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

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.