Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 CHEBYSHEV'S THEOREM For any set of data and for any number k, greater than one, the.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1

CHEBYSHEV'S THEOREM

For any set of data and for any number k,

greater than one, the proportion of the

data that lies within k standard

deviations of the mean is at least:

2k

11


CHEBYSHEV'S THEOREM for k = 2CHEBYSHEV'S THEOREM for k = 2

According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 2) standard deviations of the mean?

At least

of the data falls within 2 standard deviations of the mean.

%7543

21

12


CHEBYSHEV'S THEOREM for k = 3CHEBYSHEV'S THEOREM for k = 3


At least


%9.8898

31

12


CHEBYSHEV'S THEOREM for k =4CHEBYSHEV'S THEOREM for k =4


At least


%8.931615

41

12


Using Chebyshev’s Theorem

A mathematics class completes an examination and it is found that the class mean is 77 and the standard deviation is 6.

According to Chebyshev's Theorem, between what two values would at least 75% of the grades be?


Mean = 77 Standard deviation = 6

At least 75% of the grades would be in the interval:

s2xtos2x

77 – 2(6) to 77 + 2(6)

77 – 12 to 77 + 12

65 to 89


Mean and Standard Deviation of Grouped Data

• Make a frequency table

• Compute the midpoint (x) for each class.

• Count the number of entries in each class (f).

• Sum the f values to find n, the total number of entries in the distribution.

• Treat each entry of a class as if it falls at the class midpoint.


Sample Mean for a Frequency Distribution

x = class midpoint

n

xfx


Sample Standard Deviation for a Frequency Distribution

1

)( 2

n

fxxs


Calculation of the mean of grouped data

Calculation of the mean of grouped data

Ages: f

30 – 34 4

35 – 39 5

40 – 44 2

45 - 49 9

x 3237

42

47

xf 128

185

84

423

xf = 820

f = 20


Mean of Grouped Data

f

xf

n

xfx

0.4120

820


Calculation of the standard deviation of grouped data


Ages: f 30 - 34 4

35 - 39 5

40 - 44 2

45 - 49 9

x

32

37

42

47

x –mean – 9

– 4

1

6

Mean

(x –mean)2 81

16

1

36

(x – mean)2 f 324

80

2

324

f =20 (x – mean)2 f

= 730




f = n = 20

20.642.38

120

730

1

)( 2

n

fxxs

7302 xx


Weighted Average

Average calculated where some of the numbers are assigned more

importance or weight


Weighted Average

x. value data the ofweight the w

AverageWeighted

where

w

xw


Compute the Weighted Average:

• Midterm grade = 92

• Term Paper grade = 80

• Final exam grade = 88

• Midterm weight = 25%

• Term paper weight = 25%

• Final exam weight = 50%


Compute the Weighted Average:

x w xw

• Midterm 92 .25 23

• Term Paper 80 .25 20

• Final exam 88 .50 44

1.00 87

Average Weighted8700.1

87

w

xw


Percentiles

For any whole number P (between 1 and 99), the Pth percentile of a distribution is a value such that P% of the data fall at or below it.

The percent falling above the Pth percentile will be (100 – P)%.


Percentiles

40% of data

Low

est

valu

e

Hig

hes

t va

lueP 40

60% of data


Quartiles

• Percentiles that divide the data into fourths

• Q1 = 25th percentile

• Q2 = the median

• Q3 = 75th percentile


Quartiles

Q1

Median = Q2

Q3

Inter-quartile range = IQR = Q3 — Q1

Low

est

valu

e

Hig

hes

t va

lue


Computing Quartiles

• Order the data from smallest to largest.

• Find the median, the second quartile.

• Find the median of the data falling below Q2. This is the first quartile.

• Find the median of the data falling above Q2. This is the third quartile.


Find the quartiles:

12 15 16 16 17 18 22 22

23 24 25 30 32 33 33 34

41 45 51

The data has been ordered.

The median is 24.


Find the quartiles:

12 15 16 16 17 18 22 22

23 24 25 30 32 33 33 34

41 45 51

The data has been ordered.

The median is 24.


Find the quartiles:

12 15 16 16 17 18 22 22

23 24 25 30 32 33 33 34

41 45 51

For the data below the median, the median is 17.

17 is the first quartile.


Find the quartiles:

12 15 16 16 17 18 22 22

23 24 25 30 32 33 33 34

41 45 51

For the data above the median, the median is 33.

33 is the third quartile.


Find the interquartile range:

12 15 16 16 17 18 22 22

23 24 25 30 32 33 33 34

41 45 51

IQR = Q3 – Q1 = 33 – 17 = 16


Five-Number Summary of Data

• Lowest value

• First quartile

• Median

• Third quartile

• Highest value


Box-and-Whisker Plot

a graphical presentation of the five-number summary of data


Making a Box-and-Whisker Plot

• Draw a vertical scale including the lowest and highest values.

• To the right of the scale, draw a box from Q1 to Q3.

• Draw a solid line through the box at the median.

• Draw lines (whiskers) from Q1 to the lowest and from Q3 to the highest values.


Construct a Box-and-Whisker Plot:

12 15 16 16 17 18 22 22

23 24 25 30 32 33 33 34

41 45 51

Lowest = 12 Q1 = 17

median = 24 Q3 = 33

Highest = 51


Box-and-Whisker Plot

Lowest = 12

Q1 = 17

median = 24

Q3 = 33

Highest = 51

60 -

55 -

50 -

45 -

40 -

35 -

30 -

25 -

20 -

15 -

10 -

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 CHEBYSHEV'S THEOREM For any set of data and for any number k, greater than one, the.

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