5. Tales by Dots and Lines

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

It is not always the midpoint of the smallest and largest values. For example, for 10, 10, 11, 17 the mean is 12: distances on the left are 2 + 2 + 1 = 5, and on the right 17 − 12 = 5. But the midpoint of 10 and 17 is 13.5.

Removing a value greater than the mean makes the mean decrease. Removing a value less than the mean makes it increase.

Including or removing a value equal to the mean does not change the mean. In the fair-share idea, that value already has exactly the fair share, so nothing needs to be shared.

Yes. The total distance above the mean must equal the total distance below it.

For a mean of 10: include 12 and 8 (2 above and 2 below). For three values: include 11, 12 (distances 1 + 2 = 3 above) and 7 (distance 3 below). The mean stays 10.

Sum = 79, number of values = 11. Mean = 79 ÷ 11 ≈ 7.18.

When every value increases by 10, the mean also increases by 10. New mean ≈ 17.18.

New average = (x₁ − 2) + (x₂ − 2) + … + (x_n − 2)n = x₁ + … + x_n − 2nn = a − 2.

So the average decreases by 2. In fair-share terms, if everyone gives back 2, each fair share is 2 less.

Farooq’s Maths marks are in cell E5.

B7 has Gowri’s Odia marks, 27

Ashwin scored more than 30 in Telugu (31), English (33) and Maths (34)

=AVERAGE(G2:G23)

Class averages: Odia ≈ 31.23, Telugu ≈ 33.59, English ≈ 32.64, Maths ≈ 34.14, Social Science ≈ 31.36, Science ≈ 33.14.

No, the Odia average is less than the Telugu average.

Total marks of a student, e.g. Ratna: =SUM(B2:G2) = 200.

(i) (1 + 50) ÷ 2 = 25.5

(ii) (1 + 99) ÷ 2 = 50

(iii) (4 + 200) ÷ 2 = 102

Observation: For numbers that go up by the same step, the mean is the average of the first and the last number.

The plot shows 4, 7, 8, 8, 9, 9, 9, 9, 9, 11 (10 values, sum 83).

With 11 values and mean 9, the sum must be 99.

Missing value = 99 − 83 = 16

(i) No. Every value is 1 cm more, so the mean is also 1 cm more. Just subtract 1.

(ii) Correct average = (d) 149.2 cm

Album A. All its dots lie between 5 and 6.5 minutes, so its mean is near 5.6. Album B’s dots are mostly between 0.5 and 5, and Album C’s between 3.5 and 4.5, so their means are much smaller.

There are 16 values. Median = average of the 8th and 9th values = (41 + 41) ÷ 2 = 41

(i) Including any one value keeps the median 41, because the 8th and 9th values are both 41.

(ii) Include one value ≤ 41 and one value ≥ 41 (for example 30 and 60, or 41 and 41).

(iii) Removing any one value keeps the median 41 (the new middle value is still a 41).

Sum = 10.375 × 8 = 83. Known values add up to 70.

y = 83 − 70 = 13

Sum = 15 × 134 = 2010

Sorted without p: 8, 8, 12, 18, 25, 29, 35, 39, 47, 73. With p there are 11 values, so the median is the 6th value.

The 6th value is 29 only if p ≥ 29.

So p can be (iii) 40, (iv) 100, (v) 29, (vi) 47, (vii) 30

Counts: 0 → 3, 1 → 1, 2 → 4, 3 → 7, 4 → 7, 5 → 5, 6 → 4, 7 → 6, 8 → 3, 10 → 2. Total = 42 students.

(i) Average = 193 ÷ 42 ≈ 4.6 times

(ii) Median = average of the 21st and 22nd values = 4 times

(iii) (a) Not valid — 3 students did not cycle at all. (b) Valid — most students cycled a few times. (c) Valid — those who cycled 8 or 10 times in 7 days must have cycled more than once on some days. (d) Not valid — at least 5 students did so; others (like those who cycled 7 times) may also have cycled twice on some day.

(e) Each value increases by 1, so the average becomes ≈ 5.6 and the median becomes 5

Total students = 62. Minimum = 1 trial, maximum = 10 trials.

Mean = (1 + 4 + 20 + 54 + 84 + 120 + 90 + 100) ÷ 62 = 473 ÷ 62 ≈ 7.6

Median = average of the 31st and 32nd values = 8

(a) Not valid — the count did not increase every year (for example, it fell in 2024).

(b) Valid — in 2022–24 the USA launched about 2000–2250 of 2500–2900 objects, about 34.

(c) Not valid — Nepal is not shown in the graph, so we cannot say.

(d) Valid — China (≈ 270) and Russia (≈ 100) together are about 400.

The count roughly doubled from 2016 to 2017 and from 2019 to 2020.

The data is probably collected from the objects that countries register with the United Nations each year.

Both lines are highest on Sunday. The gap between visiting and purchasing is largest on Tuesday and Sunday.

(i) Rainy days are counted every month for many years, and the average for each month is found.

(iii) New Delhi (from the graph): Jan 1, Feb 1.5, Mar 1.5, Apr 1, May 1.5, Jun 4, Jul 10, Aug 10, Sep 4, Oct 1, Nov 0.3, Dec 1.

(iv) Most rainy days: Port Blair (about 126 days). Least: New Delhi (about 36 days).

(v) New Delhi: July–August (June to September). Rameswaram: October–December.

(i) The number of births goes up and down in the same way every year. It is highest around September–November (about 1.9–2 million) and lowest around April–June (about 1.5 million).

(ii) July 2017: about 1.7 million

(iii) About 3 years, from around April 2017 to April 2020.

(iv) The January values in 2018, 2019 and 2020 are nearly the same, about 1.7–1.8 million each.

(v) About 1.75 million per month × 12 ≈ 2.1 crore (21 million) births in 2019.

(i) Karnataka lies between Maharashtra (−15) and Andhra/Tamil Nadu (+85 to +92), near Goa (+57). Its value is likely about +60 to +70.

(ii) Rice most popular: Manipur (+100), Nagaland (+99), Mizoram (+97), Tripura (+96), Meghalaya (+95).

(iii) Wheat most popular: Rajasthan (−93), Haryana (−81), Punjab (−78), Madhya Pradesh (−60), Delhi (−45).

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

(i) Light blue = sleeping, green = eating, orange = friends/hobbies/media/family, yellow = classes and studying, purple = showering/dressing/yoga, grey = travelling.

(ii) The strip with a long yellow block from 9:30 am to about 4:30 pm is a school day — Friday. The strip with classes only in the morning and a free afternoon is Saturday. The strip with no classes and long leisure time is Sunday.

(iii) On Sunday afternoon (about 2 pm to 5 pm): he travels at about 1:30 pm, has a long orange block and travels back at about 5 pm.

(iv) The school lunch break is at about 12:30 pm – 1:00 pm (the green box in the middle of the school day).

(v) He sleeps about 8–9 hours, wakes at about 7 am and sleeps by about 9 pm. He exercises every morning and takes a nap on weekends.

Each row, column and diagonal adds up to 30, so the average is 10.

(ii) Yes. Another grid: 9, 14, 7 / 8, 10, 12 / 13, 6, 11.

(i) Mean 8: 7, 8, 9 and 2, 10, 12.

(ii) Median 15.5: 10, 15, 16, 20 and 1, 14, 17, 30.

(iii) Mean 13.6 (sum 68): 10, 12, 14, 15, 17 and 4, 8, 16, 20, 20.

(iv) Mean = median: 1, 2, 3, 4, 5, 6 and 10, 10, 10, 10, 10, 10.

(v) Mean > median: 1, 2, 3, 4, 5, 15 (mean 5, median 3.5) and 2, 2, 2, 2, 2, 20.

With 6 values, the median is the average of the 3rd and 4th values, so they must add up to 26.

Example: 1, 12, 30 → sorted 1, 5, 12, 14, 21, 30. Median = (12 + 14) ÷ 2 = 13.

One blank can be any number 14 or more (like 30, 31, 32, …). So there are infinitely many possibilities.

Sum must be 6.5 × 6 = 39. Known values add up to 35. So the two blanks add up to 4.

Possibilities: (1, 3), (2, 2), (3, 1) → 3 possibilities.

(i) Only (c) is correct. We need the heights of the two new students. The others need not be measured again.

(ii) The new students add 149 + 152 = 301 cm, which is more than 2 × 150.2 = 300.4 cm. So (b) the average will increase (slightly, to about 150.22 cm).

(iii) (d) — we do not know the class median or the other heights, so we cannot say how the median changes.

Counts: 14 → 2, 15 → 2, 16 → 3, 17 → 5, 18 → 4, 19 → 4, 20 → 3, 21 → 1, 23 → 1 (25 values).

Total distance of the values below 17 = 6 + 4 + 3 = 13. Total distance of the values above 17 = 4 + 8 + 9 + 4 + 6 = 31. They are not equal, so 17 is not the balancing point.

Mean = 443 ÷ 25 = 17.72. So No, 17 is not the average.

Total change = −2 + 1 + 1 = 0. So the mean stays 65.3 kg.

The median may or may not change. It depends on where these people are in the order. We cannot say without the data.

(i) Draw a line for each chosen state with years on the horizontal axis and price on the vertical axis.

(ii) Prices rose in most states, but not every year. Mizoram has the highest prices.

(iii) Gujarat’s price stayed almost the same (about ₹13–15) until a jump to ₹19.2 in 2025. Uttar Pradesh’s price rose steadily from ₹16 to about ₹27 and then dropped slightly.

(iv) West Bengal — from ₹9.47 to ₹23.99 (an increase of about ₹14.5, more than double).

(v) For example: Why are prices so different in different states?

(i) Valid — in 1983 about 84% of rural homes used kerosene, and about 64% of urban homes used electricity.

(ii) Valid — the kerosene line goes down in both graphs.

(iii) Not valid — in 2000 about 90% of urban homes used electricity (about 10% used kerosene).

(iv) Not valid — the graph shows the main lighting source, not power cuts.

(i) About 2 hours for urban children aged 10.

(ii) (c) 12 years

(iii) (a) Not correct — the time at 15 is less than at 10. (b) Not correct — an average does not tell us about every child.

The lower lines (around 05:00–08:00) are sunrise and the upper lines (around 16:00–20:00) are sunset.

(i) In January the sun rises earliest at Kibithu (about 5:50 am). Sunset is about 4:30 pm, so the day is about 10.5 to 11 hours long.

(ii) Srinagar has the longest day (in June, about 14.5 hours).

(iii) Kanyakumari’s day length hardly changes through the year, but places in the north have much longer summer days.

(i) Purnima: when the moon rises around sunset (about 6 pm) — around the 15th. Amavasya: when the moon rises around sunrise (about 6 am) — around the 29th–30th.

(ii) The moon rises and sets about 50 minutes later each day.