Mean, median, mode, and IQR (IB Math AA SL)

Key ideas

• Choose the measure to fit the data: with outliers or a skewed distribution, the median and IQR describe the data better than the mean and standard deviation.
• IQR = Q₃ − Q₁. Outliers: values below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR.
• Use the GDC's one-variable statistics for the mean, standard deviation and quartiles. IB questions treat the data as the whole population, so use σx, not sx.
• Adding k to every value adds k to the mean, median and quartiles but leaves the standard deviation and IQR unchanged. Multiplying every value by k multiplies all of them by k.
• For grouped data, use the mid-interval values: the mean you get is an estimate, so say so.

Worked example

The times, in minutes, that nine students took to finish a puzzle were 4, 7, 8, 9, 10, 12, 13, 15 and 31. Find the median and the IQR, and decide whether 31 is an outlier.

1. The data are in order and n = 9, so the median is the 5th value: 10 minutes.
2. Q₁ is the median of 4, 7, 8, 9: (7 + 8) ÷ 2 = 7.5. Q₃ is the median of 12, 13, 15, 31: (13 + 15) ÷ 2 = 14.
3. IQR = 14 − 7.5 = 6.5.
4. Upper fence = 14 + 1.5 × 6.5 = 23.75. Since 31 > 23.75, 31 is an outlier.
5. The mean is 109 ÷ 9 = 12.1, above the median because the outlier pulls it up.

Answer: Median 10 minutes, IQR 6.5 minutes; 31 is an outlier.

Exam-style SL AA questions on this topic

• Measures of Central Tendency questions — 24 questions with mark schemes, plus free worked examples
• All SL AA Statistics and Probability questions by topic

Do it on your GDC

A calculator is allowed on AA Paper 2, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• 1-variable and 2-variable statistics on your calculator
• Try it: standard deviation calculator · mean, median and mode calculator (free, shows the working)

Maths IA ideas that use mean, median, mode, and iqr

• Do more doctors mean fewer infant deaths?
• Are our summers getting wetter or drier?
• One snake queue or a queue at every till? Simulating supermarket checkouts
• Which method measures a building's height most accurately?
• IA statistics, step by step: descriptive statistics and box plots
• IA statistics, step by step: cleaning data and outliers
• All SL AA IA ideas

Related in Statistics and Probability

• Probability — Venn diagrams and independence
• Binomial distribution — B(n,p)
• Normal distribution and z-scores

Revise Mean, median, mode, and IQR for the exam

• SL AA revision notes — every unit summarised, with the key results to learn
• SL AA past papers — specimen papers and timed practice papers
• Formulas for Statistics and Probability — on the one-page SL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• SL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

TOK link: Can a larger set of data ever give less reliable knowledge than a smaller one?

Stretch yourself: original probability problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S101 · Problem S29 · Problem S102 · Problem S30. All 20 →

Formulas in this topic: Standard deviation. All formulas A–Z