KBAT Problems
KBAT Problem: Which Data Set Is More Consistent?
A KBAT grouped-data problem where you compute and compare spread from two frequency tables and justify a conclusion.
Estimate from grouped data
With grouped data you estimate the mean using class midpoints and read the spread from the interquartile range on an ogive. Doing this for two data sets gives a fair basis for comparison.
Judge on the spread
Two sets with a similar mean can differ sharply in spread. The one with the smaller interquartile range is more consistent.
A full answer computes both, compares, and explains the conclusion in words.
Understand the problem
An original grouped-data comparison. Two classes sit the same test, both with 26 pupils and both averaging 50 marks, using midpoints 10, 30, 50, 70, 90.
Class A frequencies are 2, 6, 10, 6, 2; Class B frequencies are 1, 5, 14, 5, 1. Equal means say nothing about consistency, so the question is really about spread, which class's marks cluster more tightly around the mean.
Plan and solve
- Confirm both means: Σfx ÷ Σf = 1300 ÷ 26 = 50 for each class.
- Deviations from 50 are −40, −20, 0, +20, +40, so squared deviations are 1600, 400, 0, 400, 1600.
- Class A: Σf(x−x̄)² = 2(1600)+6(400)+0+6(400)+2(1600) = 11,200; variance = 11,200 ÷ 26 ≈ 430.8; SD = √430.8 ≈ 20.8.
- Class B: Σf(x−x̄)² = 1(1600)+5(400)+0+5(400)+1(1600) = 7,200; variance = 7,200 ÷ 26 ≈ 276.9; SD = √276.9 ≈ 16.6.
- Compare: Class B's SD (16.6) is smaller than Class A's (20.8), so Class B is more consistent.
Check and a variant
Check with the other variance formula, Σfx² ÷ Σf − x̄²: for Class A, Σfx² = 76,200, so 76,200 ÷ 26 − 50² = 2930.8 − 2500 = 430.8, the same variance, so the working holds. A twist: the examiner asks you to compare using interquartile range instead of standard deviation, or drops one high outlier into Class B to see whether your conclusion survives.
The method, quantify the spread, then judge, stays the same.
Frequently asked questions
If two sets have the same mean, why compare spread at all?
Because the mean only tells you the centre, not how varied the results are. Two classes can both average 50 while one has marks packed near 50 and the other scattered from 10 to 90.
Consistency is about spread, so you measure it with range, interquartile range, variance or standard deviation.
Which measure of spread should I use?
Use whatever the question names. Standard deviation uses every value and is the usual choice for 'consistency', while interquartile range resists a single extreme mark better.
If a set has an outlier, mention that the IQR may give a fairer comparison. Always state which measure your conclusion rests on.
Does a smaller standard deviation mean better results?
No, it means more consistent results, not higher ones. A class could be consistently low.
Standard deviation measures how tightly values cluster around their own mean, so pair it with the mean when you judge performance. Here both means are equal, so the smaller spread fairly shows which class is steadier.