Exercise 13.2 answers: Statistics

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1. Add the 8 numbers: 6+7+10+12+13+4+8+12 = 72.
2. Mean = 72 ÷ 8 = 9.
3. Find how far each number is from 9: −3, −2, 1, 3, 4, −5, −1, 3.
4. Square each of these and add: 9+4+1+9+16+25+1+9 = 74.
5. Variance = 74 ÷ 8 = 37/4.

AnswerMean = 9, variance = 37/4.

Watch this explained “Ten observations”, 3:37 into Taking the root to get the spread back into the units of the data

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1. The numbers are 1, 2, 3, …, n.
2. Their sum is n(n+1)/2, so the mean is (n+1)/2.
3. The sum of their squares is n(n+1)(2n+1)/6.
4. Variance = sum of squares ÷ n, minus the square of the mean.
5. Variance = (n+1)(2n+1)/6 − (n+1)²/4, which simplifies to (n² − 1)/12.

AnswerMean = (n + 1)/2, variance = (n² − 1)/12.

Watch this explained “Three hundred and thirty”, 5:37 into Taking the root to get the spread back into the units of the data

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1. The numbers are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
2. Add them: 3+6+…+30 = 165. Mean = 165 ÷ 10 = 33/2 = 16.5.
3. Find how far each number is from 16.5, square each distance, and add them up: the total is 742.5.
4. Variance = 742.5 ÷ 10 = 297/4 = 74.25.

AnswerMean = 33/2, variance = 297/4.

Watch this explained “Three hundred and thirty”, 5:37 into Taking the root to get the spread back into the units of the data

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1. N = total of the frequencies = 2+4+7+12+8+4+3 = 40.
2. Multiply each xi by its fi and add: the total is 760.
3. Mean = 760 ÷ 40 = 19.
4. Find how far each xi is from 19, square it, multiply by its frequency, and add: the total is 1736.
5. Variance = 1736 ÷ 40 = 217/5.

AnswerMean = 19, variance = 217/5.

Watch this explained “The table, end to end”, 1:38 into Carrying the frequencies through, for discrete and for grouped data

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1. N = total of the frequencies = 3+2+3+2+6+3+3 = 22.
2. Multiply each xi by its fi and add: the total is 2200.
3. Mean = 2200 ÷ 22 = 100.
4. Find how far each xi is from 100, square it, multiply by its frequency, and add: the total is 640.
5. Variance = 640 ÷ 22 = 320/11.

AnswerMean = 100, variance = 320/11.

Watch this explained “The table, end to end”, 1:38 into Carrying the frequencies through, for discrete and for grouped data

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1. N = total of the frequencies = 2+1+12+29+25+12+10+4+5 = 100.
2. Take an assumed mean A = 64 and let di = xi − 64: −4, −3, −2, −1, 0, 1, 2, 3, 4.
3. Multiply each di by its fi and add: the total is 0.
4. Mean = 64 + 0/100 = 64.
5. Multiply each di² by its fi and add: the total is 286.
6. Variance = 286/100 − (0/100)² = 286/100 = 2.86.
7. Standard deviation = √2.86 = √286⁄10 ≈ 1.69.

AnswerMean = 64, standard deviation = √286⁄10 ≈ 1.69.

Watch this explained “Two more tables”, 16:50 into Shifting and scaling the observations to make the arithmetic small

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1. Take the mid-point of each class: 15, 45, 75, 105, 135, 165, 195.
2. N = total of the frequencies = 2+3+5+10+3+5+2 = 30.
3. Multiply each mid-point by its frequency and add: the total is 3210.
4. Mean = 3210 ÷ 30 = 107.
5. Find how far each mid-point is from 107, square it, multiply by its frequency, and add: the total is 68280.
6. Variance = 68280 ÷ 30 = 2276.

AnswerMean = 107, variance = 2276.

Watch this explained “Intervals, not values”, 3:36 into Carrying the frequencies through, for discrete and for grouped data

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1. Take the mid-point of each class: 5, 15, 25, 35, 45.
2. N = total of the frequencies = 5+8+15+16+6 = 50.
3. Multiply each mid-point by its frequency and add: the total is 1350.
4. Mean = 1350 ÷ 50 = 27.
5. Find how far each mid-point is from 27, square it, multiply by its frequency, and add: the total is 6600.
6. Variance = 6600 ÷ 50 = 132.

AnswerMean = 27, variance = 132.

Watch this explained “Intervals, not values”, 3:36 into Carrying the frequencies through, for discrete and for grouped data

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1. Take the mid-point of each class: 72.5, 77.5, 82.5, 87.5, 92.5, 97.5, 102.5, 107.5, 112.5.
2. Take an assumed mean A = 92.5 and class width h = 5. Let yi = (xi − 92.5) ÷ 5: −4, −3, −2, −1, 0, 1, 2, 3, 4.
3. N = total of the frequencies = 3 + 4 + 7 + 7 + 15 + 9 + 6 + 6 + 3 = 60.
4. fi·yi for each class: −12, −12, −14, −7, 0, 9, 12, 18, 12. They add to 6.
5. Mean = 92.5 + 5 × (6/60) = 92.5 + 0.5 = 93.
6. fi·yi² for each class: 48, 36, 28, 7, 0, 9, 24, 54, 48. They add to 254.
7. Variance = 5² × [254/60 − (6/60)²] = 25 × (127/30 − 1/100) = 25 × 1267/300 = 1267/12 ≈ 105.58.
8. Standard deviation = √(1267/12) = √3801/6 ≈ 10.28.
9. The answer key at the back of the book prints 10.27 for the standard deviation; √105.583… = 10.275…, which rounds to 10.28, so the key cut the decimal short.

AnswerMean = 93, variance = 1267/12 ≈ 105.58, standard deviation = √3801/6 ≈ 10.28

Watch this explained “Two more tables”, 16:50 into Shifting and scaling the observations to make the arithmetic small

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1. Make the classes continuous as the hint says: 32.5-36.5, 36.5-40.5, 40.5-44.5, 44.5-48.5, 48.5-52.5.
2. Take the mid-point of each class: 34.5, 38.5, 42.5, 46.5, 50.5.
3. N = total of the frequencies = 15+17+21+22+25 = 100.
4. Multiply each mid-point by its frequency and add: the total is 4350.
5. Mean diameter = 4350 ÷ 100 = 43.5.
6. Find how far each mid-point is from 43.5, square it, multiply by its frequency, and add: the total is 3084.
7. Variance = 3084 ÷ 100 = 30.84.
8. Standard deviation = √30.84 = √771⁄5 ≈ 5.55 mm.

AnswerMean diameter = 87/2 = 43.5 mm, standard deviation = √771⁄5 ≈ 5.55 mm.

Watch this explained “Which route each one wants”, 14:52 into Carrying the frequencies through, for discrete and for grouped data