Understand · explore · practise
Bring together averages, quartiles, grouped interpolation, variance, coding and corrected summaries in sixteen original questions with hints and worked answers.
Before you startThe preceding location and spread lessons. Use descriptive variance with denominator n unless a question states otherwise.
01 / Identify what is known
Work independently first; reveal a hint only when needed.
Keep estimates distinct from exact results and retain units.
For listed-data quartiles here, use n/4 and 3n/4: round a non-integer position up; for an integer position average that rank and the next. Grouped interpolation instead retains the fractional cumulative target.
An ordered list is supplied; find quartiles and IQR.
Use the specified rank convention, read values, then subtract Q₁ from Q₃. Ranks are not quartile values.
02 / Summarise a complete list
For 3, 5, 7, 9, 11, 13, 15, 17, find the mean, median, Q₁, Q₃ and IQR.
There are eight values with symmetric pairs.
Mean = 10; median = (9 + 11)/2 = 10. Q₁ = (5 + 7)/2 = 6; Q₃ = (13 + 15)/2 = 14; IQR = 8.
For the same data, Σx² = 968. Find descriptive variance and SD.
Use n = 8 and mean 10.
Variance = 968/8 − 10² = 121 − 100 = 21. SD = √21 ≈ 4.583.
Replace 17 by 41 in the same list. Find the new mean, range and IQR.
The total rises by 24, while quartile ranks still use unchanged values.
Old total 80; new total 104, so mean = 13. Range = 41 − 3 = 38. Q₁ = 6 and Q₃ = 14 still, so IQR = 8.
03 / Use weighted totals
Values 0, 2, 4 have frequencies 2, 3, 5. Find mean, median and mode.
n = 10; locate ranks five and six separately.
Σfx = 26, so mean = 2.6. Rank 5 is 2 and rank 6 is 4, so median = 3. The mode is 4.
For that table, find descriptive variance and SD.
Σfx² = 2×0 + 3×4 + 5×16 = 92.
Variance = 92/10 − (26/10)² = 9.2 − 6.76 = 2.44. SD = √2.44 ≈ 1.562.
Values 1, 3, 5 have frequencies 2, k, 2 and descriptive variance 1. Find k.
Mean is three, and the total squared deviation stays sixteen.
16/(k + 4) = 1 gives k = 12, a valid non-negative integer.
04 / Estimate from intervals
Classes [0,10), [10,20), [20,40) have frequencies 3, 5, 2. Estimate mean and descriptive SD.
Use midpoints 5, 15 and 30.
Σfm = 15 + 75 + 60 = 150; n = 10, so estimated mean 15. Σfm² = 75 + 1125 + 1800 = 3000. Estimated variance = 300 − 225 = 75; SD = √75 ≈ 8.660.
Estimate the median for that table.
Target rank 5 is in [10,20); cumulative count before it is three.
Median ≈ 10 + (5 − 3)/5 × 10 = 14. This assumes uniform spread within the class.
Estimate the IQR for the same table.
Q₁ target 2.5 is in the first class; Q₃ target 7.5 is in the second.
Q₁ ≈ 0 + 2.5/3 × 10 = 25/3. Q₃ ≈ 10 + (7.5 − 3)/5 × 10 = 19. Estimated IQR = 19 − 25/3 = 32/3 ≈ 10.667.
05 / Decode the variable
Pause, replay or seek freely. The notes explain the same idea and stay in view.
For y = (x − 40)/3, mean(y) = 2 and variance(y) = 4. Find mean(x), variance(x) and SD(x).
Invert to x = 3y + 40.
Mean(x) = 46; variance(x) = 9×4 = 36; SD(x) = 6.
For z = 5 − 2x, mean(x) = 3 and SD(x) = 4. Find mean(z) and SD(z).
Use the absolute multiplier for SD.
Mean(z) = −1. SD(z) = 8, not −8 and not −3.
06 / Correct and combine
Four observations have T = 22 and U = 154. An entry 10 should be 6. Find the corrected mean and descriptive variance.
Subtract 10 and add 6 to T; subtract 100 and add 36 to U.
Tnew = 18; Unew = 90. Mean = 4.5; variance = 90/4 − 4.5² = 2.25; SD = 1.5.
A has four observations with mean 2 and variance 1. B has six with mean 7 and variance 4. Find combined mean and descriptive variance.
TA = 8, UA = 4(1 + 4); TB = 42, UB = 6(4 + 49).
Combined n = 10, T = 50, U = 20 + 318 = 338. Mean = 5; variance = 338/10 − 25 = 8.8.
A dataset has n = 10 and Sxx = 88. Give descriptive variance and sample variance using n − 1.
The same squared-deviation sum has different denominators.
Descriptive variance = 88/10 = 8.8. Sample variance = 88/9 ≈ 9.778. State which is intended before reporting an SD.
07 / Make claims the summaries support
Group A has median test score 68 and IQR 8. Group B has median 73 and IQR 15. What can you reasonably say?
Separate typical score from spread, and avoid a causal claim.
B has the higher median score, while A has the smaller spread between quartiles. These summaries alone do not prove which teaching approach caused the difference or describe every pupil’s result.
A dataset of twenty has mean 50 and SD 10. Can you determine exactly how many observations exceed 60?
Mean and SD do not fix the distribution.
No. Many datasets share those summaries with different upper-tail counts. A distributional model could support an estimate or probability, but it would be an extra assumption.
08 / Choose your next practice
If a result was wrong, identify whether the issue was the data form, rank convention, weighting, class boundary, denominator, squared-sum correction or units. Return to that focused lesson, then retry the question without its solution.
Section 1 of 8 · Identify what is known