Statistics is a deceptively high-yield topic in IB Mathematics Analysis & Approaches HL. The formulas look gentle next to calculus, but the marks are routinely lost on small notational details — picking the wrong regression line, forgetting that adding a constant doesn't change the standard deviation, or multiplying 1.5 by the wrong quantity in the outlier fence. Examiners deliberately test these traps every session because they reveal whether a student really understands what each statistic measures, or is just feeding numbers into a GDC.
This cheatsheet condenses every formula, transformation rule, and exam-day decision from SL 4.1–4.4 and AHL 4.10–4.12 into one page you can revise from. It covers data classification, sampling, central tendency and dispersion, cumulative frequency and box plots, scatter diagrams and Pearson's $r$, the two regression lines, and the full attack plan for Paper 2 statistics questions. Scroll to the bottom for the printable PDF and the gated full library.
§1 — Data Types & Sampling SL 4.1
Data classifications
- Qualitative (categorical): described in words.
- Quantitative: always a number.
- Discrete: counted; takes specific values only.
- Continuous: measured; any value in a range.
- Population: entire group under study.
- Sample: representative subset of the population.
Five sampling methods
- Simple random (SRS) — random number generator; equal probability for every member.
- Systematic — every $k$-th member, where $k = N/n$.
- Stratified — random sample from each stratum.
- Quota — proportional to stratum size.
- Convenience — easiest accessible members.
Bias sources & reliability
| Bias source | Reliable data | Sufficient data |
|---|---|---|
| Sampling-frame exclusion | Repeatable results | Enough data points |
| Non-response | Similar findings | Supports conclusions |
| Poor question design | Controlled collection | Representative sample |
| Respondent dishonesty | Errors minimised | Errors logged |
TrickIn a 1-mark sampling question, name the method and its defining feature: "systematic — every $k$-th member". Both are needed for the mark.
TrapQuota $\neq$ Stratified. Quota is proportional to stratum size; stratified uses any fixed number from each stratum.
§2 — Measures of Central Tendency SL 4.3
Mean
Sample mean:$\bar{x} = \dfrac{\sum f_i x_i}{n}$
Population mean:$\mu = \dfrac{\sum f_i x_i}{n}$
Grouped data: use the midpoint of each class as $x_i$. The result is an estimate — the exact mean is unknowable.
Median & mode
- Median (ordered data): if $n$ is odd, the $\dfrac{n+1}{2}$-th term; if $n$ is even, the mean of the $\dfrac{n}{2}$-th and $\left(\dfrac{n}{2}+1\right)$-th terms.
- Mode: the most frequent value.
- Modal class (grouped data): the class with the highest frequency.
TrickIf the question gives "$n$ values, sum $= S$, mean $= \mu$", use $n = S/\mu$ to find $n$ in one step.
TrapFor grouped data you cannot find the exact mean — write "estimate" or use $\approx$. Also, the modal class is not the same as the mode.
§3 — Dispersion & Transformations SL 4.3
Key measures
Range:$x_{\max} - x_{\min}$
IQR:$Q_3 - Q_1$
Population SD:$\sigma = \sqrt{\dfrac{\sum (x_i - \mu)^2}{n}}$
Variance:$\sigma^2$
GDC symbols: $\sigma_x$ is the population SD; $s_x$ is the sample SD. Use $\sigma_x$ unless told "sample standard deviation".
Transformation rules
| Transform | Effect on $\bar{x}$ | Effect on $\sigma$ |
|---|---|---|
| Add constant $k$ | $\bar{x} + k$ | unchanged |
| Subtract $k$ | $\bar{x} - k$ | unchanged |
| Multiply by $k$ | $k\bar{x}$ | $k\sigma$ |
| Divide by $k$ | $\bar{x}/k$ | $\sigma/k$ |
Variance follows the same rules but squared: adding $k$ leaves variance unchanged; multiplying by $k$ scales variance by $k^2$.
Trick"Each value multiplied by 10" $\Rightarrow$ new $\bar{x} = 10\bar{x}$, new $\sigma = 10\sigma$, new variance $= 100\sigma^2$. Apply the scaling factor squared to the variance.
TrapAdding a constant changes the mean but never changes the standard deviation or variance. Students routinely add $k$ to $\sigma$ — do not.
§4 — GDC TI-Nspire: Statistics SL 4.3, 4.4
1-Variable Statistics
- Path: Lists & Spreadsheet. Col A header =
data; Col B =freq. - Enter values and frequencies.
- Menu $\to$ 4 $\to$ 1 $\to$ 1 (1-Var Stats).
- X1 List =
data; Freq List =freq.
| $\bar{x}$ | mean |
| $\sigma_x$ | population SD |
| $s_x$ | sample SD |
| $n$ | count |
| $Q_1$, Med, $Q_3$ | quartiles |
| Min, Max | extremes |
Linear Regression ($y$ on $x$)
- Path: Lists & Spreadsheet. Col A =
xdata; Col B =ydata. - Menu $\to$ 4 $\to$ 1 $\to$ 3 (Lin Reg $mx + b$).
- X List =
xdata, Y List =ydata. Output: $m$, $b$, $r^2$, $r$, $\bar{x}$, $\bar{y}$. - For the $x$-on-$y$ line: swap the lists (old $y$ becomes X List), re-run regression. This gives $x = ay + b$.
- Scatter plot: Home $\to$ Data & Statistics $\to$ Menu $\to$ 4 $\to$ 6 $\to$ 1 (Show Linear).
TrapThe GDC gives two SDs: $\sigma_x$ (population) and $s_x$ (sample). IB Math AA HL defaults to population — read $\sigma_x$ unless the question says "sample standard deviation".
Note2-Variable Stats: Menu $\to$ 4 $\to$ 1 $\to$ 2. Gives $\bar{x}$, $\bar{y}$, $\sigma_x$, $\sigma_y$ simultaneously — useful for confirming the mean point $(\bar{x}, \bar{y})$.
§5 — Cumulative Frequency SL 4.2
Construction rules
- Plot cumulative frequency on the $y$-axis.
- Plot each cf value against the upper class boundary.
- Start at $(x_{\min},\, 0)$; end at $(x_{\max},\, n)$.
- Join with a smooth S-shaped curve (the ogive).
Reading off the curve
| Statistic | Read from $y$-axis at | Read $x$ |
|---|---|---|
| $Q_1$ | $n/4$ | $\to$ curve $\to$ down |
| Median $Q_2$ | $n/2$ | $\to$ curve $\to$ down |
| $Q_3$ | $3n/4$ | $\to$ curve $\to$ down |
| $p$-th percentile | $pn/100$ | $\to$ curve $\to$ down |
TrickTo estimate the number of values above a threshold $t$: read off the cf at $t$, then subtract from $n$. Works for "more than" questions in reverse.
TrapAlways plot against the upper boundary — not the midpoint, not the lower boundary. Using the wrong boundary shifts the entire curve and every quartile will be wrong.
§6 — Box & Whisker Plots SL 4.2
Five-number summary
Min — $Q_1$ — Median — $Q_3$ — Max. The IQR $= Q_3 - Q_1$ is the box width.
Outlier rule
A value $x$ is an outlier if:
- $x < Q_1 - 1.5 \cdot \text{IQR}$ (below the lower fence), or
- $x > Q_3 + 1.5 \cdot \text{IQR}$ (above the upper fence).
Outliers are plotted as $\times$. Whiskers extend to the last non-outlier.
What box plots show — and don't show
Shows: Min, $Q_1$, Median, $Q_3$, Max, IQR, outliers, skewness. Does NOT show: mean, $\sigma$, exact values, or the number of data points in each region (only the proportion — always 25% per quarter).
Skewness from a box plot
| Symmetric | Positive skew | Negative skew |
|---|---|---|
| Whiskers equal length; median centred in box | Right whisker longer; median closer to $Q_1$ | Left whisker longer; median closer to $Q_3$ |
| Data approx normal | Long tail to right | Long tail to left |
TrickBox plots are compared using: median (centre), IQR (middle spread), range (total spread), and symmetry / skewness. Always address all four when comparing two distributions.
TrapEach quarter of a box plot contains exactly 25% of the data — but can span very different widths on the axis. A wide box half does not mean more data; it means more spread. A narrow box half means data is tightly clustered, not that fewer values are there.
§7 — Scatter Plots & Pearson's $r$ SL 4.4
Describing a scatter diagram
Always describe three features:
- Direction: positive ($+$) / negative ($-$).
- Form: linear / non-linear (curved).
- Strength: strong / moderate / weak.
Independent variable ($x$) on the $x$-axis; dependent variable ($y$) on the $y$-axis.
Pearson's correlation coefficient $r$
$-1 \le r \le 1$ (use the GDC to compute).
| $r = +1$ | perfect positive |
| $|r| > 0.75$ | strong |
| $0.5 < |r| < 0.75$ | moderate |
| $|r| < 0.5$ | weak |
| $r = 0$ | no linear correlation |
| $r = -1$ | perfect negative |
Only valid for linear relationships! Always check the scatter plot before trusting $r$.
Trick$r^2$ (the coefficient of determination) is given by the GDC alongside $r$. It tells you what fraction of the variation in $y$ is explained by $x$. Not required for IB but useful for interpreting strength.
Trap$r$ close to $\pm 1$ does NOT mean causation. Never write "$x$ causes $y$" based on correlation alone — always say "there is a (strong / positive) correlation between $x$ and $y$".
§8 — The Two Regression Lines SL 4.4, AHL 4.10
The two lines
- $y$-on-$x$ line: $y = ax + b$. Use when given $x$, predict $y$.
- $x$-on-$y$ line: $x = cy + d$. Use when given $y$, predict $x$.
Both lines pass through $(\bar{x},\, \bar{y})$ — solve simultaneously to recover $\bar{x}$ and $\bar{y}$.
Which line to use?
| Known | Want | Line |
|---|---|---|
| $x$ value | predict $y$ | $y$-on-$x$: $y = ax + b$ |
| $y$ value | predict $x$ | $x$-on-$y$: $x = cy + d$ |
To get the $x$-on-$y$ line on the GDC: swap the lists (old $y$ $\to$ X List, old $x$ $\to$ Y List), then re-run linear regression.
Interpreting regression coefficients in $y = ax + b$
| Gradient $a$ | Mean change in $y$ for each one-unit increase in $x$. Always state units. Must be contextualised: e.g. "population increases by 0.358 million per year". |
| Intercept $b$ | Predicted $y$ when $x = 0$. May have no physical meaning in context (e.g. negative price). Mention this if relevant. |
TrickBoth lines pass through $(\bar{x}, \bar{y})$. If you have both equations and need the means, solve them simultaneously — you get exact values of $\bar{x}$ and $\bar{y}$ in one step.
TrapNever rearrange the $y$-on-$x$ equation to predict $x$ — you will get the wrong answer. The $x$-on-$y$ line has a completely different gradient because it minimises horizontal residuals. These are different lines unless $|r| = 1$.
§9 — Interpolation, Extrapolation & Causation SL 4.4
| Interpolation | Extrapolation | Correlation $\neq$ Causation |
|---|---|---|
| Predict within data range | Predict outside data range | High $|r| \neq$ one causes other |
| More reliable | Less reliable / unreliable | Confounding variable may exist |
| Reliability $\propto |r|$ | Assume trend continues | Coincidence is possible |
| State if it is interp. | Must say "extrapolation" + "unreliable" | Never say "$x$ causes $y$" |
TrickThe 1-mark reliability question almost always wants two words: "extrapolation" and "unreliable". Write both explicitly. If it is interpolation, say "interpolation" and add a comment on the value of $r$.
TrapDo not say "unreliable because $r$ is not close to 1" for extrapolation — the reliability issue is the data range, not the correlation. State: "$x = k$ is outside the range of the data, so this is extrapolation and the prediction may be unreliable."
§10 — Exam Attack Plan All sections
When you see this in the question — reach for that:
| Question trigger | Reach for |
|---|---|
| "Describe the sampling method" | Name + defining feature (e.g. "systematic — every $k$-th") |
| "Identify a source of bias" | Name the type + give a specific example from context |
| "Sum $= S$, mean $= \mu$, find $n$" | $n = S/\mu$ directly |
| "Each value multiplied by $k$" | New mean $= k\bar{x}$, new $\sigma = k\sigma$, new variance $= k^2 \sigma^2$ |
| "Each value increased by $k$" | New mean $= \bar{x} + k$; $\sigma$ unchanged; variance unchanged |
| "Find the largest non-outlier" | $Q_3 + 1.5 \times \text{IQR}$ (upper fence); check if actual max $>$ fence |
| "Find minimum value of $Q_3$" | Set fence $=$ max, solve: $Q_3 + 1.5 \times \text{IQR} = \max$ |
| "Comment on reliability" | State interp./extrap. + "reliable/unreliable" + reason |
| "Estimate $y$ for given $x$" | Use the $y$-on-$x$ line |
| "Estimate $x$ for given $y$" | Use the $x$-on-$y$ line (swap lists on GDC) |
| "Find the mean of the data" | Both regression lines pass through $(\bar{x}, \bar{y})$ — solve simultaneously |
| "Interpret the value of $a$ in $y = ax + b$" | Change in $y$ per unit increase in $x$, with units, in context |
| "Describe the correlation" | Direction (positive/negative) + strength + linear |
| "Determine whether $X$ is an outlier" | Compute fence, compare $X$ to fence, state $X >$ or $<$ fence |
| "Compare two box plots" | Median, IQR, range, symmetry/skewness — address all four |
| "Modal class" | Class with highest frequency (grouped data only) |
| "Estimate the mean from grouped data" | Use midpoints; result is an estimate |
Top 5 marks lost — based on past papers
- Using the wrong regression line for prediction. If given $y$, use $x$-on-$y$. If given $x$, use $y$-on-$x$. Rearranging the wrong line costs all prediction marks.
- Outlier-fence formula error: writing $Q_3 + 1.5 \times Q_3$ instead of $Q_3 + 1.5 \times \text{IQR}$. Always compute IQR first, then multiply 1.5 by IQR — not by a quartile.
- Not saying "extrapolation" when asked about reliability. The exam expects the word explicitly. "The prediction may not be accurate" alone earns zero.
- Reading $s_x$ instead of $\sigma_x$ from GDC, or confusing variance and SD. Variance $= \sigma^2$. If the question asks for variance and you have $\sigma$, square it.
- Adding a constant and incorrectly changing the SD. "Each value increased by 5" changes only the mean. Standard deviation and variance are completely unaffected.
Worked Example — IB-Style Regression Analysis
Question (HL Paper 2 style — 8 marks)
For ten students at a Singapore IB school, the time $x$ (in hours) spent revising and the percentage score $y$ on the mock exam are recorded. The two regression lines obtained are $y = 4.2x + 28$ ($y$-on-$x$) and $x = 0.18y - 2.1$ ($x$-on-$y$). The data range is $4 \le x \le 18$.
(a) Find the mean revision time $\bar{x}$ and the mean score $\bar{y}$. (b) A student plans to revise for 12 hours. Estimate her score and comment on the reliability. (c) A student scored 95%. Estimate her revision time and comment.
Solution
- Both lines pass through $(\bar{x}, \bar{y})$. Substitute the second equation into the first: $y = 4.2(0.18y - 2.1) + 28$ (M1)
- Expand: $y = 0.756y - 8.82 + 28 \Rightarrow 0.244y = 19.18 \Rightarrow \bar{y} \approx 78.6$. Then $\bar{x} = 0.18(78.6) - 2.1 \approx 12.0$ (A1)(A1)
- (b) Given $x = 12$, use $y$-on-$x$: $\hat{y} = 4.2(12) + 28 = 78.4\%$ (M1)(A1)
- Reliability: $x = 12$ lies within the data range $[4, 18]$, so this is interpolation and the prediction is reasonably reliable. (R1)
- (c) Given $y = 95$, use $x$-on-$y$: $\hat{x} = 0.18(95) - 2.1 = 15.0$ hours (M1)(A1)
- Reliability: $y = 95$ is well outside the achieved range; substituting back, predicted $x = 15$ is still inside the range, but the score 95 is far from typical observed scores — borderline extrapolation. (R1)
Examiner's note: The trap in part (c) is using the $y$-on-$x$ line and rearranging $95 = 4.2x + 28$. That gives $x \approx 16.0$, which is wrong. You must use the $x$-on-$y$ line because $y$ is the known and $x$ is the prediction.
Common Student Questions
Which regression line should I use to predict $y$ from a known $x$?
Use the $y$-on-$x$ line, $y = ax + b$. The $y$-on-$x$ line minimises vertical residuals and is the only line designed to predict $y$ from $x$. Never rearrange the $y$-on-$x$ equation to predict $x$ — the $x$-on-$y$ line has a different gradient because it minimises horizontal residuals. The two lines are only the same when $|r| = 1$.
How do I tell whether a value is an outlier in IB Math AA HL?
Compute IQR $= Q_3 - Q_1$, then build the fences: lower $= Q_1 - 1.5 \times \text{IQR}$, upper $= Q_3 + 1.5 \times \text{IQR}$. Any data point below the lower fence or above the upper fence is an outlier. The most common error is multiplying $1.5$ by $Q_3$ instead of by IQR — always compute IQR first, then multiply.
What is the difference between $\sigma_x$ and $s_x$ on the GDC?
$\sigma_x$ is the population standard deviation (divides by $n$) and $s_x$ is the sample standard deviation (divides by $n - 1$). IB Math AA HL defaults to the population SD, so read $\sigma_x$ from the GDC unless the question specifically says "sample standard deviation". Picking the wrong one will silently lose every dispersion mark in the question.
If every data value is multiplied by $k$, how do the mean, SD and variance change?
The mean and SD scale by $k$: new mean $= k \bar{x}$, new SD $= k \sigma$. Variance scales by $k^2$ because variance is SD squared: new variance $= k^2 \sigma^2$. If you instead add a constant to every value, the mean shifts by that constant but the SD and variance are completely unchanged.
When IB asks me to comment on the reliability of a prediction, what should I write?
Two words must appear: state whether it is interpolation or extrapolation, and state whether it is reliable or unreliable. For extrapolation, the reason is that the $x$-value lies outside the range of the data, not that $r$ is small. For interpolation, comment on the magnitude of $|r|$ — the closer $r$ is to $\pm 1$, the more reliable the prediction.
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