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CIE A-Level Maths Study Notes

5.3.4 Determining Medians and Percentiles

The concept of determining medians and percentiles from Probability Density Functions (PDFs) is crucial for a comprehensive understanding of continuous random variables. This aspect of statistics enables us to quantify and interpret the behavior of a dataset or distribution, providing a deeper insight into its characteristics.

Continuous Random Variables and PDFs

  • Continuous Random Variables: Can take infinite values within a range.
  • Probability Density Function (PDF): A curve where the area under it between two points represents the probability of the variable falling within that range. Characteristics include:
    • Non-Negativity: Always ≥ 0.
    • Total Area = 1: Integral over its range equals 1.

Medians and Percentiles

  • Median: The point dividing the distribution so 50% is to its left.
    • Calculation: Find the value where the area under the PDF to the left is 0.5.
  • Percentiles: Points below which a certain percentage of data falls.
    • Calculation: For the pth p^{th} percentile, find the value where the area under the PDF to the left equals p100\frac{p}{100} .
Median and Percentile

Example Problems

Example 1: Finding the Median

Calculate the median of a uniform distribution where ( X ) is distributed evenly between 0 and 10.


  • PDF: f(x)=0.1f(x) = 0.1 for 0x100 \leq x \leq 10.
  • Median: For a uniform distribution, the median is the midpoint of the range.
  • Computation: Median =0+102=5= \frac{0 + 10}{2} = 5.

Conclusion: The median of the distribution is 5.

Uniform Distribution Graph

Example 2: Calculating the 25th Percentile

Determine the 25th percentile for a continuous random variable YY with PDF f(y)=2yf(y) = 2y for 0y10 \leq y \leq 1.


  • 25th Percentile: The point where 25% of the values lie, or the CDF is 0.25.
  • Integral Setup: Integrate f(y)=2yf(y) = 2y from 0 to yy to find the CDF.
  • Calculation: Solve 0y2ydy=0.25\int_{0}^{y} 2y \, dy = 0.25.
  • Solve for yy: Find yy when y2=0.25y^2 = 0.25, which gives y=0.25=0.5y = \sqrt{0.25} = 0.5.

Conclusion: The 25th percentile of YY is 0.5.

Distribution Graph
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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