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Median, quartiles and percentiles of a continuous random variable

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For the density on 2 to 6, solve the normalised area equation for the lower quartile root u

\[u^2=\frac{1}{4},\quad0\le u\le1\]
Year 12, VCE Units 3 & 4, Chapter 12, 4 difficulty levels, Calculator allowedBuild a worksheet

Percentiles from accumulated area

A percentile is a value with a specified amount of probability to its left. Find it by setting the area from the lower support endpoint equal to the target probability, then keep the admissible root

\[\int_a^m f(x)\,dx=p,\quad Q_1:\ p=\frac{1}{4},\quad\operatorname{median}:\ p=\frac{1}{2}\]
  1. Set an accumulated area A quantile is found by integrating from the lower support endpoint to the required value \(\displaystyle \int_a^m f(x)\,dx=p\)
  2. Use the target probability The area to the left of the quantile equals its cumulative probability \(\displaystyle \Pr(X\le m)=\frac{1}{4}\;\Rightarrow\;\int_a^m f(x)\,dx=\frac{1}{4}\)
  3. Keep the root inside the support If an area equation gives more than one root, keep the root in the normalised support \(\displaystyle u^2=\frac{1}{4},\quad0\le u\le1\;\Rightarrow\;u=\frac{1}{2}\)
  4. Subtract the quartiles The interquartile range is the upper quartile minus the lower quartile \(\displaystyle Q_1=3,\quad Q_3=7\;\Rightarrow\;\operatorname{IQR}=Q_3-Q_1=4\)

Worked example

Find the lower quartile for this density

\[f(x)=\frac{x-2}{8}\text{ for }2\le x\le6,\quad f(x)=0\text{ elsewhere};\qquad\int_2^{Q_1}\frac{x-2}{8}\,dx=\frac{1}{4}\]
  1. \(\displaystyle u=\frac{Q_1-2}{4},\quad0\le u\le1\)Scale the support so that its lower endpoint is 0 and its upper endpoint is 1.
  2. \(\displaystyle \int_0^u2t\,dt=u^2=\frac{1}{4}\)The accumulated area for the lower quartile is 1/4.
  3. \(\displaystyle u=\pm\frac{1}{2},\quad0\le u\le1\;\Rightarrow\;u=\frac{1}{2}\)Reject the negative root because it is outside the normalised support
  4. \(\displaystyle Q_1=2+4\left(\frac{1}{2}\right)=4\)Convert the normalised root back to the original support

Common mistakes

Using density height as probability
✕\(\displaystyle f(x)=2x,\quad0\le x\le1\;\qquad f(Q_1)=\frac{1}{4}\;\Rightarrow\;2Q_1=\frac{1}{4}\;\Rightarrow\;Q_1=\frac{1}{8}\)✓\(\displaystyle \int_0^{Q_1}2x\,dx=\frac{1}{4}\;\Rightarrow\;Q_1^2=\frac{1}{4}\;\Rightarrow\;Q_1=\frac{1}{2}\)
A quantile is determined by accumulated area, not by the height of the density at one value
Keeping a root outside the support
✕\(\displaystyle u=-\frac{1}{2}\)✓\(\displaystyle 0\le u\le1\;\Rightarrow\;u=\frac{1}{2}\)
The quantile must lie on the stated support, so a root below 0 or above 1 is rejected after normalising
Reversing the area limits
✕\(\displaystyle \int_{Q_1}^{2}f(x)\,dx=\frac{1}{4}\)✓\(\displaystyle \int_2^{Q_1}f(x)\,dx=\frac{1}{4}\)
The accumulated area starts at the lower support endpoint and ends at the quantile

Difficulty levels

introFind a median from a linear density with an exact rational root
standardFind quartiles or an interquartile range from a polynomial or piecewise-linear density
challengeFind an arbitrary percentile and report an exact surd or stated-precision value
examSelect the correct area equation and calculate an interquartile range

Questions about this topic

What does a percentile represent?+

The pth percentile is the value with cumulative probability p/100, so probability p/100 lies at or below that value

Why can an area equation have an unwanted root?+

Algebra can produce roots outside the density's support. Only a root inside the support represents a possible value of the random variable

How is the interquartile range found?+

Find the lower quartile at cumulative probability 1/4 and the upper quartile at cumulative probability 3/4. Subtract Q1 from Q3.

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 12: Continuous probability distributions

In this chapter

12A Continuous random variables and density functions: Validate and normalise a continuous probability density function12A Continuous random variables and density functions: Estimate probabilities from grouped continuous frequency dataGroup numerical data into class intervalsConstruct a histogram from a frequency table12A Continuous random variables and density functions: Finite events, probability assignments and complements12A Continuous random variables and density functions: Conditional probability from exact representationsEvaluate and determine function rules12B Probability density validity and normalisation: Validate and normalise a continuous probability density function12B Probability density validity and normalisation: Estimate probabilities from grouped continuous frequency dataSolve a linear equation12B Probability density validity and normalisation: Evaluate a polynomial or recover a coefficient parameterWork with sets and interval notation12B Probability density validity and normalisation: Round to decimal places and significant figuresInterval and conditional probabilities from a densityCompound events, the addition rule and mutual exclusivity12C Interval and conditional probabilities: Finite events, probability assignments and complements12C Interval and conditional probabilities: Conditional probability from exact representationsMean, variance and standard deviation of a continuous random variableMedian, quartiles and percentiles of a continuous random variableExpand and factor polynomial expressions12D Mean, variance, standard deviation and percentiles: Evaluate a polynomial or recover a coefficient parameterMove between roots, factors, polynomial rules and equationsSolve a quadratic exactly with the quadratic formula12D Mean, variance, standard deviation and percentiles: Round to decimal places and significant figuresLinear transformations and totals of independent continuous random variables