Exercise 13.1 answers: Statistics

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1. Add the 8 numbers: 4+7+8+9+10+12+13+17 = 80.
2. Mean = 80 ÷ 8 = 10.
3. Find how far each number is from 10, ignoring sign: 6, 3, 2, 1, 0, 2, 3, 7.
4. Add these distances: 6+3+2+1+0+2+3+7 = 24.
5. Mean deviation = 24 ÷ 8 = 3.

Answer3

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1. Add the 10 numbers: 38+70+48+40+42+55+63+46+54+44 = 500.
2. Mean = 500 ÷ 10 = 50.
3. Find how far each number is from 50, ignoring sign: 12, 20, 2, 10, 8, 5, 13, 4, 4, 6.
4. Add these distances: 12+20+2+10+8+5+13+4+4+6 = 84.
5. Mean deviation = 84 ÷ 10 = 42/5 = 8.4.

Answer42/5

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1. Sort the 12 numbers: 10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18.
2. With 12 numbers (an even count), the median is the average of the 6th and 7th: (13 + 14) ÷ 2 = 13.5.
3. Find how far each number is from 13.5, ignoring sign: 3.5, 2.5, 2.5, 1.5, 0.5, 0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5.
4. Add these distances: the total is 28.
5. Mean deviation about the median = 28 ÷ 12 = 7/3 ≈ 2.33.

AnswerMean deviation about the median = 7/3 ≈ 2.33

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1. Sort the 10 numbers: 36, 42, 45, 46, 46, 49, 51, 53, 60, 72.
2. With 10 numbers, the median is the average of the 5th and 6th: (46+49)÷2 = 95/2 = 47.5.
3. Find how far each number is from 47.5, ignoring sign, and add them up: the total is 70.
4. Mean deviation = 70 ÷ 10 = 7.

Answer7

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1. N = total of the frequencies = 7+4+6+3+5 = 25.
2. Multiply each xi by its fi and add: 5×7+10×4+15×6+20×3+25×5 = 350.
3. Mean = 350 ÷ 25 = 14.
4. Find |xi − 14| for each value: 9, 4, 1, 6, 11.
5. Multiply each by its frequency and add: 9×7+4×4+1×6+6×3+11×5 = 63+16+6+18+55 = 158.
6. Mean deviation = 158 ÷ 25.

Answer158/25

Watch this explained “A table of six values”, 1:32 into The same procedure once the data arrive already grouped

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1. N = total of the frequencies = 4+24+28+16+8 = 80.
2. Multiply each xi by its fi and add: 10×4+30×24+50×28+70×16+90×8 = 4000.
3. Mean = 4000 ÷ 80 = 50.
4. Find |xi − 50| for each value: 40, 20, 0, 20, 40.
5. Multiply each by its frequency and add: 40×4+20×24+0×28+20×16+40×8 = 160+480+0+320+320 = 1280.
6. Mean deviation = 1280 ÷ 80 = 16.

Answer16

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1. N = total of the frequencies = 8+6+2+2+2+6 = 26. Half of N is 13.
2. Running totals of the frequencies: 8, 14, 16, 18, 20, 26.
3. The first running total to reach 13 is 14, on the row xi = 7, so the median is 7.
4. Find |xi − 7| for each value: 2, 0, 2, 3, 5, 8.
5. Multiply each by its frequency and add: 2×8+0×6+2×2+3×2+5×2+8×6 = 16+0+4+6+10+48 = 84.
6. Mean deviation = 84 ÷ 26 = 42/13.

Answer42/13

Watch this explained “Eight values, thirty observations”, 4:20 into The same procedure once the data arrive already grouped

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1. N = total of the frequencies = 3+5+6+7+8 = 29. Half of N is 14.5.
2. Running totals of the frequencies: 3, 8, 14, 21, 29.
3. The first running total to reach 14.5 is 21, on the row xi = 30, so the median is 30.
4. Find |xi − 30| for each value: 15, 9, 3, 0, 5.
5. Multiply each by its frequency and add: 15×3+9×5+3×6+0×7+5×8 = 45+45+18+0+40 = 148.
6. Mean deviation = 148 ÷ 29.

Answer148/29

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1. Take the mid-point of each class: 50, 150, 250, 350, 450, 550, 650, 750.
2. N = total of the frequencies = 4+8+9+10+7+5+4+3 = 50.
3. Multiply each mid-point by its frequency and add: the total is 17900.
4. Mean = 17900 ÷ 50 = 358.
5. Find how far each mid-point is from 358, weight by frequency, and add: the total is 7896.
6. Mean deviation = 7896 ÷ 50 = 3948/25.

Answer3948/25

Watch this explained “Worked about its mean”, 8:41 into The same procedure once the data arrive already grouped

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1. Take the mid-point of each class: 100, 110, 120, 130, 140, 150.
2. N = total of the frequencies = 9 + 13 + 26 + 30 + 12 + 10 = 100.
3. Multiply each mid-point by its frequency and add: 900 + 1430 + 3120 + 3900 + 1680 + 1500 = 12530.
4. Mean = 12530 ÷ 100 = 125.3.
5. Distance of each mid-point from 125.3, ignoring sign: 25.3, 15.3, 5.3, 4.7, 14.7, 24.7.
6. Multiply each distance by its frequency: 227.7, 198.9, 137.8, 141, 176.4, 247. They add to 1128.8.
7. Mean deviation about the mean = 1128.8 ÷ 100 = 11.288 = 1411/125.
8. The answer key at the back of the book prints 11.28; the exact value is 1128.8 ÷ 100 = 11.288, and the key cut off the last digit instead of rounding, so to two decimal places the answer is 11.29.

AnswerMean deviation about the mean = 1411/125 = 11.288

Watch this explained “Worked about its mean”, 8:41 into The same procedure once the data arrive already grouped

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1. N = total of the frequencies = 6 + 8 + 14 + 16 + 4 + 2 = 50. Half of N is 25.
2. Running totals: 6, 14, 28, 44, 48, 50. The first running total to reach 25 is 28, so the median class is 20-30.
3. For the median class: l = 20, C = 14 (running total before it), f = 14, h = 10.
4. Median = 20 + ((25 − 14) ÷ 14) × 10 = 20 + 55/7 = 195/7 ≈ 27.86.
5. Take the mid-point of each class: 5, 15, 25, 35, 45, 55.
6. Distance of each mid-point from 195/7, ignoring sign: 160/7, 90/7, 20/7, 50/7, 120/7, 190/7.
7. Multiply each by its frequency: 960/7, 720/7, 280/7, 800/7, 480/7, 380/7. They add to 3620/7 ≈ 517.14.
8. Mean deviation about the median = (3620/7) ÷ 50 = 362/35 ≈ 10.34.

AnswerMean deviation about the median = 362/35 ≈ 10.34

Watch this explained “The median of a banded table”, 13:05 into The same procedure once the data arrive already grouped

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1. Make the classes continuous, as the hint says: 15.5-20.5, 20.5-25.5, 25.5-30.5, 30.5-35.5, 35.5-40.5, 40.5-45.5, 45.5-50.5, 50.5-55.5.
2. Take the mid-point of each new class: 18, 23, 28, 33, 38, 43, 48, 53.
3. N = total of the frequencies = 5+6+12+14+26+12+16+9 = 100. Half of N is 50.
4. Running totals: 5, 11, 23, 37, 63, 75, 91, 100. The 50th value falls in the class 35.5-40.5.
5. Median = 35.5 + ((50 − 37) ÷ 26) × 5 = 35.5 + 2.5 = 38.
6. Find how far each mid-point is from 38, weight by frequency, and add: the total is 735.
7. Mean deviation = 735 ÷ 100 = 147/20.

Answer147/20

Watch this explained “When the classes do not touch”, 16:14 into The same procedure once the data arrive already grouped