Measures of Central Tendency (AA SL)
Mean, median and mode all describe where the "centre" of a data set sits, but they answer slightly different questions and each has its own failure mode when the data comes as a frequency table rather than a raw list. This page covers the weighted-mean method examiners expect, with worked examples and the mistakes that cost marks. It's part of the broader Descriptive Statistics & Correlation topic.
21 questions on this sub-topic.
The key methods
Covered under IB syllabus reference SL4.3: measures of central tendency (mean, median, mode) and estimation of the mean from grouped data, including the modal class.
Weighted mean
\(\bar x = \dfrac{\sum fx}{\sum f}\)
Use whenever data comes as a frequency table. Each value \(x\) is repeated \(f\) times, so you can't just average the distinct values - you have to weight by frequency.
Median (ordered data)
Middle value if \(n\) is odd; mean of the two middle values if \(n\) is even.
For grouped data, the median is estimated from a cumulative frequency graph or by interpolation within the class that contains it.
Need the full syllabus wording and formula-booklet reference table? See Descriptive Statistics & Correlation.
Worked examples
Marks and frequencies are shown.
| Mark | 5 | 6 | 7 | 8 |
|---|---|---|---|---|
| Frequency | 2 | 4 | 3 | 1 |
Find the mean mark.
Worked solution
With a frequency table each mark \(x\) is repeated \(f\) times, so \(\bar x=\dfrac{\sum fx}{\sum f}\) (not the plain average of the distinct marks). M1
\(\sum fx = 5(2)+6(4)+7(3)+8(1)=10+24+21+8=63.\) A1
\(\sum f = 2+4+3+1=10\) marks in total. A1
\(\bar x=\dfrac{63}{10}=6.3.\) A1
For the data \(3, 5, 5, 6, 8, 8, 8, 10\):
(a) Find the mode.
(b) Find the median.
Worked solution
(a) The mode is the most frequent value; \(8\) appears three times (more than any other), so the mode is \(8.\) A1
(b) With \(n=8\) (even) the median is the mean of the 4th and 5th values: \(\dfrac{6+8}{2}\) M1
\(=7.\) A1
Common mistakes
- Averaging the distinct values instead of weighting by frequency. Given the table above, averaging \(5,6,7,8\) gives \(6.5\), not the correct \(6.3\) - the frequencies have to multiply into the total first.
- Forgetting to reorder the data before finding the median. The median only works on ordered data - if the list is given in the order it was collected, sort it first or the "middle" value is meaningless.
- Assuming every data set has a mode. If every value occurs the same number of times there is no mode, and if two values tie for most frequent the set is bimodal - don't force a single answer where none exists.
Ready to practise properly?
22 central-tendency questions, marked instantly like the real exam.
Quick answers
How do you find the mean from a frequency table?
Use the weighted mean \(\bar x = \dfrac{\sum fx}{\sum f}\): multiply each value by its frequency, add the results, then divide by the total frequency. On a GDC, enter values and frequencies into two lists and run 1-Variable Statistics with a frequency list.
What is the difference between the mean, median and mode?
The mean is the total divided by the number of values, the median is the middle value when the data is ordered, and the mode is the most frequently occurring value. A data set can have no mode, one mode, or several. For grouped data, see the parent topic's GDC guidance.