Averages from an ungrouped frequency table
Use frequencies as counts of repeated values when finding mean and median.
Section Paper 2 · Extended · Non-calculator
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Calculate mean and median of a list.
- Read frequency tables.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does frequency five for value two mean?
What you will learn
- Calculate Σfx / Σf for exact ungrouped values.
- Locate median positions and the mode from frequencies.
Weight each value by its count
A table records one book for three learners, two books for five and three books for two. The total frequency is 3 + 5 + 2 = 10. The total books is 1×3 + 2×5 + 3×2 = 19, so mean is 19/10 = 1.9 books.
Do not average the three distinct labels one, two and three as if each occurred once. Their frequencies differ. A table with exact discrete values preserves the original total, so this mean is exact for the recorded data rather than a grouped midpoint estimate.
Locate observations using cumulative counts
In the same table, positions one to three are value one, positions four to eight are value two and positions nine to ten are value three. The fifth and sixth observations are both two, so the median is two. The mode is two because it has the largest frequency five.
Range uses largest value minus smallest value, here 3 − 1 = 2, rather than largest frequency minus smallest frequency. When identifying a median position, count observations, not rows. A frequency table is a compressed list, not a list of frequencies to average.
Pause and explain
Values 1,2,3 have frequencies 3,5,2. What is the mean?
Worked example
Values 1,2,3 have frequencies 3,5,2. Find mean and median.
Show the worked solution
- Σf = 10 and Σfx = 19.
- Mean = 19/10 = 1.9.
- Positions five and six both contain value two, so median is two.
Answer Mean 1.9; median 2.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Values 2 and 5 have frequencies 3 and 2. Find the mean.
Give me a hint
Multiply each value by its frequency.
Compare my reasoning
- The total value is 2×3 + 5×2 = 16.
- The total count is five.
- Mean is 16/5 = 3.2.
3.2.
Look for these in your work
- Weighted the values.
- Used the total frequency.
Values 0,1,2 have frequencies 2,4,3. Find median, mode and range.
Give me a hint
Locate the fifth observation among nine.
Compare my reasoning
- Zero occupies positions one to two; one occupies three to six.
- The fifth value and the most frequent value are both one.
- Range is 2−0 = 2.
Median 1; mode 1; range 2.
Look for these in your work
- Counted observation positions.
- Used data values for range.
A frequency table records value 4 five times and value 10 once. Explain why the mean is not seven.
Give me a hint
The two distinct labels have unequal weights.
Compare my reasoning
- Total value is 4×5 + 10 = 30.
- Six observations give mean five.
- Seven would incorrectly give each distinct label equal weight.
Mean 5, because the value 4 occurs five times.
Look for these in your work
- Used repeated observations.
- Explained the incorrect shortcut.
Common mistakes
- Averaging the row labels without weighting.
- Taking the median frequency.
- Using frequency differences as range.
What can you explain now?
Does the largest frequency equal the mode?
Compare with the explanation
No; the value with that frequency is the mode.
A frequency is a count, while the mode is the most common observed value.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, expand a short frequency table into a list and verify its summaries both ways.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.