Measures of Dispersion for Ungrouped Data · 8.2.1
Range, IQR, variance and SD
Students calculate four numerical measures of spread from a list of individual (ungrouped) values. Range is the largest value minus the smallest.
Interquartile range is Q3 minus Q1. Variance is the average squared deviation from the mean, and standard deviation is its square root, using the formula σ² = Σx²/n − (Σx/n)².
The official learning standard (8.2.1)
“Determine the range, interquartile range, variance and standard deviation as measures to describe the dispersion of ungrouped data.”
What it means
Students calculate four numerical measures of spread from a list of individual (ungrouped) values. Range is the largest value minus the smallest.
Interquartile range is Q3 minus Q1. Variance is the average squared deviation from the mean, and standard deviation is its square root, using the formula σ² = Σx²/n − (Σx/n)².
How it is examined
This is a major SPM Paper 2 topic, usually a full question requiring students to compute the mean, variance and standard deviation using the formula σ² = Σx²/n − x̄², along with the range and interquartile range from ordered data. Paper 1 may test a shorter range or IQR calculation.
Worked example
A set of data is 4, 6, 8, 10, 12. Find the range, interquartile range, variance and standard deviation of the data.
- Range = 12 - 4 = 8.
- Median = 8; lower half = 4, 6 so Q1 = (4+6)/2 = 5; upper half = 10, 12 so Q3 = (10+12)/2 = 11; IQR = 11 - 5 = 6.
- Mean, x̄ = (4+6+8+10+12)/5 = 40/5 = 8.
- Σx² = 4² + 6² + 8² + 10² + 12² = 16+36+64+100+144 = 360.
- Variance, σ² = Σx²/n - x̄² = 360/5 - 8² = 72 - 64 = 8.
- Standard deviation, σ = √8 = 2.83 (3 s.f.).
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What is the difference between variance and standard deviation?
Variance is the average of the squared deviations from the mean, so its unit is squared (e.g. m²).
Standard deviation is the square root of variance, bringing the value back to the original unit (e.g. m), which is why standard deviation is easier to interpret alongside the data.
How do I find Q1 and Q3 for an odd number of data values?
First find the median, which splits the ordered data into a lower half and an upper half (the median itself is not included in either half). Q1 is the median of the lower half, and Q3 is the median of the upper half.
Which formula for variance should I use in the exam?
Use σ² = Σx²/n − (Σx/n)², which only needs Σx and Σx², making it faster and less error-prone than summing squared deviations from the mean one by one. Both formulae give the same correct answer, but this version is standard practice for SPM.