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IB DP Maths AA · Topic 4 Statistics and probability · SL 4.1–4.3, 4.5, 4.6, 4.11 · AHL 4.13 · ~9 min read

Statistics and probability

Sampling and data, mean, median, mode, quartiles, standard deviation, outliers, box plots and cumulative frequency, the effect of transforming data, probability rules, Venn and tree diagrams, conditional probability and independence, and Bayes’ theorem for HL.

Sampling & biasMean, median, modeQuartiles & IQRStandard deviationOutliersBox plotsCumulative frequencyVenn diagrams
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No tag = SL and HLHL ONLY HL students onlyFORMULA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • Centre: mean, median, mode. Spread: range, IQR = Q3 − Q1, standard deviation σ. An outlier is more than 1.5 × IQR beyond a quartile.
  • Adding a constant to every value shifts the mean and median but leaves the spread alone. Multiplying by k multiplies both the centre and the spread by k.
  • Probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B), P(A ∣ B) = P(A ∩ B)/P(B). A and B are independent ⇔ P(A ∩ B) = P(A)P(B).
  • HL Bayes: reverse a conditional probability with a tree diagram: P(A ∣ B) = P(A ∩ B)/P(B), with P(B) summed over all branches.

By the end you can

  • Identify types of data, sampling methods and sources of bias.
  • Find and interpret measures of centre and spread, including from grouped data.
  • Draw and interpret box plots and cumulative frequency graphs, and identify outliers.
  • Describe the effect of adding or multiplying every value by a constant.
  • Use Venn diagrams, tree diagrams and sample space diagrams to find probabilities.
  • Use conditional probability and test for independence.
  • Use Bayes' theorem with two or three events. HL
Syllabus noteFrom first exams in 2029, bivariate statistics (scatter diagrams, correlation and regression lines) leaves the AA syllabus. If your exams are before 2029, you still need it: the short section near the end covers it.

1Data and sampling

  • Discrete data is counted (number of siblings); continuous data is measured (height, time).
  • Sampling methods: simple random, systematic (every kth), stratified (proportional to group sizes), quota, convenience. Know one advantage and one problem for each.
  • Bias comes from the method, not the size: a large convenience sample is still biased.
Common mistakeCalling a sample "random" just because you didn't choose it carefully. Random means every member had an equal chance of being picked.

2Centre and spread

x̄ = Σ f xn FORMULA BOOKLET
mean; for grouped data use the class midpoints
  • σ (population standard deviation) measures the typical distance from the mean. In IB exams, use the GDC for σ; you are not expected to use the formula by hand.
Worked example

Find the mean, median, quartiles and IQR of 3, 5, 7, 8, 9, 11, 12, 15, 18.

  1. mean = 88/9 = 9.78 A1
  2. median = 5th value = 9 A1
  3. Q1 = 6, Q3 = 13.5, so IQR = 7.5 A1A1
Worked example

Are there any outliers in that data set?

  1. Q1 − 1.5(7.5) = −5.25 and Q3 + 1.5(7.5) = 24.75 M1
  2. every value is between these, so no outliers A1
0510152025min 3Q₁ 6median 9Q₃ 13.5max 18IQR = 13.5 − 6 = 7.5. Outliers lie beyond Q₁ − 1.5 × IQR or Q₃ + 1.5 × IQR.
The box holds the middle 50% of the data; the whiskers reach the smallest and largest non-outliers.
Worked example

The ages of 30 people are grouped: 0–10 (4), 10–20 (10), 20–30 (12), 30–40 (4). Estimate the mean.

  1. midpoints 5, 15, 25, 35 M1
  2. (20 + 150 + 300 + 140)/30 M1
  3. ≈ 20.3 years A1
  • Why "estimate"? You only know which class each value is in, so the midpoint stands in for every value in the class.
Change to every valueMean, medianSD, IQR
add b+ bno change
multiply by a× a× ǀaǀ
y = ax + ba x̄ + bǀaǀσ
Worked example

The mean of a data set is 12 and the SD is 3. Each value is doubled and then 5 is added. Find the new mean and SD.

  1. new mean = 2(12) + 5 = 29 A1
  2. new SD = 2(3) = 6 (adding 5 doesn't change spread) A1

3Cumulative frequency

time (min)cum. freq.0102030405020406080≈ 26≈ 19≈ 35n = 80: read acrossfrom 40 (median),20 (Q₁) and 60 (Q₃)
Plot cumulative frequency at the upper class boundary, join with a smooth curve or straight lines, then read across and down.
  • Reading off: median at n/2, Q1 at n/4, Q3 at 3n/4. For "how many took longer than 30 minutes", read up from 30 and subtract from n.
  • Percentiles: the 90th percentile is read from 0.9n.

4Probability rules

P(A′) = 1 − P(A) FORMULA BOOKLET
complement
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) FORMULA BOOKLET
addition rule; for mutually exclusive events P(A ∩ B) = 0
P(A ∣ B) = P(A ∩ B)P(B) FORMULA BOOKLET
conditional probability
P(A ∩ B) = P(A)P(B) FORMULA BOOKLET
independent events only
UAB0.350.150.250.25P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.15, so P(A ∪ B) = 0.75
Fill a Venn diagram from the middle outwards, then make sure everything adds to 1.
Worked example

P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.15. Find P(A ∪ B) and P(A ∣ B), and decide whether A and B are independent.

  1. P(A ∪ B) = 0.5 + 0.4 − 0.15 = 0.75 A1
  2. P(A ∣ B) = 0.15/0.4 = 0.375 A1
  3. P(A)P(B) = 0.2 ≠ 0.15, so not independent R1
Common mistakeConfusing "mutually exclusive" with "independent". Mutually exclusive events can't happen together (P(A ∩ B) = 0); independent events don't affect each other. Two events with non-zero probability can't be both.

5Tree diagrams

RB5/83/8R4/7RR 20/56B3/7RB 15/56R5/7BR 15/56B2/7BB 6/565 red, 3 blue: two picks without replacement
Multiply along the branches; add the end results you want.
Worked example

A bag has 5 red and 3 blue counters. Two are taken without replacement. Find P(both the same colour).

  1. P(RR) = 58 × 47 = 2056 M1
  2. P(BB) = 38 × 27 = 656 M1
  3. total = 2656 = 1328 A1
Worked example

Given that both counters are the same colour, find the probability that they are both red.

  1. P(RR ∣ same) = 20/5626/56 M1
  2. = 1013 A1
Exam tip"Given that" means conditional probability: the denominator is the probability of the condition, here P(same colour).

6Bayes' theorem HL

P(B ∣ A) = P(B)P(A ∣ B)P(B)P(A ∣ B) + P(B′)P(A ∣ B′) FORMULA BOOKLET
the tree-diagram version of P(B ∣ A) = P(A ∩ B)/P(A)
DD′0.020.98+0.950.019−0.050.001+0.100.098−0.900.882Disease test: 2% have it, 95% detected, 10% false positives
Most positive tests come from the large healthy group, which is why the answer is so low.
Worked example

2% of people have a disease. A test detects 95% of cases but gives a positive result for 10% of healthy people. Find P(disease ∣ positive).

  1. P(+) = 0.02(0.95) + 0.98(0.10) = 0.019 + 0.098 = 0.117 M1A1
  2. P(D ∣ +) = 0.0190.117 M1
  3. = 0.162 A1
  • Three or more events: the same idea works with three first-stage branches (say, three factories making a part): the denominator adds three products.

7Bivariate statistics (before 2029 only)

  • Pearson's r (−1 ≤ r ≤ 1) measures linear correlation; use the GDC. r close to ±1 is strong, close to 0 is weak. Correlation is not causation.
  • Regression line y on x predicts y from x; x on y predicts x from y. Both pass through (x̄, ȳ). Only interpolate: predictions outside the data range are unreliable.
GDC (calculator) methodPaper 2: enter data as a list (with a frequency list for grouped data) and use 1-variable statistics for x̄, σ, Q1, median and Q3. Write down the values you use; "from GDC" with the numbers is accepted.
Linking questionProbability distributions (Random variables, binomial and normal) are built on these rules, and the mean and SD of a distribution are the theoretical versions of x̄ and σ here.
Idea for your IA / explorationCollect your own data (reaction times, commute times, sleep) and compare two groups with box plots and summary statistics, or test a claim using conditional probability. Explain the sampling method and possible bias.

Test yourself — cover the answers first

Quick check

  1. Is "time to run 100 m" discrete or continuous?
  2. Find the median of 4, 9, 2, 7, 5, 1.
  3. Every value is increased by 10. What happens to the IQR?
  4. P(A) = 0.3. Find P(A′).
  5. A and B are independent, P(A) = 0.4 and P(B) = 0.5. Find P(A ∩ B).
  6. HL Give the denominator in Bayes' theorem in words.

Exam-style

  1. In a class of 30, 18 study French, 12 study Spanish and 5 study both. A student is chosen at random. Find the probability that they study French given that they study Spanish, and the probability that they study neither.
  2. HL Machines A, B and C make 50%, 30% and 20% of the parts in a factory. 2%, 3% and 5% of their parts are faulty. A part is faulty. Find the probability it came from C.
Show answers
  1. continuous
  2. 4.5
  3. no change
  4. 0.7
  5. 0.2
  6. the total probability of the given event
  7. P(F ∣ S) = 512
    neither = 30 − (18 + 12 − 5) = 5
    P(neither) = 16
  8. P(F) = 0.5(0.02) + 0.3(0.03) + 0.2(0.05) = 0.01 + 0.009 + 0.01 = 0.029
    P(C ∣ F) = 0.01/0.029
    = 0.345
Practise this chapter
IB Maths AA practice papers
Next chapterRandom variables, binomial and normal distributions →
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by the International Baccalaureate Organization. Syllabus codes refer to the IB DP subject guides.