Whether adding a value raises or lowers the median

What to assume they know

• The median as the middle value of sorted data, and that with an even count it is the average of the two middle values — recalled at the head of the chapter and restated at Part II p.108, with the even-count rule itself at Part II p.109
• Sorting a short list of whole numbers
• The mean's response to an insertion, for contrast — Which added values move the mean, and in which direction
• Averaging two numbers, including when the answer ends in .5
• Counting positions in a list, and knowing which position is the middle one for a given count

What they should be able to do

• State the median of a small collection, choosing correctly between the single middle value and the average of two
• Predict whether inserting a stated value raises the median, lowers it, or leaves it alone, before sorting the new list
• Explain why a value above the median cannot lower it, in terms of how many values now sit above the old middle
• Compute the new median after an insertion and locate it between the old median and its neighbour
• Produce a value whose insertion leaves the median unchanged, and say when every value has that property
• Produce a pair of values, and a removal, that leave a stated median unchanged
• Contrast the median's response with the mean's under the same insertion, and say which is the more stable and why
• Judge "always true / sometimes true / never true" claims about the median
• Recognise a situation in which the given information does not determine the new median, and state what would be needed

Where it usually goes wrong

• "The median is the middle of the range." It is the middle of the list. For 4, 7, 8, 12, 14 the middle of the range is 9 and the median is 8, and no amount of stretching the largest value changes the median at all.
• "A really large new value must pull the median up a long way." It pulls it up by one step, to the average of the old median and its neighbour, and no further — whether the newcomer is 11 or 11,000.
• "With an even count you pick one of the two middle values." You average them. The chapter is explicit about this at Part II p.109, and the answer 9.5 is not a value in the data — which is fine, and worth saying.
• "Inserting a value equal to the median might still shift it." It cannot. The new pair of middle values are the median and itself.
• "The median always changes when the data changes." Item 5 is the counter-example the chapter chose deliberately: a repeated middle value makes the median survive any single insertion and any single removal.
• "If the mean is unchanged the median is unchanged." Item 8 is exactly this trap: the total is preserved, so the mean is safe, and the median is not determined at all.
• "More values above the median means the median is too low." Half the values sit above it by construction. What matters in the argument on Part II p.109 is that inserting above changes that count from equal to unequal, which is what disqualifies the old middle.
• "Not enough information" is a cop-out answer. It is printed as an option twice in Part II p.128 item 6, and it is the correct one for the median both times. Recognising an underdetermined question is part of what is being assessed.

Questions to check understanding

• Find the median of a listed collection, with an odd and with an even count
• State whether a given insertion raises, lowers or preserves the median, then check
• Compute the new median after an insertion and say between which two old values it had to fall
• Give a value whose insertion leaves a stated median unchanged; give a pair; give a removal
• Find every value of an unknown that makes a stated median come out right, and state the condition rather than list cases
• Judge always/sometimes/never claims about the median with a counter-case for each rejection
• Compare the effect of one extreme new value on the mean and on the median
• Identify a question about a median that the given data cannot answer, and say what is missing

Examples worth working on the board

Values marked printed appear on the page. Values marked not in the book are worked out here on the chapter's stated inputs. The chapter prints no answers to any exercise item and Part II has no answer appendix.

• The three median diagrams (Part II p.108, printed as three dot plots on an axis labelled 3 to 13, the first captioned Median = 8 and the third Median = 9.5). Read off the printed page: the first plot carries 4, 7, 8, 12, 14 with a dashed rule at the median; the second plot is the same five dots plus one drawn in a second colour at 11; the third is the six dots with the dashed rule now standing between 9 and 11. The middle panel's coloured dot is the insertion the running text on Part II p.109 describes.
• The printed argument (Part II p.109, printed). Because the new value is larger than the old median, there are now more values above 8 than below it, so 8 cannot be the middle any more; with six values the median becomes the average of the two middle ones, 8 and 11, which is 9.5. The page then states that the mirror-image argument covers a value inserted below the median.
• The distance test (not in the book, on the chapter's own data). Insert 100 instead of 11 into 4, 7, 8, 12, 14: the median is still the average of 8 and 12, which is 10, while the mean jumps from 9 to about 24.2. Insert 12 instead: the median is again 10. The median's answer depends only on the fact that the newcomer landed above 8, and its size decides nothing.
• How far the median can travel (not in the book). Adding one value to an odd-sized collection puts the new median halfway between the old median and one of its neighbours, so for 4, 7, 8, 12, 14 any single insertion above the middle lands the median between 8 and 12, and any insertion below it lands the median between 7 and 8. That bracket is a good thing to show before any arithmetic.
• Item 5 (Part II p.114, printed data, already sorted): 8, 10, 19, 23, 26, 34, 40, 41, 41, 48, 51, 55, 70, 84, 91, 92 — sixteen values. Not in the book: the eighth and ninth values are both 41, so the median is 41, and because those two middle values coincide the median here is remarkably hard to disturb.
  • (i) Any single inserted value leaves the median at 41. With seventeen values the median is the ninth, and nine of the sixteen are already at or below 41, so a newcomer on either side leaves 41 in the ninth place.
  • (ii) Two values work provided one is at most 41 and the other at least 41 — for instance 30 and 60, or 41 twice. Two values both below 41 drop the median to 40.5; two both above 48 raise it to 44.5.
  • (iii) Any single removal also leaves the median at 41, for the same reason read backwards. These three answers together are the strongest case in the chapter for the claim that the median attends to counts, not sizes.
• Item 9 (Part II p.115, printed). The data is 12, 47, 8, 73, 18, 35, 39, 8, 29, 25 and an eleventh value p, and the median has to be 29. The offered candidates are 10, 25, 40, 100, 29, 47 and 30. Not in the book: sorted without p the ten values run 8, 8, 12, 18, 25, 29, 35, 39, 47, 73; with eleven values the median is the sixth. Any p of 29 or more puts 29 in the sixth place, so 40, 100, 29, 47 and 30 all work, while 10 and 25 push 29 into seventh place and leave 25 as the median. The general condition — p ≥ 29 — is a better answer than the list.
• Item 6, parts (i), (iii) and (iv) (Part II p.115, printed as always/sometimes/never claims). Added verdicts: removing a value below the median can move the median up or leave it where it is, but can never bring it down, so (i) is never true. Adding four values need not disturb the median — two below and two above leave it exactly where it was — but four values on one side will move it, so (iii) is sometimes true. Four values below the median can only drag the middle down or leave it, so (iv) is never true. Part (ii) of the same item is about the mean and belongs to Which added values move the mean, and in which direction.
• Item 6(iii) (Part II p.128, printed). Two students join a class whose average height was 150.2 cm and whose individual heights are not given; the candidates are that the median stays, rises, falls, or that there is not enough information. Not in the book: the last one. Even with the two new heights given as 149 cm and 152 cm, the median depends on where they land among the other twenty-four, which the item never tells you. The mean, by contrast, is fully determined — that pairing is why this item and Part II p.128 item 6 parts (i) and (ii) should be taught by two videos that both know about it.
• Item 8 (Part II pp.128–129, printed). A group's mean weight was 65.3 kg and its median 67 kg; this month one member is 2 kg lighter and two are 1 kg heavier. Not in the book: the median may stay at 67 or may move, and nothing in the question fixes which, because we are not told whose weights changed. The mean is settled at 65.3 kg — the total is unchanged — and that half of the item belongs to Which added values move the mean, and in which direction.

Figures to have open

• The three-panel median figure (Part II p.108). Redraw with the same structure: the same axis three times, the newcomer in a second colour, the dashed median rule moving between panels one and three.
• A sorted-strip diagram: the values as a row of labelled cells with the middle cell or middle pair highlighted, so that insertion is visibly a re-indexing rather than an arithmetic operation. Not in the chapter, and it is what makes sections 3 and 7 land.
• A bracket figure showing the region between the old median and its neighbour. Standard schematic.
• The sixteen-value list from Part II p.114 item 5, set as a strip with both 41s marked. Standard schematic; no textbook art needed.
• A side-by-side pair of number lines for section 9, one carrying the median's move and one the mean's, with an insertion far out to the right.
• No photograph or dataset beyond the chapter's own numbers is needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 5, "Tales by Dots and Lines", §5.1 "The Balancing Act", the unnumbered subsection "Tinkering with Median", Part II pp.108–109. The three dot plots and the restated definition are at the foot of Part II p.108; the argument and the value 9.5 are at the top of Part II p.109, immediately above "Finding the Unknown".
• SUMMARY, Part II p.133, second bullet: after stating the direction-of-change result for the mean, it adds that similar behaviour can be seen with the median. That one clause is the whole of the chapter's summary treatment of this topic.
• Exercise items owned by this topic: Part II p.114 item 5; Part II p.115 item 6 parts (i), (iii) and (iv); Part II p.115 item 9; Part II p.128 item 6 part (iii); Part II pp.128–129 item 8, median half only. Every one of those sets is shared with a sibling topic.
• Sibling topics: the mean's response to the same changes is Which added values move the mean, and in which direction; recovering an unknown value from a stated median is Working backwards from an average to a missing value; finding a median from a frequency table without writing the list out is Mean and median from a frequency table, by hand and in a spreadsheet, Part II p.110.