Cumulative distributions and percentiles

IB Mathematics AA SL· Unit 4: Statistics & Probability· 15 min read

1. 1. Cumulative Distribution Function (CDF) Basics★★☆☆☆⏱ 5 min

📘 Definition

Cumulative Distribution Function (CDF)

$F (x) = P (X \leq x)$

For any random variable $X$ , the CDF gives the total probability that $X$ takes a value less than or equal to a given input $x$ , and it ranges from 0 to 1.

Example:

For a fair 6-sided die, $F (3) = P (X \leq 3) = 0.5$

For discrete random variables, the CDF is a step function built by summing individual probabilities up to each value of $x$ . For continuous random variables, the CDF is a continuous increasing function found by integrating the probability density function (PDF) from negative infinity up to $x$ .

📐 Worked Example

A discrete random variable $X$ has probability distribution: $P (X = 1) = 0.2$ , $P (X = 2) = 0.3$ , $P (X = 3) = 0.4$ , $P (X = 4) = 0.1$ . Calculate the full CDF $F (x)$ .

1
For $x$ less than the smallest outcome 1: no probability is accumulated:
2
$F(x) = 0 \quad \text{for } x < 1$
3
For $1 \leq x < 2$ , sum probabilities up to 1:
4
$F(x) = 0.2$
5
For $2 \leq x < 3$ , add the probability of $X = 2$ :
6
$F(x) = 0.2 + 0.3 = 0.5$
7
For $3 \leq x < 4$ , add the probability of $X = 3$ :
8
$F(x) = 0.5 + 0.4 = 0.9$
9
For $x \geq 4$ , all probabilities are accumulated:
10
$F(x) = 0.9 + 0.1 = 1 \quad \text{for } x \geq 4$

2. 2. Calculating Percentiles and Median★★★☆☆⏱ 6 min

📘 Definition

k-th Percentile

The $k$ -th percentile $p_{k}$ is the value of $X$ that satisfies $F (p_{k}) = \frac{k}{100}$ . The median is the 50th percentile.

For discrete random variables, if $F (x)$ never exactly equals $\frac{k}{100}$ , the percentile is the smallest value of $x$ where $F (x) \geq \frac{k}{100}$ . For continuous random variables, you solve the equation $F (x) = \frac{k}{100}$ directly to get an exact value.

📐 Worked Example

Find the 50th percentile (median) for the CDF calculated in the previous section.

1
Write the condition for the 50th percentile:
2
$F(p_{50}) = \frac{50}{100} = 0.5$
3
Check the CDF values: $F (2) = 0.5$ , which matches the condition exactly.
4
Therefore, the 50th percentile (median) is 2.

📐 Worked Example

A continuous random variable $X$ has CDF $F (x) = 0.2 x$ for $0 \leq x \leq 5$ . Find the 25th percentile.

1
Set up the percentile condition:
2
$F(p_{25}) = 0.25$
3
Substitute the CDF expression and solve:
4
$0.2 p_{25} = 0.25 \implies p_{25} = \frac{0.25}{0.2} = 1.25$

3. 3. Interquartile Range (IQR) from CDF★★★☆☆⏱ 4 min

The interquartile range is a common measure of spread that ignores outliers. It is calculated as $I QR = Q_{3} - Q_{1}$ , where $Q_{1}$ (first quartile) is the 25th percentile and $Q_{3}$ (third quartile) is the 75th percentile. To find IQR from a CDF, you just calculate both quartiles then subtract.

📐 Worked Example

A continuous random variable $X$ has CDF $F (x) = \frac{x ^{2}}{16}$ for $0 \leq x \leq 4$ . Calculate the interquartile range.

1
Find $Q_{1}$ (25th percentile): set $F (Q_{1}) = 0.25$ :
2
$\frac{Q_1^2}{16} = 0.25 \implies Q_1^2 = 4 \implies Q_1 = 2$
3
Find $Q_{3}$ (75th percentile): set $F (Q_{3}) = 0.75$ :
4
$\frac{Q_3^2}{16} = 0.75 \implies Q_3^2 = 12 \implies Q_3 = 2\sqrt{3} \approx 3.464$
5
Calculate IQR:
6
$IQR = Q_3 - Q_1 = 2\sqrt{3} - 2 \approx 1.46$

4. Common Pitfalls

Wrong move:

Using $P (X < x)$ instead of $P (X \leq x)$ for discrete CDFs

Why:

This changes cumulative probability values for discrete distributions, leading to incorrect percentiles

Correct move:

Always include the current value of $x$ when calculating cumulative probability for discrete CDFs

Wrong move:

Using PDF directly to find percentiles

Why:

PDF values do not represent cumulative probability, so solving $f (x) = 0.5$ will not give the median

Correct move:

Always integrate the PDF to get the CDF first, then use the CDF to find percentiles

Wrong move:

Picking the wrong x for discrete percentiles when F(x) ≠ k/100

Why:

Many students pick the last x with F(x) < k/100, which contradicts the definition

Correct move:

Always pick the smallest x where F(x) ≥ k/100 for discrete distributions

Wrong move:

Forgetting to check that CDF ends at 1

Why:

A summation or integration error that leaves F(x) not equal to 1 will throw off all percentile calculations

Correct move:

Always verify that F(max(X)) = 1 before calculating any percentiles or quartiles

5. Quick Reference Cheatsheet

Concept	Discrete Random Variable	Continuous Random Variable
CDF	$F (x) = \sum_{t \leq x} P (X = t)$	$F (x) = \int_{- \infty}^{x} f (t) d t$
k-th Percentile Rule	Smallest $x$ with $F (x) \geq \frac{k}{100}$	Solve $F (x) = \frac{k}{100}$ for $x$
Quartiles	$Q_{1} = 2 5^{t h}, Q_{3} = 7 5^{t h}$ percentile	$Q_{1} = 2 5^{t h}, Q_{3} = 7 5^{t h}$ percentile
IQR Formula	$I QR = Q_{3} - Q_{1}$	$I QR = Q_{3} - Q_{1}$

6. Frequently Asked

What is the difference between CDF and PDF?

A PDF (probability density/mass function) gives the probability of $X$ taking a specific value, while a CDF gives the cumulative probability of $X$ being at or below a specific value. CDFs are always used to calculate percentiles.

How do I find percentiles for discrete CDFs when F(x) never equals k/100?

By definition, you select the smallest value of $x$ where $F (x) \geq \frac{k}{100}$ , which is the convention used in IB exams.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · 1
Find median from continuous CDF
2022 · 2
Calculate IQR from discrete CDF

Going deeper

review videoIB AA SL CDF and Percentiles WalkthroughCovers common exam question types

What's Next

Understanding cumulative distributions and percentiles is foundational for more advanced statistical concepts you will encounter in IB AA SL, including normal distribution probability calculations, confidence intervals, and descriptive data analysis. These skills are regularly tested in both paper 1 and paper 2 of the IB exam, often combined with other topics like discrete probability distributions and box plots. Mastery of CDFs and percentiles will also help you when interpreting and summarizing statistical results for your internal assessment project.