IB Math AI HL Univariate Statistics — Complete Cheatsheet
Every measure of centre and spread — mean, median, standard deviation and IQR — plus grouped-data estimates, cumulative-frequency graphs, box plots and the 1.5×IQR outlier test for IB Mathematics Applications & Interpretation HL. Built by an IBO-certified Singapore tutor, with a print-ready PDF to download.
Univariate statistics is where the Statistics & Probability strand of IB Mathematics Applications & Interpretation HL begins, and it is pure AI in spirit: you are handed real data and asked to summarise, picture and interpret it rather than to prove anything. A calculator is always allowed, so almost every statistic — mean, median, standard deviation, quartiles — is read straight off the GDC's 1-variable statistics screen. The marks live in choosing the right measure (centre versus spread, mean versus median), reading the correct value off the calculator, and interpreting it in context — not in hand-grinding arithmetic.
This cheatsheet condenses SL 4.1–4.3 onto one page — sampling and data types, measures of central tendency and spread, grouped-data estimates, cumulative-frequency graphs, box plots and the $1.5\times\text{IQR}$ outlier rule — and flags the traps that quietly cost marks ($\sigma_x$ versus $s_x$, dividing by $\sum f$, boundary outliers). The print-ready PDF is at the bottom, free to download.
§1 — Sampling & types of data SL 4.1
Before you calculate anything, classify the data and check how it was collected — a biased sample makes every statistic that follows meaningless.
Data types:Discrete data are counted (whole numbers); continuous data are measured (any value in a range).
Population vs sample:A sample is a subset used to estimate a feature of the whole population.
Systematic sample:$k=\dfrac{\text{population}}{\text{sample size}}$, then take every $k$-th item from a random start $r$: $\;r,\,r+k,\,r+2k,\dots$
TrapThe sampling interval is exactly population $\div$ sample size — no rounding up and no $+1$.
NoteSystematic sampling is quick but becomes biased if the list hides a periodic pattern that lines up with $k$. Other techniques you should recognise: simple random, stratified (sample each group in proportion), quota and convenience.
§2 — Mean, median & mode SL 4.3
The three "averages" each summarise the centre of a data set with a single number.
Mean:$\bar{x}=\dfrac{\sum x}{n}$ — the balance point; every value counts.
Median:The middle of the ordered list; for $n$ values it sits in position $\dfrac{n+1}{2}$.
Mode:The most frequent value — a data set can have none, one or several.
TrapThe mean is dragged towards extreme values: a single outlier can pull it well away from the median. For skewed data or data with outliers, the median is the more representative centre.
TrickAdd first, divide last — and on the GDC, 1-Var Stats returns $\bar{x}$ and the median together. Sanity-check that $\bar{x}$ lies between the smallest and largest value.
§3 — Mean & standard deviation from the GDC SL 4.3
Real data usually arrive in a frequency table. Enter the values in one list and their frequencies in another, then read everything off 1-Var Stats.
Mean:$\bar{x}=\dfrac{\sum fx}{\sum f}$ — multiply each value by its frequency, add, then divide by $\sum f$.
Standard deviation:$\sigma=\sqrt{\dfrac{\sum f\,(x-\bar{x})^{2}}{\sum f}}$ — the typical distance of a value from the mean.
On the GDC:Read $\bar{x}$ and $\sigma_x$ directly; here $\sum f=n$ is the number of data points.
TrapIB uses the population standard deviation: take $\sigma_x$ from the calculator, never $s_x$ (the sample value, which divides by $n-1$).
NoteA larger $\sigma$ means less consistent data. The variance is simply $\sigma^{2}$.
§4 — Range, quartiles & the IQR SL 4.3
Centre alone hides the story — two data sets with the same mean can look completely different once you measure their spread.
Range:$x_{\text{max}}-x_{\text{min}}$ — simple, but wrecked by a single outlier.
Quartiles:$Q_1$ is the median of the lower half; $Q_3$ is the median of the upper half.
IQR:$\text{IQR}=Q_3-Q_1$ — the spread of the middle $50\%$ of the data.
NoteThe IQR ignores the extreme quarter at each end, so it is a resistant measure of spread — untroubled by outliers, unlike the range.
TrapSort the data first, and build the IQR from the quartiles ($Q_3-Q_1$) — never from the maximum and minimum.
§5 — Grouped data: estimated mean & modal class SL 4.3
When data are grouped into classes the raw values are lost, so we represent each class by its midpoint.
Midpoint:$m=\dfrac{\text{lower}+\text{upper}}{2}$ — e.g. the class 20–30 has midpoint 25.
Estimated mean:$\bar{x}=\dfrac{\sum fm}{\sum f}$ — enter the midpoints and frequencies into 1-Var Stats.
Modal class:The class with the highest frequency.
TrapDivide by $\sum f$ (the total frequency), never by the number of classes. And the answer is only an estimate — the original values are gone.
NoteIf the classes have unequal widths, the modal class is the one with the greatest frequency density (frequency $\div$ width), not simply the greatest frequency.
§6 — Cumulative frequency & percentiles SL 4.2
A cumulative-frequency (CF) graph — the ogive — lets you estimate the median, quartiles and any percentile of grouped data.
Plot:Cumulative frequency against the upper class boundary, joined by a smooth curve.
Read-off positions:Median at $\dfrac{N}{2}$, $\;Q_1$ at $\dfrac{N}{4}$, $\;Q_3$ at $\dfrac{3N}{4}$, and the $p$-th percentile at $\dfrac{p}{100}\,N$.
Interpolation:$\text{value}=L+\dfrac{\text{pos}-\text{CF}_{\text{before}}}{f}\times w$, with $L$ the lower class boundary and $w$ its width.
TrickGo up from the CF axis at the required position, across to the curve, then down to read the value. For grouped data the GDC will not give an exact median — the graph or interpolation is the method.
NoteLinear interpolation assumes the values are spread evenly across each class, so any percentile read this way is an estimate.
§7 — Box-and-whisker plots SL 4.2
A box plot is the five-number summary drawn to scale — the fastest way to see centre, spread and skew at once.
A box-and-whisker plot of the five-number summary. The box spans Q₁ to Q₃ (its width is the IQR) with the median inside; the marked point lies beyond the upper fence Q₃ + 1.5×IQR, so it is flagged as an outlier.
Box:Spans $Q_1$ to $Q_3$, so its width is the IQR, with a line drawn at the median.
Whiskers:Reach to the smallest and largest values — or, on a modified box plot, to the last value inside the outlier fences.
GDC:1-Var Stats lists $\min X,\ Q_1,\ \text{Med},\ Q_3,\ \max X$ — the whole summary on one screen.
TrickTo compare two data sets, draw parallel box plots on the same scale and compare medians (centre) and box widths (spread) at a glance.
NoteA long whisker or an off-centre median signals skew; roughly equal whiskers with the median in the middle suggests a symmetric distribution.
§8 — Outliers: the 1.5×IQR rule SL 4.2
IB uses one precise definition of an outlier, built entirely from the IQR.
Lower fence:$Q_1-1.5\,\text{IQR}$
Upper fence:$Q_3+1.5\,\text{IQR}$
Outlier:Any value below the lower fence or above the upper fence.
TrapIt is $1.5\times$ the IQR, not $1.5\times Q_3$. And a value sitting exactly on a fence is not an outlier — the inequality is strict.
NoteFind the IQR first, then the two fences. On a modified box plot each outlier is plotted as a separate point and the whisker stops at the last value still inside the fence.
§9 — Transformations & combining data sets SL 4.3
Two exam favourites: what a linear change does to your statistics, and how to merge or reverse-engineer means.
Transformation
Mean $\bar{x}$
Std dev $\sigma$
IQR / range
Add a constant $c$ ($x\to x+c$)
$\bar{x}+c$
unchanged
unchanged
Multiply by $k$ ($x\to kx$)
$k\bar{x}$
$|k|\,\sigma$
multiplied by $|k|$
Total trick:$\text{total}=\bar{x}\times n$ — reverse a mean, or update it when a value is added or changed.
Combined mean:$\dfrac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2}$ — the weighted mean of two groups.
Mean of an AP:$\bar{x}=\dfrac{S_n}{n}=a+\dfrac{(n-1)d}{2}=\dfrac{u_1+u_n}{2}$ — a symmetric data set, so its mean equals its median.
TrapNever average two means directly unless the groups are the same size — always weight by the counts $n_1$ and $n_2$.
TrickAdding a constant shifts the centre but leaves every measure of spread ($\sigma$, IQR, range) unchanged — a fact examiners love to test.
§10 — Exam attack plan All sections
Question cue
What to do
Watch for
"Find the mean / median / mode"
Enter the data in 1-Var Stats; read $\bar{x}$ and Med
Mean is dragged by outliers; median is resistant
"Find the standard deviation"
1-Var Stats, read $\sigma_x$
Use $\sigma_x$ (population), never $s_x$
"…from a frequency table"
Values in one list, frequencies in another; 1-Var Stats
Divide by $\sum f$, not the number of rows
"Estimate the mean of grouped data"
Use class midpoints as $x$ in 1-Var Stats
Only an estimate; divide by $\sum f$, not the number of classes
Sort first if by hand; IQR uses quartiles, not $\max-\min$
"Median / percentile from a graph"
Read across the CF curve at $\tfrac{N}{2},\tfrac{N}{4},\tfrac{3N}{4}$
Plot CF against the upper class boundary
"Is $X$ an outlier?"
Compute $Q_1-1.5\,\text{IQR}$ and $Q_3+1.5\,\text{IQR}$
$1.5\times$ IQR (not $Q_3$); on the boundary is not an outlier
"Effect of $+c$ or $\times k$"
$+c$ shifts the centre only; $\times k$ scales centre and spread
Adding $c$ leaves $\sigma$ and IQR unchanged
Worked Example — Reading Statistics off the GDC
Question (AI HL Paper 2 style — 8 marks)
The masses, in grams, of 11 apples from a crate are: 142, 150, 155, 158, 160, 162, 165, 168, 170, 175, 210. (a) Use your GDC to find (i) the mean mass and (ii) the standard deviation. (b) Find the median, the lower quartile $Q_1$, the upper quartile $Q_3$, and hence the interquartile range. (c) Using the $1.5\times\text{IQR}$ criterion, determine whether the 210 g apple is an outlier.
Solution
(a) Enter the 11 masses into 1-Var Stats. (M1) The calculator gives the mean $\bar{x}=165$ g (A1) and the population standard deviation $\sigma_x=16.8$ g (3 s.f.) (A1). (Read $\sigma_x$, not $s_x$.)
(b) With $n=11$ the median is the 6th ordered value, so $\text{median}=162$ g (A1). From 1-Var Stats, $Q_1=155$ g and $Q_3=170$ g (A1), hence $\text{IQR}=Q_3-Q_1=170-155=15$ g (A1).
(c) Upper fence $=Q_3+1.5\times\text{IQR}=170+1.5\times15=170+22.5=192.5$ g. (M1) Since $210>192.5$, the value lies beyond the upper fence, so the 210 g apple is an outlier. (A1)
Examiner's note: keep the units (g) and round to 3 s.f. unless told otherwise. Two marks are routinely thrown away here — by quoting $s_x$ instead of $\sigma_x$, and by testing against $1.5\times Q_3$ instead of $1.5\times\text{IQR}$. A value landing exactly on a fence would not count as an outlier.
Common Student Questions
Which average should I use — mean, median or mode?
Use the mean when the data are roughly symmetric with no outliers, since it uses every value. Switch to the median when there are outliers or the data are skewed, because the median is resistant and is not dragged towards an extreme value. Quote the mode for categorical data or when a question asks for the most common value. In AI, read both the mean and the median off 1-Var Stats and let the shape of the data decide which to report.
Should I use $\sigma_x$ or $s_x$ from my calculator?
Use $\sigma_x$. IB works with the population standard deviation, which divides by $n$ — that is the $\sigma_x$ output. The $s_x$ value divides by $n-1$ (the sample estimate) and is not what the mark scheme wants. The two are close for large data sets but differ enough to cost the accuracy mark on small ones, so pick $\sigma_x$ every time.
How do I find the quartiles — my calculator and the by-hand method sometimes disagree?
Different textbooks split the data slightly differently, but IB expects the calculator's values, so read $Q_1$, $\text{Med}$ and $Q_3$ straight from 1-Var Stats. If you must work by hand, sort the data, find the median, then take $Q_1$ as the median of the lower half and $Q_3$ as the median of the upper half. A tiny disagreement with a friend's answer is almost always this convention, not an arithmetic slip.
When is a value officially an outlier?
When it lies more than $1.5\times\text{IQR}$ beyond a quartile — that is, below $Q_1-1.5\,\text{IQR}$ or above $Q_3+1.5\,\text{IQR}$. Work out the IQR first, then the two fences, then compare. Two things to watch: it is $1.5\times$ the IQR, not $1.5\times Q_3$, and a value sitting exactly on a fence is not counted as an outlier.
Why are the mean and median from grouped data only "estimates"?
Because grouping throws away the raw values. For the mean we assume every value sits at its class midpoint; for the median from a cumulative-frequency graph we assume the values are spread evenly across each class (linear interpolation). Both assumptions are approximations, so the results are estimates — which is also why you cannot read an exact grouped median off the GDC and must use the midpoint method or the CF graph instead.
What's NOT in this cheatsheet
This page gives you the formulas and traps for free. The full Photon Academy IB Math AI HL library (enrolled students, or the lifetime resource library) adds:
Notes PDF — every definition, formula and GDC keystroke sequence, with fully worked grouped-data and outlier examples.
Tutorial booklet — 25+ AI HL-style questions from one-variable-stats fluency to full data-analysis applications (box plots, ogives, outliers).
Tutorial Solutions — mark-scheme-style solutions with M1/A1 annotations and the exact GDC screens to reproduce.
Predicted-paper questions — the univariate-statistics question types most likely to appear in the next AI HL session.
