Chebyshevs
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Here are the steps to solve these problems using Chebyshev's theorem: 1) Mean = P2M Standard deviation = P250K k = 2 (for at least 75% of values) Range = Mean ± 2 * Standard deviation = P2M ± 2 * P250K = P2M ± P500K So at least 75% of house prices will be between P2M - P500K and P2M + P500K 2) Mean = P4.00 Standard deviation = P0.50 Range = P4.00 ± 2 * P0.50 = P4.00 ± P1.00 k = 2 By Chebys
Chebyshevs
- 1. Frequency Distribution CLASS CLASS f LIMITS BOUNDARIES 23 - 27 22.5 - 27.5 2 28 - 32 27.5 - 32.5 6 33 - 37 32.5 - 37.5 8 38 - 42 37.5 - 42.5 14 43 - 47 42.5 - 47.5 20 48 - 52 47.5 - 52.5 13 53 - 57 52.5 - 57.5 10 58 - 62 57.5 - 62.5 5 63 - 67 62.5 - 67.5 2
- 2. Frequency Distribution CLASS CLASS f Xm LIMITS BOUNDARIES 23 - 27 22.5 - 27.5 2 28 - 32 27.5 - 32.5 6 33 - 37 32.5 - 37.5 8 38 - 42 37.5 - 42.5 14 43 - 47 42.5 - 47.5 20 48 - 52 47.5 - 52.5 13 53 - 57 52.5 - 57.5 10 58 - 62 57.5 - 62.5 5 63 - 67 62.5 - 67.5 2
- 3. Frequency Distribution CLASS CLASS f Xm fXm LIMITS BOUNDARIES 23 - 27 22.5 - 27.5 2 25 28 - 32 27.5 - 32.5 6 30 33 - 37 32.5 - 37.5 8 35 38 - 42 37.5 - 42.5 14 40 43 - 47 42.5 - 47.5 20 45 48 - 52 47.5 - 52.5 13 50 53 - 57 52.5 - 57.5 10 55 58 - 62 57.5 - 62.5 5 60 63 - 67 62.5 - 67.5 2 65
- 4. Frequency Distribution CLASS CLASS f Xm fXm Xm2 LIMITS BOUNDARIES 23 - 27 22.5 - 27.5 2 25 50 28 - 32 27.5 - 32.5 6 30 180 33 - 37 32.5 - 37.5 8 35 280 38 - 42 37.5 - 42.5 14 40 560 43 - 47 42.5 - 47.5 20 45 900 48 - 52 47.5 - 52.5 13 50 650 53 - 57 52.5 - 57.5 10 55 550 58 - 62 57.5 - 62.5 5 60 300 63 - 67 62.5 - 67.5 2 65 130
- 5. Frequency Distribution CLASS CLASS f Xm fXm Xm2 fXm2 LIMITS BOUNDARIES 23 - 27 22.5 - 27.5 2 25 50 625 28 - 32 27.5 - 32.5 6 30 180 900 33 - 37 32.5 - 37.5 8 35 280 1225 38 - 42 37.5 - 42.5 14 40 560 1600 43 - 47 42.5 - 47.5 20 45 900 2025 48 - 52 47.5 - 52.5 13 50 650 2500 53 - 57 52.5 - 57.5 10 55 550 3025 58 - 62 57.5 - 62.5 5 60 300 3600 63 - 67 62.5 - 67.5 2 65 130 4225
- 6. Frequency Distribution CLASS CLASS f Xm fXm Xm2 fXm2 LIMITS BOUNDARIES 23 - 27 22.5 - 27.5 2 25 50 625 1250 28 - 32 27.5 - 32.5 6 30 180 900 5400 33 - 37 32.5 - 37.5 8 35 280 1225 9800 38 - 42 37.5 - 42.5 14 40 560 1600 22400 43 - 47 42.5 - 47.5 20 45 900 2025 40500 48 - 52 47.5 - 52.5 13 50 650 2500 32500 53 - 57 52.5 - 57.5 10 55 550 3025 30250 58 - 62 57.5 - 62.5 5 60 300 3600 18000 63 - 67 62.5 - 67.5 2 65 130 4225 8450
- 7. Frequency Distribution CLASS CLASS f Xm fXm Xm2 fXm2 LIMITS BOUNDARIES 23 - 27 22.5 - 27.5 2 25 50 625 1250 28 - 32 27.5 - 32.5 6 30 180 900 5400 33 - 37 32.5 - 37.5 8 35 280 1225 9800 38 - 42 37.5 - 42.5 14 40 560 1600 22400 43 - 47 42.5 - 47.5 20 45 900 2025 40500 48 - 52 47.5 - 52.5 13 50 650 2500 32500 53 - 57 52.5 - 57.5 10 55 550 3025 30250 58 - 62 57.5 - 62.5 5 60 300 3600 18000 63 - 67 62.5 - 67.5 2 65 130 4225 8450 80 405 3600 19725 168550
- 8. Solutions: fX m 2 2 fX m n fX m x s2 n n 1 3600 36002 x 168550 80 80 s2 80 1 x 45 s2 223.73 s 15
- 9. 2 4 6 8 10 12 14 16 18 20 22.5 27.5 32.5 37.5 42.5 47.5 52.5 Histogram 57.5 62.5 67.5
- 10. 2 4 6 8 10 12 14 16 18 20 22.5 27.5 X s 32.5 37.5 42.5 X 47.5 52.5 Histogram 57.5 X s 62.5 67.5
- 11. 2 4 6 8 10 12 14 16 18 20 22.5 27.5 32.5 37.5 42.5 45 47.5 52.5 Histogram 57.5 62.5 67.5
- 12. 2 4 6 8 10 12 14 16 18 20 22.5 27.5 30 32.5 37.5 42.5 45 47.5 52.5 Histogram 57.5 60 62.5 67.5
- 13. Histogram X 3s X 2s X s X X s X 2s X 3s
- 14. Pafnuty Lvovich Chebyshev He specified the proportions of the spread in terms of the standard deviation. Born: 16 May 1821 in Okatovo, Russia Died: 8 Dec 1894 in St. Petersburg, Russia
- 15. Chebyshev's theorem The proportion of values from a data set that will fall within k standard deviations of the mean will be at least 1 1 k2 where k is a number greater than 1 (k is not necessarily an integer).
- 16. Chebyshev's theorem Let k = 2: 1 1 1 3 1 1 1 75% k2 22 4 4 This means that at least three-fourths or 75% of the data values fall within 2 standard deviations of the mean of the data set.
- 17. Chebyshev's theorem Let k = 3: 1 1 1 8 1 1 2 1 88.89% k2 3 9 9 This means that at least eight-ninths or 88.89% of the data values fall within 3 standard deviations of the mean of the data set.
- 18. Chebyshev's theorem Example: If two variables measured in the same units have the same mean, say 70, and variable A has a standard deviation of 1.5 while variable B has a standard deviation of 5, then the data for variable B will be more spread out than the data for variable A.
- 19. Chebyshev's theorem For variable A: mean = 70 standard deviation = 1.5 X 2s : 70 + 2(1.5) = 73 70 – 2(1.5) = 67 At least three-fourths, or 75%, of the data values fall between 67 and 73.
- 20. Chebyshev's theorem At least 88.89% At least 75% 65.5 67 68.5 70 71.5 73 74.5 X 3s X 2s X s X X s X 2s X 3s
- 21. Chebyshev's theorem At least 88.89% At least 75% 55 60 65 70 75 80 85 X 3s X 2s X s X X s X 2s X 3s
- 22. Try this out 1. The average price of house and lot in a certain subdivision is P2 M, and the standard deviation is P250K. Find the price range for which at least 75% of the house and lots will sell. 2. A survey of local companies found that the mean amount of travel allowance for executives was P4.00 per kilometer. The standard deviation was P0.50. Using Chebyshev's theorem, find the minimum percentage of the data values that will fall between P2.75 and P5.25.