1

a value at the

center or middle of a data set

Measures of Center

2

Mean

(Arithmetic Mean)

AVERAGE

the number obtained by adding the values and dividing the total by the number of values

Definitions

3

Notation

denotes the addition of a set of values

x is the variable usually used to represent the individual

data values

n represents the number of data values in a sample

N represents the number of data values in a population

4

Notationis pronounced ‘x-bar’ and denotes the mean of a set of sample values

x =n

xx

5

Notation

µ is pronounced ‘mu’ and denotes the mean of all values in a population

is pronounced ‘x-bar’ and denotes the mean of a set of sample values

Calculators can calculate the mean of data

x =n

xx

Nµ =

x

6

Definitions Median

the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude

7

Definitions Median

the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude

often denoted by x (pronounced ‘x-tilde’)~

8

Definitions Median

the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude

often denoted by x (pronounced ‘x-tilde’)

is not affected by an extreme value

~

9

6.72 3.46 3.60 6.44

3.46 3.60 6.44 6.72 no exact middle -- shared by two numbers

3.60 + 6.44

2

(even number of values)

MEDIAN is 5.02

10

6.72 3.46 3.60 6.44 26.70

3.46 3.60 6.44 6.72 26.70

(in order - odd number of values)

exact middle MEDIAN is 6.44

6.72 3.46 3.60 6.44

3.46 3.60 6.44 6.72 no exact middle -- shared by two numbers

3.60 + 6.44

2

(even number of values)

MEDIAN is 5.02

11

Definitions Mode

the score that occurs most frequently

Bimodal

Multimodal

No Mode

denoted by M

the only measure of central tendency that can be used with nominal data

12

Measures of Variation

Range

valuehighest lowest

value

13

Midrange

the value midway between the highest  and lowest values in the original data set

Definitions

14

Midrange

the value midway between the highest and lowest values in the original data set

Definitions

Midrange =highest score + lowest score

2

15

5% trimmed mean

the mean of the middle 90% of the scores

Definitions

16

Carry one more decimal place than is present in the original set of values

Round-off Rule for Measures of Center

17

a measure of variation of the scores about the mean

(average deviation from the mean)

Measures of Variation

Standard Deviation

18

Sample Standard Deviation Formula

calculators can compute the sample standard deviation of data

(x - x)2

n - 1S =

19

Sample Standard Deviation Shortcut Formula

n (n - 1)

s =n (x2) - (x)2

calculators can compute the sample standard deviation of data

20

Population Standard Deviation

calculators can compute the population standard deviation

of data

2 (x - µ)

N =

21

Symbolsfor Standard DeviationSample Population

x

xn

s

Sx

xn-1

Book

Some graphicscalculators

Somenon-graphicscalculators

Textbook

Some graphicscalculators

Somenon-graphics

calculators

22

Measures of Variation

Variance

standard deviation squared

23

Measures of Variation

Variance

standard deviation squared

s

2

2

}

use square keyon calculatorNotation

24

SampleVariance

PopulationVariance

Variance

(x - x )2

n - 1s2 =

(x - µ)2

N 2 =

25

Find the range, midrange, mean, median, and 5% trimmed

mean

Data Set A

Range = 98Mean = 50

Midrange = 50Median = 50

5% trimmed = 50

Data Set B

Range = 98Mean = 50

Midrange = 50Median = 50

5% trimmed= 50

A B

1 1

2 2

40 3

40 4

40 5

60 95

60 96

60 97

98 98

99 99

26

A x- µ (x - µ)2

1 -49 2401

2 -48 2304

40 -10 100

40 -10 100

40 -10 100

60 10 100

60 10 100

60 10 100

98 48 2304

99 49 2401

(x - µ)2

N =

10

=

= 31.63610010

Find the Standard Deviation

27

A x- µ (x - µ)2

1 -49 2401

2 -48 2304

3 -47 2209

4 -46 2116

5 -45 2025

95 45 2025

96 46 2116

97 47 2209

98 48 2304

99 49 2401

(x - µ)2

N =

10

=

= 47.02122110

Find the Standard Deviation

28

Coefficient of Variance

Used to compare the variability of two data setsthat do not measure the same thing.

vmeanpopulation

deviationstandardpopulation

varianceoft coefficien

v

29

Coefficient of Variance

v

Age Shoe Size

14 7

18 9

16 9 1/2

15 8

20 10

Use the coefficient of variance todetermine which varies more.

Age µ = 16.6 = 2.154

Shoe Size µ = 8.7

= 1.077

Age varies more than shoe size

.13016.6

2.154v

.1248.7

1.077v

30

Histograma bar graph in which the horizontal scale represents classes and the vertical scale

represents frequencies

Sturgess’ RuleThe number of intervals in a histogram be

approximately 1 +3.3 log(N)•Less-Than cummulative frequency curve ( ] •Greater-Than cummulative frequency curve [ )

•Frequency Polygon A straight line graph that is formed by connecting the midpoints of the top of the bars of the

histogram•Frequency Curve A smoothed out frequency polygon•Ogive A frequency curve formed from a cummulative frequency histogram by connecting the corners with a

smooth curve

31

Qwertry Keyboard Word RatingsTable 2-1

2 2 5 1 2 6 3 3 4 2

4 0 5 7 7 5 6 6 8 10

7 2 2 10 5 8 2 5 4 2

6 2 6 1 7 2 7 2 3 8

1 5 2 5 2 14 2 2 6 3

1 7

32

Frequency Table of Qwerty Word RatingsTable 2-3

0 - 2 20

3 - 5 14

6 - 8 15

9 - 11 2

12 - 14 1

Rating Frequency

33

Histogram of Qwerty Word Ratings

0 - 2 20

3 - 5 14

6 - 8 15

9 - 11 2

12 - 14 1

Rating Frequency

34

Relative Frequency Histogram of Qwerty Word Ratings

Figure 2-3

0 - 2 38.5%

3 - 5 26.9%

6 - 8 28.8%

9 - 11 3.8%

12 - 14 1.9%

RatingRelative

Frequency

35

Histogram and

Relative Frequency Histogram

Figure 2-2 Figure 2-3

36

Figure 2-4

Frequency Polygon

37

Figure 2-5

Ogive

38

Bachelor High SchoolDegree Diploma

Figure 1-1 Salaries of People with Bachelor’s Degrees and with High School Diplomas

$40,000

30,000

25,000

20,000

$40,500

$24,400

35,000

$40,000

20,000

10,000

0

$40,500

$24,40030,000

Bachelor High SchoolDegree Diploma

(a) (b)

39

SymmetricData is symmetric if the left half of its histogram is roughly a mirror of its right half.

SkewedData is skewed if it is not symmetric and if it extends more to one side

than the other.

Definitions

40

Skewness

Mode = Mean = Median

SYMMETRIC

Figure 2-13 (b)

41

Skewness

Mode = Mean = Median

SKEWED LEFT(negatively)

SYMMETRIC

Mean Mode Median

Figure 2-13 (b)

Figure 2-13 (a)

42

Skewness

Mode = Mean = Median

SKEWED LEFT(negatively)

SYMMETRIC

Mean Mode Median

SKEWED RIGHT(positively)

Mean Mode Median