Median and Quartiles: Position Measures Made Simple

The median is the middle value of data arranged in order — half the observations lie above it and half below. Quartiles extend this idea: the lower quartile (Q1), median (Q2) and upper quartile (Q3) split ordered data into four equal quarters, each containing 25% of the observations.

They matter because they describe position without being dragged around by extreme values. A neighbourhood's "typical" house price is better given by the median than the mean, since one mansion would distort the mean but cannot move the median.

Key Concepts

Median: the middle observation of ordered data; unaffected by outliers. Q1 (lower quartile): the value below which 25% of data falls. Q3 (upper quartile): the value below which 75% of data falls. Second quartile Q2 is the median itself. Position rule: for n observations, the p-th percentile position is (n+1)·p/100 in ordered data. Quartiles divide the range into four equal-count parts, not equal-width parts.

Formulas

Median of ungrouped data: if n is odd, median = value at position (n+1)/2; if n is even, median = average of values at positions n/2 and (n/2)+1. Quartile positions: Q1 at (n+1)/4, Q3 at 3(n+1)/4. Median of grouped data: Median = L + ((n/2 − cf)/f)·w, where L is the lower boundary of the median class, cf the cumulative frequency before it, f its frequency, and w the class width. Units are the units of the data.

Worked Example

Marks: 42, 55, 61, 48, 73, 58, 66. Step 1 — order them: 42, 48, 55, 58, 61, 66, 73 (n = 7, odd). Step 2 — median: position (7+1)/2 = 4th value = 58. Step 3 — quartiles: Q1 at (7+1)/4 = 2nd value = 48; Q3 at 3(7+1)/4 = 6th value = 66. So half the marks lie between 48 and 66. Grouped check: with classes 40–49 (f=3), 50–59 (f=7), 60–69 (f=6), 70–79 (f=4), n = 20: n/2 = 10 falls in 50–59 (cumulative before = 3). Median = 49.5 + ((10−3)/7)·10 = 49.5 + 10 = 59.5. Matches the ogive reading from the earlier example.

Common Misconceptions

1. "The median is always one of the data values." For an even count it is the average of two middle values and may never appear in the data. 2. "Quartiles split the value range into equal parts." They split the counts into equal quarters — Q1 to median and median to Q3 can span very different widths. 3. "Median needs sorted data but the mean doesn't." True for the median — sorting (or cumulative frequencies) is mandatory; an unsorted middle pick is meaningless.

Real-World Applications

Medians report typical house prices, typical salaries and typical waiting times — all skewed by extremes. Quartiles power box plots used in data science, set grade boundaries (top 25% get A), analyse income inequality, and help banks assess typical loan sizes without distortion from a few giant loans.

The median is the middle ordered value and the quartiles cut ordered data into equal quarters of 25% each. Resistant to outliers and readable from ogives, they describe where data sits far more honestly than the mean whenever extremes are present.