Statistics and Probability — IB Math AI SL Study Guide

For: IB Mathematics Applications & Interpretation SL candidates.

Covers: IB AI SL Topic 4 — descriptive statistics (mean, median, variance), correlation and Pearson's $r$, linear regression, probability rules, conditional probability, normal distribution, chi-squared test for independence and goodness of fit.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the IB AI SL style for educational use. They are not reproductions of past IBO papers.

1. Why Stats & Prob in AI SL

Topic 4 is one of the largest in AI SL (about 25% of assessment) and the IA modelling almost always involves statistical analysis. Get fluent with your GDC's statistics functions — most calculations are 1-2 button presses.

2. Descriptive statistics

For a dataset $x_1,x_2,\dots ,x_n$:

• Mean: $\bar{x}=\frac{1}{n}\sum x_i$.
• Median: middle value when sorted (or average of two middles).
• Mode: most frequent value(s).
• Variance: ${\sigma }^2=\frac{1}{n}\sum (x_i-\bar{x}{)}^2$. (Sample variance uses $n-1$ instead.)
• Standard deviation: $\sigma =\sqrt{{\sigma }^2}$.
• Range: max − min. IQR: $Q_3-Q_1$.

Box plot uses median, $Q_1$, $Q_3$, min, max. Outliers (any of: <$Q_1-1.5\text{IQR}$ or >$Q_3+1.5\text{IQR}$) shown as crosses.

3. Correlation and regression

For paired data $(x_i,y_i)$:

Pearson's correlation coefficient $r$ measures linear association: $-1\le r\le 1$.

• $r=1$: perfect positive linear; $r=-1$: perfect negative; $r\approx 0$: no linear association.
• Strong $r$ does not imply causation.

Linear regression fits $y=ax+b$ via least-squares (your GDC computes $a,b,r$ directly).

Prediction within data range = interpolation (reliable). Prediction outside range = extrapolation (less reliable, especially as $r$ is far from $\pm 1$).

4. Probability rules

For events $A$ and $B$:

• $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
• $P(A^{\prime })$ (complement) $=1-P(A)$.
• Mutually exclusive: $P(A\cap B)=0$, so $P(A\cup B)=P(A)+P(B)$.
• Independent: $P(A\cap B)=P(A)\cdot P(B)$.
• Conditional: $P(A|B)=\frac{P(A\cap B)}{P(B)}$.

Tree diagrams are heavily examined — start with the first event, branch by outcome, then continue. Multiply along branches; add across branches.

5. Normal distribution

If $X\sim N(\mu ,{\sigma }^2)$, $X$ is bell-shaped, symmetric about $\mu$.

Standard normal: $Z=(X-\mu )/\sigma$, with $\mu =0$, $\sigma =1$. Tables and GDC give $P(Z<z)$.

68-95-99.7 rule: about 68% within 1$\sigma$, 95% within 2$\sigma$, 99.7% within 3$\sigma$.

To find $P(a<X<b)$: standardise to $z_a,z_b$, then $P(z_a<Z<z_b)=\Phi (z_b)-\Phi (z_a)$.

Inverse normal: given probability, find the $x$. GDC's invNorm function.

6. Chi-squared test (AI SL)

The ${\chi }^2$ test of independence checks whether two categorical variables are associated.

1. Set up null hypothesis: variables are independent.
2. Compute expected frequencies: $E_{ij}=(\text{row total})(\text{col total})/(\text{grand total})$.
3. Compute test statistic: ${\chi }^2=\sum (O-E{)}^2/E$.
4. Degrees of freedom: $(r-1)(c-1)$ for an $r\times c$ table.
5. Compare to critical value or use $p$-value from GDC. Reject if ${\chi }^2>$ critical, or $p<$ significance level (usually 0.05).

Conditions: all expected frequencies $\ge 5$ (or pool categories if not).

7. Worked Example

A survey of 100 students gives data on whether they cycle to school and whether they live within 2 km. Test at 5% significance whether cycling and proximity are independent.

Solution.

Expected frequencies: $E_{11}=(40)(35)/100=14$, $E_{12}=(40)(65)/100=26$, $E_{21}=(60)(35)/100=21$, $E_{22}=(60)(65)/100=39$.

${\chi }^2=\frac{(25-14{)}^2}{14}+\frac{(15-26{)}^2}{26}+\frac{(10-21{)}^2}{21}+\frac{(50-39{)}^2}{39}$ $=\frac{121}{14}+\frac{121}{26}+\frac{121}{21}+\frac{121}{39}$ $=8.64+4.65+5.76+3.10=22.16$

Degrees of freedom: $(2-1)(2-1)=1$. Critical ${\chi }^2$ at $\alpha =0.05$, df = 1: 3.841.

22.16 > 3.841, so reject $H_0$. Conclusion: cycling and proximity are not independent at 5% significance.

8. Common Pitfalls

• Sample vs population variance: AI SL uses population formula ($n$ in denominator) by default. Some questions specify sample (use $n-1$).
• Tree diagram with replacement: probabilities along the branches stay the same; without replacement, they change after each draw.
• Extrapolation reliability: report $r$ before predicting; if $r$ is weak, even interpolation is shaky.
• Chi-squared expected frequency: must be $\ge 5$ for the test to be valid. If not, pool categories or use Fisher's exact (out of AI SL scope).

9. Practice Questions

1. Find the mean and variance of {3, 7, 8, 5, 12, 4}.
2. The number of customers entering a shop per minute has $\mu =4$, $\sigma =2$ (Normal). Find $P(2<X<7)$.
3. In a sample of 200, 60 own a car. Construct the 95% confidence interval for the population proportion.

10. Quick Reference Cheatsheet

• Mean = $\sum x/n$; population SD uses $n$, sample uses $n-1$.
• IQR = $Q_3-Q_1$; outliers $\notin [Q_1-1.5\text{IQR},Q_3+1.5\text{IQR}]$.
• Pearson $r$: −1 to 1; strong = magnitude near 1.
• Independent: $P(A\cap B)=P(A)P(B)$.
• Conditional: $P(A|B)=P(A\cap B)/P(B)$.
• Normal: $P(\mu -\sigma <X<\mu +\sigma )\approx 0.68$.
• ${\chi }^2$: $\sum (O-E{)}^2/E$; df = $(r-1)(c-1)$.

11. What's Next

Stats & Prob ties to Topic 5 (Calculus) for continuous distributions and expectation as integral. IA modelling almost always involves regression or chi-squared. Use Ollie for any specific test or distribution: "Set up the chi-squared step by step for this 3×3 table" or "Why is my GDC giving me a different SD?"