Measures of Dispersion for Ungrouped Data · 8.2.4
Effect of data changes on dispersion
Students investigate how dispersion measures and their graphs change when data is transformed, for example, when every value is increased or scaled by the same amount, when an outlier or extreme value appears, or when values are added to or removed from a data set, then state the resulting effect clearly.
The official learning standard (8.2.4)
“Determine the effect of data changes on dispersion based on: (i) the value of measures of dispersion, and (ii) graphical representation. This includes the effect on dispersion when: (i) each data value is changed uniformly, (ii) an outlier or extreme value exists, and (iii) certain values are added or removed.”
What it means
Students investigate how dispersion measures and their graphs change when data is transformed, for example, when every value is increased or scaled by the same amount, when an outlier or extreme value appears, or when values are added to or removed from a data set, then state the resulting effect clearly.
How it is examined
Paper 2 questions typically give an original data set with known dispersion measures, then ask students to predict or calculate the new range, IQR, variance or standard deviation after a stated change, and to explain why the measure increased, decreased or stayed the same, often linked to outliers.
Worked example
A set of data is 3, 5, 7, 9, 11. (a) State the range.
(b) If every value is increased by 4, state the new range and explain why it does not change. (c) If the value 11 is replaced by 31, determine the new range and explain its effect on dispersion.
- Original data: 3, 5, 7, 9, 11. Range = maximum − minimum = 11 − 3 = 8.
- Adding 4 to every value gives 7, 9, 11, 13, 15. New range = 15 − 7 = 8, the same as before, since a uniform shift moves every value by the same amount, so the gap between the maximum and minimum is unchanged.
- Replacing 11 with 31 gives 3, 5, 7, 9, 31. New range = 31 − 3 = 28.
- Compare: the range jumped from 8 to 28 because of a single outlier, showing that measures like range and standard deviation are very sensitive to extreme values.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Does adding a constant to every data value change the standard deviation?
No. Adding or subtracting the same constant from every value shifts the whole data set up or down without changing how spread out the values are, so the range, IQR, variance and standard deviation all stay exactly the same.
Why is the interquartile range less affected by outliers than the range?
The range only uses the two extreme values, so a single outlier changes it directly. The IQR is based on Q1 and Q3, which sit within the middle of the data, so one extreme value usually has little or no effect on it.
What happens to the variance if every data value is multiplied by 2?
The variance is multiplied by 2² = 4, while the standard deviation is multiplied by 2, since standard deviation is the square root of variance and scales directly with the data, not with its square.