Measures of Central Tendency (AA HL)
Mean, median and mode all answer "what's typical?", but exam questions rarely just ask you to calculate one from a plain list. More often you're weighting two components, hunting for a missing frequency, or merging two groups into one combined average. This page works through those variations. It's part of the broader Descriptive Statistics topic.
17 questions on this sub-topic.
The mean, weighted and combined
Covered under IB syllabus reference SL4.3: measures of central tendency (mean, median, mode), including estimating the mean of grouped data from midpoints. The mean formula \(\bar x=\dfrac{\sum fx}{\sum f}\) is in the formula booklet; the combined-mean formula for merging two groups isn't - it's built from first principles.
Mean of raw or grouped data
\(\bar x = \dfrac{\sum fx}{\sum f}\)
For raw data every frequency is \(1\), so this reduces to \(\bar x = \tfrac{\sum x}{n}\). For grouped data, \(x\) is the midpoint of each class.
Combined mean of two groups
\(\bar x_{\text{combined}} = \dfrac{n_1\bar x_1 + n_2\bar x_2}{n_1 + n_2}\)
Each group's total contributes to a shared grand total before dividing by the combined size.
Worked examples
A course grade weights coursework 30% and the exam 70%. A student scores 80 on coursework and 65 on the exam.
Find the overall grade.
Worked solution
Overall \(= 0.30(80) + 0.70(65)\) M1
\(= 24\) A1
\(+ 45.5\) M1
\(= 69.5.\) A1
Group A has 15 values with mean 20. Group B has 25 values with mean 28.
(a) Find the combined mean of all 40 values.
(b) Group A is found to have an extra value omitted; recompute the mean of A if a value of 52 is added to it.
Worked solution
(a) Combined mean. Total \(= 15(20) + 25(28) = 1000\); M1
mean \(= \dfrac{1000}{40} = 25.\) A1
(b) New mean of A. New total \(= 300 + 52 = 352\) over 16 values; M1
mean \(= \dfrac{352}{16}\) A1
\(= 22.\)
Common mistakes
- Averaging the two group means directly. The combined mean isn't \(\tfrac{\bar x_1 + \bar x_2}{2}\) unless the groups happen to be the same size - you need to weight by \(n_1\) and \(n_2\) first.
- Using the class boundary instead of the midpoint. Estimating the mean of grouped data needs the midpoint of each class interval, not its lower or upper boundary.
- Losing track of an updated total when a value changes. If a value is added or corrected after the mean was first found, recompute the total from scratch rather than adjusting the old mean by eye.
Ready to practise properly?
17 central-tendency questions, marked instantly like the real exam. Need a calculator refresher first? See the parent topic's GDC guidance.
Quick answers
What is the formula for the mean of grouped data?
The mean of grouped or frequency data is \(\bar x = \dfrac{\sum fx}{\sum f}\), where \(x\) is each value or midpoint and \(f\) is its frequency.
How do you find a missing frequency if you're given the mean?
Write the mean formula \(\dfrac{\sum fx}{\sum f} = \) the given mean with the unknown frequency as a variable, then cross-multiply and solve the resulting linear equation.