Exercise 13.1 answers: Statistics

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1. Number of plants	Houses (f)	Class mark (x)	f × x
   0 − 2	1	1	1
   2 − 4	2	3	6
   4 − 6	1	5	5
   6 − 8	5	7	35
   8 − 10	6	9	54
   10 − 12	2	11	22
   12 − 14	3	13	39

2. Each class mark is the middle of its two limits, so the marks are 1, 3, 5, 7, 9, 11, 13.
3. Multiply every mark by its own frequency and add the column: 1+6+5+35+54+22+39 = 162.
4. The frequencies add to 20, which matches the 20 houses in the survey.
5. Mean = 162 ÷ 20 = 8.1 plants per house.
6. All the class marks and frequencies here are small, so the direct method — multiply, total, divide — needed no large numbers. That is why it was chosen.

AnswerMean number of plants per house = 8.1. The direct method was used, because the class marks and frequencies are all small enough to multiply in one line.

Watch this explained “Four sizes of one shape”, 9:26 into The direct method, and where it becomes unwieldy

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1. Daily wages (₹)	Workers (f)	Class mark (x)	f × x
   500 − 520	12	510	6120
   520 − 540	14	530	7420
   540 − 560	8	550	4400
   560 − 580	6	570	3420
   580 − 600	10	590	5900

2. The class marks are 510, 530, 550, 570, 590 — the middle of each ₹20 band.
3. Multiply each mark by its frequency: 6120, 7420, 4400, 3420, 5900. Total = 27260.
4. The frequencies add to 50, matching the 50 workers.
5. Mean = 27260 ÷ 50 = ₹545.20.

AnswerMean daily wages = ₹545.20.

Watch this explained “Four sizes of one shape”, 9:26 into The direct method, and where it becomes unwieldy

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1. Pocket allowance (₹)	Children (f)	Class mark (x)	f × x
   11 − 13	7	12	84
   13 − 15	6	14	84
   15 − 17	9	16	144
   17 − 19	13	18	234
   19 − 21	f	20	20f
   21 − 23	5	22	110
   23 − 25	4	24	96

2. Add the known frequencies: 7+6+9+13+5+4 = 44, so the total is 44 + f children.
3. Add the known f×x values: 84+84+144+234+110+96 = 752, so the f×x total is 752 + 20f.
4. The mean is given as 18, so (752 + 20f) ÷ (44 + f) = 18.
5. Multiply out: 752 + 20f = 792 + 18f, so 2f = 40.
6. f = 20.

AnswerThe missing frequency is f = 20.

Watch this explained “The same equation, read backwards”, 13:35 into The direct method, and where it becomes unwieldy

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1. Heartbeats/min	Women (f)	Class mark (x)	f × x
   65 − 68	2	66.5	133
   68 − 71	4	69.5	278
   71 − 74	3	72.5	217.5
   74 − 77	8	75.5	604
   77 − 80	7	78.5	549.5
   80 − 83	4	81.5	326
   83 − 86	2	84.5	169

2. Each class is 3 units wide, so the class marks are 66.5, 69.5, 72.5, 75.5, 78.5, 81.5, 84.5.
3. Multiply each mark by its frequency and total the column: 133+278+217.5+604+549.5+326+169 = 2277.
4. The frequencies total 30, matching the 30 women.
5. Mean = 2277 ÷ 30 = 75.9 heartbeats per minute.
6. The numbers stay small here, so the direct method is a suitable choice for this table.

AnswerMean heartbeats per minute = 75.9.

Watch this explained “The shape of the work: four actions, one of them heavy”, 1:41 into The direct method, and where it becomes unwieldy

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1. Number of mangoes	Boxes (f)
   50 − 52	15
   53 − 55	110
   56 − 58	135
   59 − 61	115
   62 − 64	25

2. The printed classes have a gap of 1 between them (52 to 53, and so on), so they are not continuous yet.
3. Close the gaps by moving every limit by half a unit: 49.5 − 52.5, 52.5 − 55.5, 55.5 − 58.5, 58.5 − 61.5, 61.5 − 64.5.
4. This does not move any class mark, since each mark is the average of its two limits and both limits moved by the same half-unit: the marks are 51, 54, 57, 60, 63.

5. Class mark (x)	Boxes (f)	f × x
   51	15	765
   54	110	5940
   57	135	7695
   60	115	6900
   63	25	1575

6. Total of f×x = 765+5940+7695+6900+1575 = 22875. Total boxes = 400.
7. Mean = 22875 ÷ 400 = 57.1875 mangoes per box.
8. The direct method was used, once the classes were made continuous.

AnswerMean number of mangoes per box = 57.1875, found by the direct method after closing the gaps between the classes.

Watch this explained “Continuity leaves the mean alone”, 12:08 into The direct method, and where it becomes unwieldy

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1. Expenditure (₹)	Households (f)	Class mark (x)	f × x
   100 − 150	4	125	500
   150 − 200	5	175	875
   200 − 250	12	225	2700
   250 − 300	2	275	550
   300 − 350	2	325	650

2. The class marks (125, 175, 225, 275, 325) are 50 units apart, so a step-deviation approach also works, but the direct total is easy here: 500+875+2700+550+650 = 5275.
3. The frequencies total 25, matching the 25 households.
4. Mean = 5275 ÷ 25 = ₹211.

AnswerMean daily expenditure on food = ₹211.

Watch this explained “The shape of the work: four actions, one of them heavy”, 1:41 into The direct method, and where it becomes unwieldy

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1. SO2 (ppm)	Localities (f)	Class mark (x)	f × x
   0.00 − 0.04	4	0.02	0.08
   0.04 − 0.08	9	0.06	0.54
   0.08 − 0.12	9	0.10	0.90
   0.12 − 0.16	2	0.14	0.28
   0.16 − 0.20	4	0.18	0.72
   0.20 − 0.24	2	0.22	0.44

2. Keep every class mark as an exact decimal — do not round any of them before multiplying.
3. Total of f×x = 0.08+0.54+0.90+0.28+0.72+0.44 = 2.96.
4. The frequencies total 30, matching the 30 localities.
5. Mean = 2.96 ÷ 30 = 0.0987 ppm (to 4 decimal places).

AnswerMean concentration of SO2 = 0.0987 ppm (37/375 exactly).

Watch this explained “Rounding once, at the end”, 11:01 into The direct method, and where it becomes unwieldy

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1. Days absent	Students (f)	Class mark (x)	f × x
   0 − 6	11	3	33
   6 − 10	10	8	80
   10 − 14	7	12	84
   14 − 20	4	17	68
   20 − 28	4	24	96
   28 − 38	3	33	99
   38 − 40	1	39	39

2. The classes here are of different widths (6, 4, 4, 6, 8, 10, 2), so there is no common step size to shrink the arithmetic — the direct method is the only practical choice.
3. Total of f×x = 33+80+84+68+96+99+39 = 499.
4. The frequencies total 40, matching the 40 students.
5. Mean = 499 ÷ 40 = 12.475 days.

AnswerMean number of days a student was absent = 12.475.

Watch this explained “What h must satisfy, and when to give the method up”, 10:47 into Dividing through by the class width to shrink the arithmetic further

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1. Literacy rate (%)	Cities (f)	Class mark (x)	f × x
   45 − 55	3	50	150
   55 − 65	10	60	600
   65 − 75	11	70	770
   75 − 85	8	80	640
   85 − 95	3	90	270

2. Total of f×x = 150+600+770+640+270 = 2430.
3. The frequencies total 35, matching the 35 cities.
4. Mean = 2430 ÷ 35 = 69.43% (to 2 decimal places).

AnswerMean literacy rate ≈ 69.43% (486/7 exactly).

Watch this explained “Thirty-five territories, run directly”, 2:28 into The direct method, and where it becomes unwieldy