Two records with the same average that describe very different players

What to assume they know

• The arithmetic mean of a list, and the sigma notation the chapter writes it in
• The median rule for a list of odd length and for a list of even length
• Arranging a list in ascending order
• Plotting numbers as points on a number line

What they should be able to do

• Compute the mean of each of the chapter's two ten-innings records and show that both come to 53
• Sort each record and apply the even-length median rule, obtaining 53 in both cases
• State what a measure of central tendency does report, and what it leaves undetermined
• Read the two dot diagrams of §13.1 and describe in words how each differs from the other
• Construct a third ten-value record that has the same mean and the same median as one of the chapter's and is visibly more strewn, and explain why this is always possible
• Explain why moving two observations equal distances in opposite directions leaves the mean unchanged
• State that the missing quantity is called a measure of dispersion, and say what it must be computed from

Where it usually goes wrong

• "Equal averages mean equal performance." This is the belief the section is built to break. A's ten innings include a duck and a century; B's never leave a fifteen-run band. The averages are identical and the two records would be selected on for completely different reasons.
• "The median would have caught what the mean missed." Here it did not. Both medians are 53 as well. Robustness to outliers is a real property of the median, but it is a property of a locator, and no locator reports spread.
• "The median is always one of the observations." For an even-length list it is generally not. A's median of 53 is the average of 42 and 64, and 53 never appears in his record.
• "Scatter just means the biggest value is big." It is about the whole arrangement. Section 8's manufactured record and B's record differ in only two entries, and the difference in spread is fourfold.
• "You could just look at the dots and skip the number." A picture works for ten values on one axis. The chapter wants a single figure that can be tabulated, compared and computed with, which is what §13.2 goes after.
• "A mean tells you what a typical innings looked like." For B it nearly does. For A no innings resembled 53 in the sense that mattered — his scores arrived as failures and as centuries.

Questions to check understanding

• Given two short records, compute both means and both medians and state whether the summaries distinguish them
• Given a list of even length, produce the median and say whether it is one of the observations
• Construct a second data set with a stated mean and median and a wider spread than a given one
• Explain in words why two data sets can share every measure of central tendency and still differ
• Read a dot diagram and rank two data sets by scatter without computing anything
• Short-answer: name the property that mean and median between them do not report

Examples worth working on the board

Values marked verified are worked out here on the data printed in §13.1.

• The two records (§13.1, p. 257). Batsman A: 30, 91, 0, 64, 42, 80, 30, 5, 117, 71. Batsman B: 53, 46, 48, 50, 53, 53, 58, 60, 57, 52. Ten innings each.
• The means. Verified: A totals 530 and B totals 530; ten innings each, so both means are 53. The chapter prints 53 in both columns of its summary and does not show the addition; the addition is worth showing, because the two routes to 530 look nothing alike — A gets there through a 117 and a 0, B through ten values that never leave the forties and fifties.
• The medians. Verified: sorted, A reads 0, 5, 30, 30, 42, 64, 71, 80, 91, 117, so the fifth and sixth entries are 42 and 64 and the median is 53. Sorted, B reads 46, 48, 50, 52, 53, 53, 53, 57, 58, 60, so the fifth and sixth entries are both 53 and the median is 53. Note: A's median is not one of A's scores — he never made 53 — whereas B's median is a score he actually made, three times. Same summary, different standing.
• Fig 13.1 and Fig 13.2 (§13.1, p. 258). Two number lines, each ticked and labelled 0, 10, 20, …, 120, with one dot per innings. On Fig 13.1 the dots run the length of the axis; on Fig 13.2 they crowd into a short block between the 40s and the 60s. Both are drawn on the same scale, which is the only reason the comparison reads at a glance. Redraw them stacked, sharing one axis.
• Two dispersion figures for the same pair, computed as inputs the later modules will recompute properly. Verified: the average distance of A's scores from 53 is 31.6 and of B's is 3.2; the average of the squared distances is 1300.6 for A and 17.4 for B. Neither number is printed in §13.1 — they are here so the explanation can say at the outset how large the gap is that the four printed summaries missed.
• A third player, built to order (not in the book). Take B's record and swap his 46 for a 0 and his 60 for a 106. The total is unchanged at 530, so the mean is still 53; the middle eight values have not moved, so the median is still 53. Verified: the new record reaches from 0 to 106, and its average distance from 53 is 12.4 against B's 3.2. Same two summaries, four times the strewing.
• The two rules §13.1 restates (p. 258). For an odd count the median is the entry in position (n + 1)/2 of the sorted list; for an even count it is the average of the entries in positions n/2 and n/2 + 1. Both records here have n = 10, so the even branch is the one used.

Figures to have open

• Fig 13.1 and Fig 13.2 redrawn as one stacked pair on a single 0–120 axis with ticks every 10. This is the chapter's own figure (p. 258) and the whole point is that both rows share a scale — redraw as a schematic rather than reproducing the printed artwork.
• A third dot row for the manufactured record of section 8, on the same axis, with the mean and median both marked at 53.
• A balance-beam schematic for section 7: a pivot at 53 with two weights sliding outward by equal amounts and the beam staying level. Standard schematic.
• No photograph is needed. The portrait of Karl Pearson printed beside §13.1 (p. 257) belongs to the Historical Note's material and is not required here.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 13 "Statistics", §13.1 "Introduction", printed pp. 257–258. The two records, the summary of means and medians, the restated median rules, and Fig 13.1 and Fig 13.2 are all inside those two pages.
• The same two records are picked up again in §13.3 (p. 259), where their ranges are computed. That is the next topic's material, deliberately left there.
• The Historical Note (pp. 287–288) supplies the biographical material for the Karl Pearson portrait on p. 257; it is the Consolidation topic's, not this one's.