Mode, median, quartiles and skewness

Median, quartiles and percentiles

• Median m: F(m) = 0.5.
• Lower quartile: F(Q1) = 0.25; upper quartile: F(Q3) = 0.75.
• The P th percentile: F(x) = P/100.

Solve the equation using the correct piece of F, and reject any root outside the range (examiners report students leaving impossible roots, such as negative values, or a value for m² instead of m). A common slip is to work out F(0.5) instead of solving F(m) = 0.5.

The mode

The mode is where f(x) is greatest. If f has a turning point inside the range, solve f'(x) = 0 (check it is a maximum). If f is increasing over the whole range, the mode is at the upper end; if decreasing, at the lower end. If the question says 'using calculus' and starts from F(x), differentiate twice: f = F', then solve f'(x) = 0.

Example: f(x) = (4/27)x²(3 - x) on 0 ≤ x ≤ 3. f'(x) = (4/27)(6x - 3x²) = 0 gives x = 2, so the mode is 2.

Skewness

• mean < median < mode: negative skew.
• mean = median = mode: symmetrical.
• mode < median < mean: positive skew.

Give a reason with the values. For f(x) = 3x²/8 on 0 ≤ x ≤ 2: mean 1.5, median 4^1/3 = 1.587, mode 2, so mean < median < mode: negative skew. Examiners give no marks for naming the skew without a reason.

Worked example

f(x) = 3x²/8 for 0 ≤ x ≤ 2. Find the median.

1. F(x) = x³/8 for 0 ≤ x ≤ 2.
2. F(m) = 0.5 gives m³/8 = 0.5, so m³ = 4.
3. m = 4^1/3

Answer: m = 1.587 (3 s.f.: 1.59)

Check yourself

1. f(x) = 3x²/8 for 0 ≤ x ≤ 2 and 0 otherwise. Find the median of X. Give your answer to 3 significant figures.
   Show answer

   1.59. m³/8 = 0.5, m = 4^1/3 = 1.5874.

2. f(x) = 3x²/8 for 0 ≤ x ≤ 2 and 0 otherwise. Find the upper quartile of X. Give your answer to 3 significant figures.
   Show answer

   1.82. x³/8 = 0.75, x³ = 6, x = 1.8171.

3. f(x) = 3x²/8 for 0 ≤ x ≤ 2 and 0 otherwise. Find the 10th percentile of X. Give your answer to 3 significant figures.
   Show answer

   0.928. x³/8 = 0.1, x³ = 0.8, x = 0.9283.

4. X has cdf F(x) = x²/16 for 0 ≤ x ≤ 4. Find the lower quartile.
   Show answer

   2. x²/16 = 0.25 gives x² = 4, so x = 2 (x = -2 is outside the range).

5. f(x) = (4/27)x²(3 - x) for 0 ≤ x ≤ 3 and 0 otherwise. Find the mode.
   Show answer

   2. f'(x) = (4/27)(6x - 3x²) = (4/27)3x(2 - x) = 0 gives x = 2 (x = 0 is a minimum).

6. For f(x) = 3x²/8 on 0 ≤ x ≤ 2, the mean is 1.5, the median is 1.59 and the mode is 2. Describe the skewness.
   • Positive skew, because mean < median < mode
   • Negative skew, because mean < median < mode
   • Symmetrical, because the values are close
   • Positive skew, because the mode is the largest
   Show answer

   Negative skew, because mean < median < mode. mean < median < mode is negative skew.

7. f(x) = x/2 for 0 ≤ x ≤ 2 and 0 otherwise. What is the mode?
   • 1, the middle of the range
   • 0, where f(x) = 0
   • There is no mode because f'(x) is never 0
   • 2, because f is increasing over the whole range
   Show answer

   2, because f is increasing over the whole range. f is greatest at the right-hand end of the range.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S2
• Pearson Edexcel IAL Statistics S2 (WST02) examiners' report, summer 2025

Last updated 8 November 2026. Spotted a mistake? Tell us.