Percentiles and Quartiles

If your score is at the 75th percentile:

• 75% of scores are at or below yours
• 25% of scores are above yours
• You performed better than approximately 3/4 of test-takers

Data (sorted): 52, 58, 63, 67, 71, 74, 78, 82, 85, 91

What percentile is a score of 74?

Values ≤ 74: 52, 58, 63, 67, 71, 74 → 6 values Total values: 10

$\text{Percentile} = \frac{6}{10} \times 100 = 60\text{th percentile}$

A score of 74 is at the 60th percentile.

Data (sorted): 12, 18, 23, 27, 31, 35, 42, 48, 55, 62 (n = 10)

Find the 40th percentile:

$L = \frac{40}{100} \times (10 + 1) = 0.4 \times 11 = 4.4$

Position 4.4 means: 40% of the way between the 4th and 5th values.

• 4th value = 27
• 5th value = 31

$P_{40} = 27 + 0.4 \times (31 - 27) = 27 + 1.6 = 28.6$

The 40th percentile is 28.6.

|-------|-------|-------|-------|
Min    Q1     Q2     Q3     Max
      (25%)  (50%)  (75%)
       ↓      ↓      ↓
     25%    50%    75%
     below  below  below

Each section contains 25% of the data.

Data (sorted): 3, 7, 12, 15, 18, 21, 24, 28, 33 (n = 9)

Step 1: Find Q2 (median)

• Middle position = (9+1)/2 = 5th value
• Q2 = 18

Step 2: Find Q1 (median of lower half)

• Lower half: 3, 7, 12, 15
• Q1 = (7 + 12)/2 = 9.5

Step 3: Find Q3 (median of upper half)

• Upper half: 21, 24, 28, 33
• Q3 = (24 + 28)/2 = 26

Quartiles: Q1 = 9.5, Q2 = 18, Q3 = 26

Data (sorted): 5, 8, 12, 15, 19, 22, 26, 31 (n = 8)

Step 1: Find Q2 (median)

• Q2 = (15 + 19)/2 = 17

Step 2: Find Q1 (median of lower half)

• Lower half: 5, 8, 12, 15
• Q1 = (8 + 12)/2 = 10

Step 3: Find Q3 (median of upper half)

• Upper half: 19, 22, 26, 31
• Q3 = (22 + 26)/2 = 24

Quartiles: Q1 = 10, Q2 = 17, Q3 = 24

Data: 12, 18, 23, 27, 31, 35, 42, 48, 55, 62, 71, 85

Interpretation:

• Range: 85 - 12 = 73
• Middle 50% of data is between 25 and 58.5
• The distribution appears slightly right-skewed (larger gap above median)

Given Q1 = 25 and Q3 = 58.5:

$IQR = 58.5 - 25 = 33.5$

The middle 50% of values spans 33.5 units.

Data: 2, 15, 18, 21, 24, 27, 30, 33, 36, 42, 98

Step 1: Find quartiles

• Q1 = 18, Q3 = 36
• IQR = 36 - 18 = 18

Step 2: Calculate fences

• Lower fence = 18 - 1.5(18) = 18 - 27 = -9
• Upper fence = 36 + 1.5(18) = 36 + 27 = 63

Step 3: Identify outliers

• Below -9? No values
• Above 63? 98 is an outlier!

Also note: 2 is low but not an outlier (2 > -9)

Student A: 720 on Math SAT (75th percentile) Student B: 28 on ACT Math (75th percentile)

Different tests, different scales, but same relative performance! Both scored better than 75% of test-takers.

     Min        Q1      Q2        Q3        Max
      |---------|========|=========|---------|
     12        25      38.5       58.5      85
               |_______ IQR _______|
                      = 33.5

Components:

• Box: From Q1 to Q3 (contains middle 50%)
• Line inside box: Median
• Whiskers: Extend to min and max (within fences)
• Points beyond whiskers: Outliers

The 3rd decile (D3) = 30th percentile

Meaning: 30% of values fall below this point

A child’s weight is at the 60th percentile for their age.

Interpretation:

• 60% of children that age weigh less
• 40% weigh more
• The child is slightly above average but within normal range

Concern thresholds:

• Below 5th percentile: May indicate underweight
• Above 95th percentile: May indicate overweight

Pediatricians track percentiles over time—the trend matters more than a single measurement.

U.S. household income percentiles (approximate):

IQR = $125,000 -$35,000 = $90,000

The large gap between median and mean (mean ≈ $95,000) indicates right skew—high earners pull the mean up.