Download x – mean

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:
1
1 
k
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2
1
CHEBYSHEV'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?
1
3
At least 1  2 2  4  75 %
of the data falls within 2 standard deviations of
the mean.
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2
CHEBYSHEV'S THEOREM for k = 3
According to Chebyshev’s Theorem, at
least what fraction of the data falls
within “k” (k = 3) standard deviations of
the mean?
1
8
At least 1  3 2  9  88 . 9 %
of the data falls within 3 standard deviations of
the mean.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
3
CHEBYSHEV'S THEOREM for k =4
According to Chebyshev’s Theorem, at
least what fraction of the data falls
within “k” (k = 4) standard deviations of
the mean?
1
15
At least 1  4 2  16  93 . 8 %
of the data falls within 4 standard deviations of
the mean.
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4
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?
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5
Mean = 77
Standard deviation = 6
At least 75% of the grades would be in the
interval:
x  2 s to x  2 s
77 – 2(6) to 77 + 2(6)
77 – 12 to 77 + 12
65 to 89
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6
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.
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7
Sample Mean for a Frequency
Distribution
xf

x 
n
x = class midpoint
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8
Sample Standard Deviation for
a Frequency Distribution
s
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 ( x  x) f
2
n 1
9
Calculation of the mean of
grouped data
Ages:
f
x
xf
30 – 34
4
32
128
35 – 39
5
37
185
40 – 44
2
42
84
45 - 49
9
47
423
f = 20
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xf = 820
10
Mean of Grouped Data
xf  xf

x

n
f
820

 41 . 0
20
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11
Calculation of the standard
deviation of grouped data
Ages:
f
x
x –mean
(x –mean)2
(x – mean)2 f
30 - 34 4
32
–9
81
324
35 - 39 5
37
–4
16
80
40 - 44 2
42
1
1
2
45 - 49 9
47
6
36
324
f =20
Mean
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 (x – mean)2 f
= 730
12
Calculation of the standard
deviation of grouped data
  x  x   730
f = n = 20
2
( x  x) f

s

2
n 1
730
20  1
 38 . 42  6 . 20
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13
Weighted Average
Average calculated where some of
the numbers are assigned more
importance or weight
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14
Weighted Average
xw

Weighted Average 
w
where w  the weight of the data value x.
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15
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%
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16
Compute the Weighted Average:
• Midterm
• Term Paper
• Final exam
x
92
80
88
w
.25
.25
.50
1.00
xw
23
20
44
87
 xw  87  87  Weighted Average
 w 1.00
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17
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)%.
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18
Percentiles
60% of data
P 40
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Highest value
Lowest value
40% of data
19
Quartiles
• Percentiles that divide the data into
fourths
• Q1 = 25th percentile
• Q2 = the median
• Q3 = 75th percentile
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20
Q1
Median
= Q2
Q3
Highest value
Lowest value
Quartiles
Inter-quartile range = IQR = Q3 — Q1
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21
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.
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22
Find the quartiles:
12
23
41
15
24
45
16
25
51
16
30
17
32
18
33
22
33
22
34
The data has been ordered.
The median is 24.
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23
Find the quartiles:
12
23
41
15
24
45
16
25
51
16
30
17
32
18
33
22
33
22
34
The data has been ordered.
The median is 24.
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24
Find the quartiles:
12
23
41
15
24
45
16
25
51
16
30
17
32
18
33
22
33
22
34
For the data below the median, the median is 17.
17 is the first quartile.
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25
Find the quartiles:
12
23
41
15
24
45
16
25
51
16
30
17
32
18
33
22
33
22
34
For the data above the median, the median is 33.
33 is the third quartile.
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26
Find the interquartile range:
12
23
41
15
24
45
16
25
51
16
30
17
32
18
33
22
33
22
34
IQR = Q3 – Q1 = 33 – 17 = 16
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27
Five-Number Summary of Data
•
•
•
•
•
Lowest value
First quartile
Median
Third quartile
Highest value
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28
Box-and-Whisker Plot
a graphical presentation of the fivenumber summary of data
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29
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.
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30
Construct a
Box-and-Whisker Plot:
12
23
41
15
24
45
16
25
51
16
30
17
32
18
33
22
33
Lowest = 12
Q1 = 17
median = 24
Q3 = 33
22
34
Highest = 51
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31
Box-and-Whisker Plot
60 55 50 45 40 35 30 25 20 15 -
Lowest = 12
Q1 = 17
median = 24
Q3 = 33
Highest = 51
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32