 25% of data falls below Q1=60Q_1 = 60

Answer

Q3=89Q_3 = 89 (75th percentile); 25% of students scored below Q1=60Q_1 = 60.
Quartiles are specific percentiles: Q1 = P25, Q2 = P50, Q3 = P75. By definition, 25% of data falls below Q1 and 25% above Q3, making quartiles ideal for summarizing where the bulk of data lies.

About Quartiles

Quartiles divide an ordered data set into four equal parts: Q1 is the 25th percentile, Q2 is the median (50th), and Q3 is the 75th percentile.

Learn more about Quartiles →

More Quartiles Examples

Example 1 easy

Find the quartiles [formula], [formula], and [formula] for the data set: [formula].

Example 2 medium

A dataset of 10 values (sorted): [formula]. Find all quartiles and the five-number summary.

Example 3 medium

Find [formula], [formula], and [formula] for the dataset: [formula].

Example 4 easy

Find [formula] and [formula] for: [formula].

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