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Range and SD of two quiz sections

Short answer

Section 1: range 19, SD 5.73. Section 2: range 17, SD 6.74. Range suggests Section 1 varies more, but it is misleading because one outlier (1) inflates range while most scores are identical. SD shows Section 2 actually varies more, giving the correct picture.

Two different sections of a statistics class take the same quiz and the scores are recorded below: Section 1: 1 20 20 20 20 20 20 20 20 20 20 Section 2: 2 3 4 5 6 14 15 16 17 18 19 a) Find the range and standard deviation for each section. b) What do the range values lead you to conclude about the variation in the two sections? c) Why is the range misleading in this case? d) What do the standard deviation values lead you to conclude about the variation in two sections?

Range = largest − smallest. The standard deviation uses every value: s = √[Σ(x − x̄)² / (n − 1)]. Both sections have n = 11 students.

a) Range and standard deviation

Section 1Section 2
Largest, smallest20, 119, 2
Range1917
Σx201119
Mean18.2710.82
Σ(x − x̄)²328.18453.64
s² (÷ 10)32.8245.36
s5.736.74

(With divisor n the SDs are 5.46 and 6.42; the comparison is the same.)

b) Conclusion from the range

The range of Section 1 (19) is larger than that of Section 2 (17), so by range alone Section 1 seems to vary more.

c) Why the range is misleading

The range depends only on the two extreme scores. In Section 1, ten students scored exactly 20 and only one student scored 1. That single odd score makes the range large although almost all students are identical. Section 2 has scores spread over the whole scale (2 to 19).

d) Conclusion from the standard deviation

The standard deviation (6.74 against 5.73) shows that Section 2 varies more, because every score contributes to it. This is the correct picture: Section 1 is concentrated at 20 apart from one outlier, while Section 2 is truly spread out.

Answer: Range: 19 and 17. SD: 5.73 and 6.74. Standard deviation is the better measure here.

✓ Answer checked by Claude Sonnet 5.5

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