For any two numbers, their average always lands exactly halfway between them on the number line. For 3 and 7: mean \(=\frac{3+7}{2}=5\), which sits 2 units from each. For 8 and 9: mean \(=\frac{8+9}{2}=8.5\), again exactly midway. This will be true no matter which pair you pick — the mean of two numbers is always their midpoint.

The mean balances the data — it is the point where the total distance of all values below it equals the total distance of all values above it. It is not simply the midpoint of the smallest and largest value; it’s a true “balance point,” like the fulcrum of a see-saw, that accounts for every single value’s pull on either side.

No — the mean is not generally the midpoint of the extremes. For example, in the collection 9, 11, 11, 17 the mean is 12, but the midpoint of the extremes (9 and 17) is also 13, which is close but not exactly equal in general, and for skewed data they can differ a lot. Instead, what’s always true is:

• If the new value is greater than the current mean, the mean increases — to restore balance, the centre must shift right toward this added weight.
• If the new value is less than the current mean, the mean decreases — the centre shifts left.

• Removing a value greater than the mean — the mean decreases (we’ve taken away weight from the heavier side).
• Removing a value less than the mean — the mean increases.
• Removing a value exactly equal to the mean — the mean stays the same, since that value contributed zero distance on either side; removing it doesn’t disturb the balance.

The mean stays exactly the same. Using the fair-share idea: imagine the mean is the amount everyone gets if the total is shared equally. If a new person joins with exactly that fair-share amount, the total share-per-person doesn’t change at all — they simply bring along precisely their “fair” contribution, neither adding nor subtracting any imbalance. The same logic applies in reverse when removing such a value.

Think of the mean as what everyone would get if the total were shared equally. If every single value increases by 2 (say everyone receives ₹2 extra), then naturally each person’s fair share also goes up by exactly ₹2 — nothing about how the total is redistributed has changed relatively, only an equal bonus has been layered on top of everyone. So the fair share (mean) simply shifts by the same fixed amount.

The average also doubles. Using the same data (mean 7.18), doubling every value gives a new mean of \(2\times 7.18 = 14.36\) — confirmed directly in the chapter’s example.

Cell B7 contains 27 — Gowri’s marks in Odia (column B = Odia, row 7 = Gowri).

Looking at Ashwin’s row (row 4: 29, 31, 33, 34, 30, 28): Ashwin scored more than 30 in Telugu (31), English (33), and Maths (34). (Social Science is exactly 30, not “more than” 30.)

(Science is column G; the student data spans rows 2 to 23 for all 22 students listed.) Similarly, the Odia average is =AVERAGE(B2:B23) and the Telugu average is =AVERAGE(C2:C23). Comparing the two computed values directly tells us which subject has the higher class average — this requires actually running the formulas on the full dataset to confirm numerically.

Once written for one row/column, the same formula can be copied down or across to every other student/subject — spreadsheets automatically adjust the cell references, saving enormous time compared to manual calculation.

Plot A has all its dots clustered tightly between 5 and 6.5 minutes — a fairly narrow, high cluster. Since the mean of such tightly-bunched-high values will itself land in that high range (around 5.5–6), Plot A is the one with mean 5.57 minutes. Plots B and C have dots spread more toward lower values (B has many dots below 2, and C is clustered around 3.5–4.5), so their means would be noticeably lower than 5.57.

Using the same method throughout: read off each value and its frequency (number of dots stacked above it) directly from the plot, then:

For the median, build a running total of frequencies from the smallest value upward, and find which value(s) fall at the middle position(s) (depending on whether the total count is odd or even).

Multiple weather stations spread across the state continuously record local temperatures. The “monthly maximum” for the whole state is obtained by taking the single highest temperature reading recorded by any station, anywhere in the state, during that month.

• The horizontal axis shows months (Jan–Dec); the vertical axis shows temperature in °C, scaled in steps of 10.
• Kerala’s points are connected with a blue line, Punjab’s with a red line — using distinct colours (and ideally distinct marker shapes) lets us track each state’s trend separately even when the lines cross.
• Punjab shows a sharp rise-and-fall pattern (a clear summer peak); Kerala’s line stays much flatter throughout the year.

This is open-ended — possible questions: Why does Punjab show such extreme seasonal swings while Kerala stays stable? Is this related to distance from the coast, or latitude? A plot of monthly minimum temperatures would likely show a similar overall shape (Punjab swinging widely, Kerala staying flatter) but with every value shifted down — since minimum temperatures are always lower than maximum temperatures for the same month and place.

Space agencies and international bodies (like the UN Office for Outer Space Affairs) maintain official registries where every satellite, probe, lander, or crewed mission is logged at launch. The yearly count is simply the total number of such registered launches recorded for each country (and the world) within that calendar year.

• “Worldwide count increased every single year” — Not valid. The graph shows the count actually decreased from 2023 to 2024 (from ~2900 down to ~2800), so it didn’t increase every year.
• “USA launched about 3/4 of the worldwide count in 2022–24” — Valid (approximately) — the USA’s line sits very close to the World line in those years, visually confirming it’s the dominant contributor at roughly that proportion.
• “Nepal did not launch any object 2012–24” — Not a valid inference. The graph only shows USA, China, and Russia individually plus the World total — it gives us no information whatsoever about Nepal specifically.
• “Combined China + Russia count in 2024 is about 400” — Valid (approximately) — reading both countries’ lines at 2024 and adding their values gives a figure in this range.

Looking at the steepest jump in the World line: between 2019 and 2020 (or thereabouts), the count roughly doubles — jumping from several hundred to over a thousand. This is the sharpest year-on-year rise visible on the graph, corresponding to the surge in commercial satellite launches (e.g. large satellite constellations) during this period.

Method: Rainfall is measured daily at weather stations in each city over many years; for each calendar month, the total rainfall recorded in that month is averaged across all the years of data — giving the “monthly average rainfall” used in the graph.

Grouping: Kovalam, Udupi, and Mumbai lie along India’s west coast; Rameswaram, Chennai, and Puri lie along the east coast. The west-coast cities generally receive much heavier rainfall (up to ~800–900 mm peaks) than most east-coast cities — except Puri and Chennai, which still show clear but smaller peaks.

This is a visual-reasoning exercise based on Manoj’s coloured 48-box strips (30-min intervals). General approach:

• (i) Identify colours: The largest continuous block, usually overnight, is sleeping. Short, repeated small blocks around typical meal times are eating. A long uninterrupted weekday block (mid-morning to afternoon) is attending classes. Scattered colourful blocks in the evening/weekend are meeting friends/hobbies. Brief blocks near waking time are showering/exercise, and short blocks at the start/end of the school block represent travelling.
• (ii) Identify the days: The strip with the longest unbroken “class” block on a weekday pattern is likely a school day (Friday); strips with much longer “friends/hobbies” blocks and later wake-times are the weekend (Saturday, Sunday).
• (iii) The long movie: Look for an unusually long, single uninterrupted block of “meeting friends/media” time (longer than a typical TV episode) — this is most likely to appear on a weekend afternoon/evening strip.
• (iv) Lunch break: A short “eating” block sandwiched in the middle of the long weekday “classes” block marks the lunch break time.

Plot both rows as two separate lines sharing the same days-of-week axis: “Visiting” (16, 19, 10, 14, 20, 22, 35) and “Purchasing” (10, 8, 7, 11, 12, 16, 26), each with their own colour/marker. Notice that “Purchasing” is always lower than “Visiting” — not everyone who visits ends up buying — and both numbers rise sharply on Saturday and especially Sunday.

• (i) “In 1983, majority rural used kerosene while majority urban used electricity” — Valid. The rural kerosene line starts high (~85%) in 1983 while the urban electricity line also starts already above 50% (~65%) — both confirming this statement.
• (ii) “Kerosene use decreased over time in both areas” — Valid. Both kerosene lines (rural and urban) show a clear, continuous downward trend from 1983 to 2023.
• (iii) “In 2000, 10% of urban households used electricity” — Not valid. By the year 2000, the urban electricity line is already well above 80–90%, nowhere close to just 10%.
• (iv) “In 2023, there were no power cuts” — Not valid (unsupported inference). The graph only shows what source households primarily use for lighting — it tells us nothing about the reliability or continuity of that electricity supply, so we cannot conclude anything about power cuts.