Mean, median, range and weighted averages.
Weight each group's value by its size: Σ(size × value) ÷ Σ(size). A simple mean of the group values is the classic error.
What was the fund's overall return, weighted by the size of each holding?
Answer: 5.1%
Weight each return by its holding: (40×8 + 25×3 + 15×1) ÷ (40+25+15) = (320+75+15)/80 = 410/80 = 5.1%. The 4.0% foil takes a simple unweighted mean — the classic error; 6.1% drops the cash holding; 3.0% picks the median; 12.0% just sums the returns.
Sort the values first; the median is the middle value (or the mean of the two middle values).
A store's monthly sales (£000s) over seven months were: 30, 22, 38, 18, 28, 24, 27. What was the median?
Answer: 27
Sort the values: 18, 22, 24, 27, 28, 30, 38. With seven values the median is the 4th: 27. The 26.7 foil is the mean; 28 is the midrange (min+max)/2; 18 takes the middle of the UNSORTED list; 38 is the maximum.
Range = maximum − minimum.
Temperatures recorded over six days (°C) were: 8, 14, 6, 19, 11, 15. What was the range?
Answer: 13
Range = maximum − minimum = 19 − 6 = 13. The 19 and 6 foils give only the max or the min; 12.2 is the mean; 12.5 is the median.
Mean = sum of the values ÷ how many there are.
Combined mean = (n₁ × mean₁ + n₂ × mean₂) ÷ (n₁ + n₂) — weight by group size, never average the two means.
Class A has 18 students with a mean score of 64. Class B has 12 students with a mean score of 74. What is the mean score across all 30 students?
Answer: 68
Weight each class mean by its size: (18×64 + 12×74) ÷ 30 = (1,152 + 888)/30 = 2,040/30 = 68. The 69 foil takes a simple average of the two means (the classic error); 70 swaps the weights; 72.9 divides by the wrong count; 2,040 is the total, not the mean.
Q1 = median of the lower half, Q3 = median of the upper half; IQR = Q3 − Q1.
For the ordered values 10, 14, 18, 20, 26, 30, 36, 42, what is the interquartile range? (Each quartile is the median of its half of the data.)
Answer: 17
Lower half 10,14,18,20 → Q1 = (14+18)/2 = 16. Upper half 26,30,36,42 → Q3 = (30+36)/2 = 33. IQR = Q3 − Q1 = 17. The 32 foil gives the full range; 33 and 16 give a single quartile; 23 gives the median.
Total = mean × count. Subtract the known values from the total to find the missing one.
The mean of five numbers is 42. Four of them are 38, 45, 40 and 50. What is the fifth number?
Answer: 37
The five numbers sum to 5 × 42 = 210. The four known sum to 38 + 45 + 40 + 50 = 173. The fifth = 210 − 173 = 37. The 43.25 foil averages the four known; 42 just repeats the mean; 87 forgets one known value; 8.4 divides the mean by five.
The mode is the value that occurs most often — its value, not its frequency.
Daily ticket sales over nine days were: 12, 15, 12, 18, 12, 15, 20, 12, 15. What was the mode?
Answer: 12
The mode is the value that occurs most often. 12 appears four times (more than 15's three). The 15 foil is the second-most-frequent; 14.6 is the mean; 4 is the frequency of the mode (not its value); 20 is the maximum.
Sort the values, find the overall median, then take the median of the relevant half for Q1 or Q3.
For the ordered values 10, 13, 15, 16, 19, 21, 24, 26, 29, 33, 38, what is the lower quartile (Q1)? (Q1 is the median of the lower half of the data.)
Answer: 15
With 11 values the overall median is 21 (the 6th). The lower half is 10, 13, 15, 16, 19; its median is 15, so Q1 = 15. The 21 foil gives the overall median; 29 is Q3; 10 is the minimum; 15.5 wrongly includes the median in the lower half.
Compute the weighted average for each period first, then the change between them — and mind points vs percent.
A fund held £40m at 5% and £10m at 2% in 2023; in 2024 it held £30m at 6% and £20m at 3%. By how many percentage points did the fund's blended return change?
Answer: 0.4 pp
Blended returns: 2023 = (40×5+10×2)/50 = 4.4%; 2024 = (30×6+20×3)/50 = 4.8%. Change = 0.4 percentage points. The 9.1% foil gives the relative change; 0.5pp averages the rates unweighted.