The cheapest measure of spread, and everything it fails to notice

What to assume they know

• Two records with the same average that describe very different players — that mean and median between them do not report spread
• Reading the largest and smallest entries out of an unsorted list
• Subtraction of integers, and the idea of a difference as a distance on a line
• The word deviation as the difference between an observation and a fixed value

What they should be able to do

• List the four measures of dispersion §13.2 names, and state which one this chapter sets aside
• Compute the range of a data set as the difference between its largest and smallest observations
• Compute and compare the ranges of the chapter's two batting records
• Explain why a range built from two observations cannot report how the remaining observations are arranged
• Construct two data sets with the same range and visibly different scatter, and compute a distance-based figure that separates them
• State the requirement §13.3 arrives at: a measure of dispersion must be built from deviations about a central value
• Say why the range still gets computed in practice despite these limitations

Where it usually goes wrong

• "The range is a bad measure." It is a narrow one, and the chapter says as much rather than dismissing it — it uses the range to separate the two batsmen before anything better exists. The right verdict is that it answers a smaller question than the one being asked.
• "A bigger range always means more scatter." The manufactured record in section 5 has exactly B's range and more than twice his average distance from the centre. Equal ranges do not imply equal scatter; the range only bounds it.
• "Quartile deviation is not in the syllabus, so it does not exist." §13.2 names it as one of four and then declines it. Say that it exists and is not developed here, rather than quietly leaving it off the list.
• "Range needs the data sorted." It needs the two extreme entries, which can be found in one pass. Sorting is the median's requirement, not the range's.
• "Every measure of dispersion is a distance from the centre." The range is not — it is a distance between two observations, and never refers to a centre at all. That is precisely the gap §13.4 opens by.
• "Range is useless for grouped data." It is computable from the class limits, but it then reports the width of the reported classes rather than of the observations. The chapter does not raise this; do not build on it.

Questions to check understanding

• Compute the range of a short list, and of a grouped distribution from its class limits
• Given two data sets with equal ranges, decide which is more scattered and justify the choice with a computation
• Name the four measures of dispersion listed in §13.2 and say which the chapter develops
• Explain why the range does not change when an interior observation is altered
• Given a data set, change one observation so that the range doubles and the mean barely moves
• Short-answer: state the property §13.3 says the range fails to report

Examples worth working on the board

Values marked verified are worked out here on data printed in this chapter.

• The list of four (§13.2, p. 259). The chapter names range, quartile deviation, mean deviation and standard deviation, in that order, and then says the quartile deviation is not taken up here. Three of the four get sections; the second one gets a name and nothing else. A teacher should show all four and grey the second, because a student meeting the phrase later needs to know it was named on purpose and passed over on purpose.
• The rule (§13.3, p. 259). Range equals the maximum value less the minimum value. One subtraction, no sorting required beyond finding two entries.
• The batsmen (§13.3, p. 259, on the data of p. 257). Verified: A's innings run from 0 to 117, so his range is 117; B's run from 46 to 60, so his range is 14. The chapter prints both subtractions. Note that the mean and the median were identical for these two records — the range is the first summary in the chapter that tells them apart at all.
• Two records with one range (not in the book). Take batsman B's smallest and largest scores, 46 and 60, and build a second ten-innings record that uses only those two values, five of each: 46, 46, 46, 46, 46, 60, 60, 60, 60, 60. Verified: this record has range 14, exactly B's; total 530, so mean 53, exactly B's; and sorted, its fifth and sixth entries are 46 and 60, so its median is 53, exactly B's. Verified: the average distance of its entries from 53 is 7, against 3.2 for B's actual record, and the average squared distance is 49 against 17.4. Same three summaries, more than twice the spread. This is the cleanest available demonstration that the range cannot see the interior.
• Where the information goes. For B's actual record, the eight scores between 46 and 60 — 48, 50, 52, 53, 53, 53, 57, 58 — never enter the subtraction. Eight of ten observations could be replaced by any values inside the interval and the range would not move. State the count out loud; it is more persuasive than the general argument.
• Sensitivity in the other direction. Verified: delete A's 0 and his 117 — one innings at each end — and the remaining eight run from 5 to 91, a range of 86. A single unusual innings at either end moves the range by tens of runs while moving the mean by a few.
• What §13.3 asks for (p. 259). The section closes by demanding that any better measure be built from how far each observation lies from a central value, and it names two that qualify: mean deviation and standard deviation. That sentence is the hinge of the whole chapter and section 8 should sit on it.

Figures to have open

• A single number line, 0 to 120, carrying the two batting records as dot rows, with the two spans drawn as measured bars beneath. Derived from Fig 13.1 and Fig 13.2 (p. 258); redraw as a schematic.
• The same axis carrying B's real record and the five-and-five record with equal span bars. An added figure; there is nothing like it in the chapter.
• A greyed list of the four named measures. Standard schematic.
• No figure is printed inside §13.2 or §13.3 themselves — the whole of printed p. 259 is running text and headings.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 13 "Statistics", §13.2 "Measures of Dispersion" and §13.3 "Range", both printed on p. 259.
• The data the ranges are taken of is the pair of batting records printed in §13.1 on p. 257; the dot diagrams are Fig 13.1 and Fig 13.2 on p. 258.
• The chapter's Summary (p. 286) repeats the range rule and lists quartile deviation among the measures of dispersion even though §13.2 excluded it from study. Cite the Summary carefully for that reason.