Outliers, and why the median survives them

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What to assume they know

The arithmetic mean as fair-share — the arithmetic mean as a total shared out equally
Dot plots: seeing spread and clustering at a glance — reading a dot plot, and marking a value on it
Sorting a short list of numbers into increasing order
Finding the average of exactly two numbers
Class 6 Ganita Prakash: decimals to one place, and comparing them

What they should be able to do

Find the median of a sorted list, for an odd and for an even number of values
Explain why an even-sized list forces you to average the two values that share the middle
Identify an outlier in a small data set and say which end it lies at
Explain, from how each is computed, why an outlier moves the mean far more than the median
Predict the direction of the shift: a low outlier pushes the mean below the median, a high one above it
Recompute both summaries with the outlier removed and describe what changed
Decide, for a given data set, which of the two better represents it, and justify the choice
Use the phrase measures of central tendency for the pair

Where it usually goes wrong

"The median is unaffected by outliers." Too strong, and the chapter is careful not to say it — its own wording is that outliers did not move it much. Deleting the 118 from Poovizhi's family moves the median from 170 to 171.5, because removing a value also changes which position is the middle. The right statement is comparative: the median moved by 1.5 cm and the mean by 10.55 cm on the same edit.
"An outlier is a mistake in the data." Sometimes it is; here it is a real child. The chapter's outliers — a young sibling, a student who read forty stories — are all genuine values. The problem is with the summary, not the observation.
"Take the middle value of the list as written." The list has to be sorted first, and the chapter's two examples both come pre-scrambled precisely so that the sorting step cannot be skipped.
"With an even count you pick either middle value." You average them, and the reason is worth a sentence: the median is defined so that as many values sit below it as above, and with an even count no listed value does that job.
"The median is always the better summary." Not the claim being made. The chapter's own newspaper example has no serious outlier and the two summaries agree; there the mean is fine, and it uses the whole data set rather than just the middle of it. Which to prefer is a judgement about the data.
"Mean below median means the data is small." It means something is pulling from the low end. Direction of pull, not size — this is the specific confusion the three-case comparison on p.108 exists to break.
"Any value at the end of the range is an outlier." Every set has a largest and a smallest value. What makes the 118 and the 40 outliers is the gap between them and everything else, which is exactly what a dot plot shows and a sorted list does not.

Questions to check understanding

Find the median of a given list, odd count and even count
Given a data set, identify the outlier and say which end it sits at
Recompute the mean and the median with a stated value removed, and describe the change in each
Given only a mean and a median, say which side the data leans towards
"Which better represents this data, the mean or the median?" — with a justification, which is where the marks are
Construct a data set to order: one whose mean and median are equal, one whose mean exceeds its median
Given a dot plot alone, judge statements about the mean, the median and the spread — the format the chapter uses at Part II, §5.4, p.129
Explain why a village's average income can be far above what most families earn

Examples worth working on the board

The two families (Part II, §5.2, p.105). Inputs, in centimetres. Yaangba's family: 169, 173, 155, 165, 160, 164 — six people. Poovizhi's family: 170, 173, 165, 118, 175 — five people. The chapter prints both means on p.106, 164.3 and 160.2, and both medians, 164.5 and 170. Both means and both medians were recomputed for this brief and are right. The observation the whole passage turns on is also printed there: the smaller mean sits below four of the five people it is meant to describe.
The two-pair plot figure (Part II, §5.2, p.106). Checked against the printed page. Two pairs of dot plots on a shared 115–175 line — the upper pair carries the means only, drawn as solid vertical rules; the lower pair adds the medians as dashed rules. The visual point is that the solid rule for Poovizhi's family lands to the left of every dot except one. Rebuild it; that gap is the argument.
Why the 118 is there (Part II, §5.2, p.107). Input: the chapter's own explanation is that the shortest member is a much younger child. That is a reason, not an error — the data is correct and the summary is still misleading.
Removing the outlier from Poovizhi's family (Part II, §5.2, p.107). The chapter asks for it and prints no answer. Working it here so the explanation does not have to guess: the four remaining heights are 165, 170, 173, 175, whose mean is 170.75 and whose median is 171.5. Against the originals — mean 160.2, median 170 — the mean moves by more than ten centimetres and the median by one and a half. State both movements. Saying the median "does not change" would be false here and is the easiest wrong sentence to write about this page.
Are you a bookworm? (Part II, §5.2, p.107). Checked against p.107: the fifteen answers are handwritten on coloured paper slips and extract as nothing. Reading them off the printed page, they are 6, 3, 0, 8, 2, 5, 7, 15, 12, 10, 40, 5, 0, 1, 8. Their total is 122 over fifteen students. The chapter itself prints only one summary for this set — the median, 6 — together with what that means in words. The 40 is named as the outlier on p.108.
Are We on the Same Page? (Part II, §5.2, p.108). Inputs: the number of pages a newspaper carried on each day from Monday to Sunday — 16, 18, 20, 22, 26, 16, 10. The plot beside it is a blank grid labelled 0 to 25 (it runs on a little past 25, so the 26 still fits) for the reader to fill; checked against p.108. This is the chapter's example of a set whose two summaries sit close together.
The three-case rule of thumb (Part II, §5.2, p.108). Inputs: the chapter asks which of its three data sets — family heights, short stories, newspaper pages — shows mean and median close together, which shows mean below median, and which shows mean above median, and then states the pattern. The pattern is the deliverable: a balanced set puts the two summaries near each other; a low outlier pulls the mean down below the median; a high outlier pushes it up above.
The open question (Part II, §5.2, p.109). Input: what happens when there are outliers at both ends. The chapter asks the reader to invent data and find out, and prints nothing further. The honest answer is that the two pulls partly cancel, so the mean can end up close to the median even though neither number describes the set well — which is a good place to stop and hand over to Why one number is never enough to describe a data set.

Figures to have open

Two family line-ups with heights labelled, one of them containing a much shorter child. Standard schematic; redraw rather than lifting the book's own illustration on p.107.
A dot plot on a 115–175 line that can carry a solid mean rule and a dashed median rule, and can have one dot removed live. This is the workhorse figure of the topic and must be re-usable across sections 2, 6 and 10.
Fifteen paper slips carrying the bookworm values, laid out unsorted and then sorted. Redraw; the book's own is a photograph of handwritten slips.
A three-panel comparison holding one plot per case — mean ≈ median, mean < median, mean > median.
No figure needs to be taken from the textbook.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part II, printed Chapter 5, "Connecting the Dots...", §5.2 "Representative Values", pp.105–109. The named sub-headings used are "Outliers and Medians" (pp.105–109), and inside it "Height of a Family" (p.105), "Are you a bookworm?" (p.107) and "Are We on the Same Page?" (p.108). None carries a number — this book prints only one level of section numbering.
The SUMMARY's one-line definition of the median: Part II, p.134.
Source file gegp205.pdf / gegp205.txt, printed pp.97–135; PDF page 1 is printed page 97.