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Learning outcomes

  • define percentile and quartile
  • identify Q1, Q2, and Q3
  • compute interquartile range
  • explain why IQR is useful in describing spread

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Percentiles

  • Percentiles divide ordered data into 100 parts.
  • The pth percentile is a value below which about p% of observations lie.
Example idea:
  • 25th percentile means roughly 25% of observations are at or below that point.

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Quartiles

  • Quartiles divide ordered data into four parts.
Important quartiles:
  • Q1 = first quartile = 25th percentile
  • Q2 = second quartile = median = 50th percentile
  • Q3 = third quartile = 75th percentile

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Interquartile range

  • IQR = Q3 - Q1
What it measures:
  • spread of the middle 50% of the data

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Why IQR is useful

  • It is less affected by extreme values than total range.
  • It gives a better picture of the central spread.

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Example

Ordered data:
  • 2, 4, 6, 8, 10, 12, 14, 16
Median:
  • Q2 = (8 + 10)/2 = 9
Lower half:
  • 2, 4, 6, 8 -> Q1 = (4 + 6)/2 = 5
Upper half:
  • 10, 12, 14, 16 -> Q3 = (12 + 14)/2 = 13
IQR:
  • 13 - 5 = 8

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Exam hints and traps

  • Quartiles require ordered data.
  • Q2 is the median.
  • IQR is subtraction, not division.
  • Percentiles describe position, not frequency count alone.

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Quick practice

Data:
  • 1, 3, 5, 7, 9, 11, 13, 15
  1. Find Q2.
  2. Find Q1.
  3. Find Q3.
  4. Find IQR.

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Answer key

  1. (7 + 9)/2 = 8
  2. (3 + 5)/2 = 4
  3. (11 + 13)/2 = 12
  4. 12 - 4 = 8