Chapter 5: Tales by Dots and Lines (Data Handling)

• 3 and 7: mean 5.
• 8 and 9: mean 8.5.
• 10 and 20: mean 15.

The mean of two numbers is always exactly halfway between them. It is as far from one as from the other.

The mean of two numbers is the midpoint between them.

The mean is not always the midpoint of the two extremes. For 2, 4, 9 the midpoint is 5.5, but the mean is 5.

The mean is the balance point: the total distance of the values below it equals the total distance of the values above it.

Means: 7, 6, 5, 10, 15, 10.5, 12, 7.5. In each case the total distance below the mean equals the total distance above it.

Only one centre. If the point moves right, every distance to the values below grows and every distance to the values above shrinks, so the two totals stop being equal. The same happens if it moves left.

Including a value:

• above the mean → the mean increases
• below the mean → the mean decreases
• equal to the mean → no change

Removing a value:

• above the mean → the mean decreases
• below the mean → the mean increases
• equal to the mean → no change

Fair share: a new person who brings more than the current fair share raises everyone's share. One who brings less lowers it. One who brings exactly the fair share changes nothing.

There is only one balance point. A value above the mean pulls it up when included (down when removed); a value below does the opposite; a value equal to the mean changes nothing.

Two values: include two values the same distance on either side of 9, e.g. 7 and 11, or 9 and 9. To remove two values, take out a pair such as 6 and 12.

Three values: this is possible as long as the distances balance: (total distance below 9) = (total distance above 9). For example, include 8, 9 and 10. The book shows another example: two 7s (distances 2 + 2) and one 13 (distance 4).

Two values above, one below: for example, 10 and 11 (distances 1 + 2 = 3) with 6 (distance 3). Or 10 and 12 (1 + 3) with 5 (4).

Yes: the included (or removed) values must have equal total distances on both sides of 9, e.g. 7 and 11; or 10, 11 and 6.

• Sum $=79$, with 11 values, so the mean $=\frac{79}{11}\approx 7.18$.
• Adding 10 to every value adds 10 to the mean: 17.18. There is no need to add the numbers again.

Subtracting 2 from each value:

$$\frac{(x_1-2)+(x_2-2)+\cdots +(x_n-2)}{n}=\frac{x_1+\cdots +x_n}{n}-\frac{2n}{n}=a-2$$

In fair-share terms, if everyone gives up 2, each fair share falls by 2.

Doubling each value:

$$\frac{2x_1+\cdots +2x_n}{n}=2\times \frac{x_1+\cdots +x_n}{n}=2a$$

7.18 and 17.18; subtracting 2 from every value lowers the mean by 2; doubling every value doubles the mean.

A value above the median moves the median up or leaves it the same. With 8 as the median of an odd number of values, including 11 makes it 9.5.

A value below the median moves it down or leaves it the same. If the values near the middle are equal, the median may not change at all.

A value above the median can only raise it (or leave it), a value below can only lower it (or leave it).

Mean:

$$\frac{3\times 3+4\times 11+5\times 9+6\times 7+7\times 3+8+9+10}{36}=\frac{188}{36}\approx 5.22$$

Median: the average of the 18th and 19th values. The running totals are $3,14,23,\dots$, so positions 15 to 23 are all 5. The median is 5.

Mean ≈ 5.22, median = 5.

• Farooq's Maths: cell E5 (42 marks).
• B7: Gowri's Odia marks, 27.
• Ashwin (row 4) scored more than 30 in Telugu (31), English (33) and Maths (34). His Social Science mark is exactly 30, which is not more than 30.

E5; B7 = 27 (Gowri, Odia); Ashwin: Telugu, English and Maths.

The Science marks are in column G, rows 2 to 23, so the formula is =AVERAGE(G2:G23).

The Odia average (31.23) is less than the Telugu average (33.59).

Totals: in H2 type =SUM(B2:G2) and copy it down to H23:

=AVERAGE(G2:G23) gives 33.14 for Science; Odia (31.23) is lower than Telugu (33.59); totals with =SUM(B2:G2) etc. (highest: Aishwarya, 273).

1. $\frac{1+50}{2}=25.5$ (the sum is $1275$)
2. $1,3,\dots ,99$: mean $=\frac{1+99}{2}=50$ (the sum is $50^2=2500$)
3. $4,8,\dots ,200$: mean $=\frac{4+200}{2}=102$

Observation: for equally spaced numbers, the mean is the average of the first and last numbers, which is also the median. The values pair up around the middle: $1+50=2+49=\cdots$.

(i) 25.5 (ii) 50 (iii) 102 (for evenly spaced data, mean = (first + last)/2).

The dots shown are 4, 7, 8, 8, 9, 9, 9, 9, 9, 11, a sum of 83. With the missing dot there are 11 values, so the total must be $11\times 9=99$. The missing value is $99-83=16$.

The missing value is 16.

(i) No. Every height is 1 cm too much, so the mean is also exactly 1 cm too much. Just subtract 1.

(ii) $150.2-1=149.2$ cm, which is option (d).

(i) No, just subtract 1 cm (ii) (d) 149.2 cm

A. Its songs are 5, 5, 5.25, 5.5, 5.75, 6 and 6.5 min:

$$\frac{39}{7}\approx 5.57$$

B has every song of 5 min or less, and C has songs between 3.5 and 4.5 min, so their means are below 5.

Album A (sum 39 minutes for 7 songs).

There are 16 values, so the median is the average of the 8th and 9th values: $\frac{41+41}{2}=$ 41.

(i) Any value. With 17 values, the median is the 9th. If the new value is below 41, the old 8th value (41) moves into 9th place. If it is 41 or more, the 9th is still 41.

(ii) One value must be ≤ 41 and the other ≥ 41, e.g. 10 and 90, or 41 and 41. Two values both below 41 would make the median 40.5, and two values both above 41 would make it 44.5.

(iii) Any value. With 15 values the median is the 8th, and it is 41 whichever value is removed, because the two middle values are both 41.

Median = 41. (i) Any value (ii) one value ≤ 41 and one ≥ 41 (iii) any value.

1. Never. Removing a smaller value shifts the middle upwards or leaves it unchanged.
2. Always. A value below the mean pulls the balance point down.
3. Sometimes. For example, including 2 values below and 2 above (with an odd number of values) keeps the median. Including 4 values all below usually changes it.
4. Never. Smaller values can only lower the median or leave it unchanged.

(i) Never (ii) Always (iii) Sometimes (iv) Never

The sum is $8\times 10.375=83$, and the known values add up to 70. So $y=83-70=13$.

y = 13

$15\times 134=2010$

2010

Without $p$, the sorted data are 8, 8, 12, 18, 25, 29, 35, 39, 47, 73. With $p$ there are 11 values, and the median is the 6th:

• If $p\ge 29$, the 6th value is 29 ✓
• If $p\le 25$, the 6th value becomes 25 ✗

p = 40, 100, 29, 47 or 30 (any p ≥ 29); not 10 or 25.

From the plot:

(i) There are 42 students and 193 rides in all, so the mean $=\frac{193}{42}\approx 4.6$.

(ii) The 21st and 22nd values are both 4 (the running total reaches 15 at 3 and 22 at 4). The median is 4.

(iii)

• (a) Not valid: 3 students never rode.
• (b) Valid: 38 of the 42 rode 2 or more times.
• (c) Valid: a week has only 7 days, so the 5 students with 8 or 10 rides must have ridden more than once on some day.
• (d) Not valid: at least 5 students did. Others with 7 rides or fewer may also have ridden twice on some day, so "exactly 5" cannot be claimed.

(e) Adding 1 to every value adds 1 to both: mean ≈ 5.6 and median 5.

(i) ≈ 4.6 (ii) 4 (iii) (b) and (c) are valid (e) mean ≈ 5.6, median 5.

There are 62 students and $1+4+20+54+84+120+90+100=473$ throws.

• Minimum 1 throw and maximum 10 throws.
• Mean $=\frac{473}{62}\approx 7.6$ throws.
• Median: the 31st and 32nd values. The running total is 27 at 7 throws and 42 at 8, so the median is 8.

Most students needed 7 to 10 throws. Only one hit the bull's eye at the first attempt.

Minimum 1, maximum 10, mean ≈ 7.6, median 8.

Yes. Both graphs show the same 24 values. The line graph joins each state's monthly values, so the trend over the year is easier to follow.

Some questions to explore:

• Why does Kerala stay between about 29 °C and 33 °C all year? (It is near the sea and close to the equator, and it gets heavy monsoon rain.)
• Why does Punjab swing from about 19 °C to 38 °C? (It is far inland, and further north.)
• Which other states behave like Punjab, e.g. Haryana or Rajasthan?
• What would the monthly minimum temperatures look like?

Yes, they show the same data; the line graph makes the trend clearer.

Method: countries register every object they launch with the United Nations, which keeps a record of all of them.

Which inferences are valid:

• "Increased every year": not valid. The count dipped in 2016, 2018 and 2024.
• "USA launched about ¾ of the worldwide count in 2022–24": valid. For example, about 2240 out of 2900 in 2023 (≈ 77%).
• "Nepal launched nothing": not valid. Only three countries are shown, so the graph says nothing about Nepal.
• "China + Russia ≈ 400 in 2024": valid. About 280 + 120.

Doubling: from 2019 (about 600) to 2020 (about 1270), the count more than doubled. From 2016 (about 220) to 2017 (about 470) it also roughly doubled.

Valid: the USA share of about ¾ and China + Russia ≈ 400. The worldwide count roughly doubled from 2019 to 2020 (and from 2016 to 2017).

Method: the rainfall each month is recorded over many years, and the totals for the same month are averaged.

Grouping:

• West coast (Kovalam, Udupi, Mumbai): heavy rain in June–August from the south-west monsoon.
• East coast: Rameswaram and Chennai get most of their rain in October–December from the north-east monsoon (Chennai peaks in November). Puri peaks in July–September.
• January–March are dry everywhere.

West-coast cities peak in June–August (south-west monsoon); Rameswaram and Chennai peak in October–December (north-east monsoon); January–March are dry.

Both lines rise towards the weekend, and Sunday is the busiest day. Wednesday is the quietest day. On every day, fewer people buy than visit.

See the graph; Sunday has the most visitors (35) and buyers (26).

(i) Count the rainy days in each month at a weather station for many years, then average them month by month.

(iii) New Delhi, read from the graph and rounded:

(ii)

(iv) Rainy days per year:

• Port Blair: about 126 (most)
• Mangaluru: about 112
• Rameswaram: about 42
• New Delhi: about 37 (fewest)

(v)

• New Delhi: July–August (June to September is the monsoon).
• Rameswaram: October–December.

(iv) Port Blair has the most rainy days (≈ 126), New Delhi the fewest (≈ 37). (v) New Delhi: July–August; Rameswaram: October–December.

(i) There is a clear yearly pattern. Births are highest from August to October (about 2 million a month) and lowest from February to April (about 1.5 million).

(ii) About 1.75 million in July 2017.

(iii) April 2017 to March 2020: 36 months.

(iv)

• January 2018: about 1.67 million
• January 2019: about 1.75 million
• January 2020: about 1.77 million

The number rose a little each year.

(v) Adding the twelve monthly values for 2019 (about 1.5 to 2.0 million each) gives about 21 million (2.1 crore).

(ii) ≈ 1.75 million (iii) April 2017 – March 2020 (iv) about 1.67, 1.75 and 1.77 million, slowly rising (v) ≈ 2.1 crore births in 2019.

(i) Karnataka's neighbours are all strongly rice: Andhra +92, Tamil Nadu +85, Kerala +79, Goa +57. Only Maharashtra (−15) leans to wheat. Karnataka is shaded purple, so its value is probably about +60 to +75.

(ii) Top 5 rice states:

• Manipur +100
• Nagaland +99
• Mizoram +97
• Tripura +96
• Meghalaya +95

(iii) Top 5 wheat states:

• Rajasthan −93
• Haryana −81
• Punjab −78
• Madhya Pradesh −60
• Delhi −45

(iv) Nearly balanced: Bihar (+3), Maharashtra (−15), Uttarakhand (−18) and Himachal Pradesh (−19).

(i) About +60 to +75 (ii) Manipur, Nagaland, Mizoram, Tripura, Meghalaya (iii) Rajasthan, Haryana, Punjab, Madhya Pradesh, Delhi (iv) Bihar, Maharashtra, Uttarakhand, Himachal.

(i) The colours, judged by when and how long each appears:

(ii)

• Middle strip = Friday: a full school day from about 9 am to 4:30 pm, with travel before and after.
• Right strip = Saturday: school only in the morning (until about 12:30), then travel home, lunch, a nap and an evening out.
• Left strip = Sunday: no school. There is study in the morning and evening and leisure in between.

(iii) On Sunday afternoon: he travels at about 1 pm, has a long orange block from about 1:30 to 4 pm, and travels back at 4 pm. That fits going out for a long movie.

(iv) The green box in the middle of Friday's school block: about 12:30 to 1 pm. A short orange break follows it.

(v) For example:

• He sleeps about 9 to 10 hours.
• He wakes up earlier on Sunday (6:30 am).
• He eats about four times a day.
• He has about 6 to 7 hours of classes on a school day.

Blue sleep, purple getting ready, green eating, grey travel, yellow classes and study, orange friends and hobbies. Middle = Friday, right = Saturday, left = Sunday; movie on Sunday about 1:30–4 pm; lunch break about 12:30–1 pm.

An average of 10 over three cells means every line must add up to 30.

(i)

(ii) Yes, many grids work. The centre must always be 10, and opposite cells must add up to 20. Another example:

E.g. 13 6 11 / 8 10 12 / 9 14 7; and yes, e.g. 16 2 12 / 6 10 14 / 8 18 4 (centre 10, opposite cells add to 20).

1. 7, 8, 9 and 2, 10, 12 (sum 24)
2. 10, 15, 16, 20 and 1, 14, 17, 30 (middle two average 15.5)
3. 10, 12, 14, 16, 16 and 1, 2, 3, 4, 58 (sum 68)
4. 1, 2, 3, 4, 5, 6 (both 3.5) and 5, 5, 5, 5, 5, 5
5. 1, 2, 3, 4, 5, 15 (mean 5, median 3.5) and 2, 2, 2, 2, 2, 8 (mean 3, median 2)

See the examples; any data with the stated sum or middle values work.

There are 6 values, so the average of the 3rd and 4th must be 13. The middle pair can only be 12 and 14 or 13 and 13.

• 12 and 14: one blank is 12, another is at most 12, and the third is 14 or more. For example, 12, 1, 50.
• 13 and 13: two blanks are 13 and the third is at most 13. For example, 13, 13, 2.

In the first case the third blank can be any counting number 14 or more.

E.g. 12, 12, 30 or 13, 13, 1. Infinitely many: the third blank can be any number ≥ 14 in the 12/14 case.

The total must be $6\times 6.5=39$, and the given numbers add up to 35. So the two blanks add up to 4.

3 possibilities: 1 and 3, 2 and 2, 3 and 1 (2 if order does not matter).

1. False: $\frac{2a+2b}{2}=a+b$, which can be odd, e.g. the average of 2 and 4 is 3.
2. False: $\frac{5a+5b}{2}$ need not be a whole number, e.g. the average of 5 and 10 is 7.5.
3. False: $\frac{5(a+b+c+d+e)}{5}=a+b+c+d+e$, which is any whole number. For example, the average of 5, 5, 5, 5 and 10 is 6.

(i) False (2, 4 → 3) (ii) False (5, 10 → 7.5) (iii) False (5, 5, 5, 5, 10 → 6)

(i) Only (c) is correct: we need only the two new heights. The old total is $24\times 150.2$, so there is no need to measure everyone again. The average may go up, down or stay the same, depending on the new heights.

(ii) The new students' total is $301=2\times 150.5$, which is $0.6$ more than $2\times 150.2$. So the new average is $\frac{3604.8+301}{26}\approx 150.22$ cm. (b): it increases slightly.

(iii) (d): we do not know the individual heights. If the old median is between 149 and 152, it may stay the same. Otherwise it may shift.

(i) (c) (ii) (b) increases slightly, to about 150.22 cm (iii) (d) not enough information.

The counts at 14 to 23 are 2, 2, 3, 5, 4, 4, 3, 1, 0, 1 (25 values):

$$14\times 2+15\times 2+16\times 3+17\times 5+18\times 4+19\times 4+20\times 3+21+23=443$$

Mean $=\frac{443}{25}=17.72$, so 17 is not the average.

Quick check by balance: the distances below 17 add up to $6+4+3=13$. The distances above add up to $4+8+9+4+6=31$. The two totals are not equal, so the mean must be above 17.

No; the mean is 17.72 (the distances above 17 outweigh those below).

The total weight is unchanged ($-2+1+1=0$), so the mean stays 65.3 kg.

The median may or may not change. It depends on whose weight changed and where those people are in the order. For example, if the middle person gained 1 kg, the median could become 68 kg.

Mean unchanged (65.3 kg); the median cannot be decided without more information.

(i)

(ii) Prices rose in almost every state, faster after 2021. Mizoram is the most expensive throughout (₹20 to ₹30). Assam is the cheapest in most years (₹12 to ₹15 after 2016).

(iii)

• Gujarat: prices stayed almost flat, about ₹13 to ₹15 from 2017 to 2024 (₹16.50 in 2016), then jumped to ₹19.20 in 2025.
• Uttar Pradesh: a steady rise from about ₹16 to ₹27 (2024), then ₹24.81 in 2025.

(iv) West Bengal: from ₹9.47 to ₹23.99, a rise of ₹14.52 (about 153%).

(v) For example: why is salt cheapest in Gujarat (it produces most of India's salt)? Why is it costliest in the north-east (transport costs)?

(iv) West Bengal increased the most (₹9.47 → ₹23.99). Gujarat stayed nearly flat until 2024, while Uttar Pradesh rose steadily.

1. Valid: rural kerosene was about 85% and urban electricity about 64%.
2. Valid: both kerosene lines fall all the way to 2023.
3. Not valid: about 90% of urban households used electricity in 2000. It was kerosene that was about 10%.
4. Not valid: the graph shows the main source of lighting, not how often the power fails.

(i) and (ii) are valid; (iii) and (iv) are not.

(i) About 2 hours (a little more than 2 h).

(ii) About (d) 14 years.

(iii)(a) Incorrect. At 15 it is about 1.1–1.3 h, while at 10 it is about 2.1–2.4 h. That is about half, not twice.

(iii)(b) Incorrect. The graph shows averages. The rural average at 15 is about 1.3 h, but some children may spend much less.

(i) ≈ 2 hours (ii) (d) 14 years (iii) both are incorrect.

These are projects, so your data will be your own. Use the method shown for Manoj:

1. Colour 48 half-hour boxes for each day.
2. Count the boxes of each colour, and divide by 2 to get hours.
3. Average each activity over the days.
4. Draw a strip for the "average day".

For the sleep data, find the mean and median for children, adults and elderly people separately, then compare them. For school timings, record the start time, end time and break time. Show the school-day lengths on a dot plot.

Projects: collect your own data and summarise it with strips, means, medians and dot plots.

In each graph, the lower lines (around 4–8 am) are the sunrises and the upper lines (around 4–8 pm) are the sunsets.

(i) Kibithu, the easternmost point of India. The sun rises there at about 5:55 am and sets at about 4:30 pm, so the day is about 10½ hours long.

(ii) Srinagar, the most northern of the four, has the longest summer days: about 5:20 am to 7:45 pm in June, roughly 14½ hours. Kibithu's longest day is about 14 hours.

(iii)

• Places further east see sunrise and sunset earlier: Kibithu is about 1¾ hours ahead of Ghuar Moti in the far west.
• Places further north have a bigger difference between summer and winter.
• Kanyakumari, near the equator, has almost the same day length all year (about 11½ to 12½ hours).

(i) Kibithu, sunrise ≈ 5:55 am; day ≈ 10½ hours (ii) Srinagar (≈ 14½ h in June).

Purnima (full moon) is about the 13th. On that day the moon rises around 6 pm, near sunset, and sets around 6:30 am.

Amavasya (new moon) is about the 28th. On that day the moon rises around 6 am and sets around 6 pm, together with the sun, so it cannot be seen.

(ii) What the graph shows:

• The moon rises about 50 minutes later each day. Its moonrise and moonset lines climb steadily and wrap round past midnight.
• On one day there is no moonrise at all (around the 21st), and on another no moonset (around the 6th).
• After about 29½ days the pattern repeats, which is one lunar month.

Full moon ≈ 13th, new moon ≈ 28th; moonrise and moonset are about 50 minutes later each day.

Tips:

• The first player has an advantage. Starting near the centre is strong.
• Use "bridges": two of your cells with two empty cells between them that touch both. If the opponent takes one of the empty cells, you take the other, so the link cannot be cut.
• Block early: a block close to the opponent's chain is easy to go round. Block a little ahead of it instead.

A remarkable fact: Hex can never end in a draw. When the board is full, exactly one player has a connecting chain.

Strategy: start near the centre, use two-bridges, and block ahead of your opponent; Hex always has a winner.