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MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH

EDITIONEDITION

ELEMENTARY STATISTICSSection 2-4 Measures of Center

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Objectives Day 1

•Given a data set, determine the mean, median, and mode.

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a value at the

center or middle of a data set

Measures of Center

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Mean

(Arithmetic Mean)

AVERAGE

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

Definitions

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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

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

x =n

xx

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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

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Definitions Median

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

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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’)~

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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

~

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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

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(even number of values)

MEDIAN is 5.02

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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

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(even number of values)

MEDIAN is 5.02

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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

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a. 5 5 5 3 1 5 1 4 3 5

b. 1 2 2 2 3 4 5 6 6 6 7 9

c. 1 2 3 6 7 8 9 10

Examples

Mode is 5

Bimodal - 2 and 6

No Mode

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Midrange

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

Definitions

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Midrange

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

Definitions

Midrange =highest score + lowest score

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

Round-off Rule for Measures of Center

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use class midpoint of classes for variable x

Mean from a Frequency Table

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use class midpoint of classes for variable x

Mean from a Frequency Table

x = Formula 2-2f

(f • x)

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use class midpoint of classes for variable x

Mean from a Frequency Table

x = class midpoint

f = frequency

f = n

x = Formula 2-2f

(f • x)

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214)( fx

Example Qwerty Keyboard Word Ratings

Word

Ratings

Interval

Midpoints

Frequency

0-2 1 20 20

3-5 4 14 56

6-8 7 15 105

9-11 10 2 20

12-14 13 1 13

x f x f

52f

2144 11 4 1 points

52. .x

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• Pages 65-66 3,5,9,11

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Homework Solutionspg 65 #3

• Mean

• Median

35 46 55 65 74 83 88 93 99 107 108 119

Occurs between the 6th and 7th data values

Mode- none

972

1281 0 seconds.

xx

n

x

x

83 8885 5 seconds

2.x

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Homework Solutionspg 65 #5

• Mean

• Median

• Mode

7 15 minutes.x

7 20.x Jefferson Valley

Providence7.70

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Homework Solutionspg 65 #9

40-49

50-59

60-69

70-79

80-89

90-99

100-109

freq x*fmidpts

200f

1487074 35 74 4 minutes

200. .x

14870 fx

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Homework Solutionspg 65 #11

• Mean

233946 78 46 8 mph

50. .x

Midptsx

Freqf x*f

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Objectives Day 2

• Given a data set where the scores vary in importance, compute a weighted mean.

• Determine how extreme values affect measures of center.

• Understand the relationship between the shape of a distribution and the relative location of the mean and median.

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Weighted Mean – used whenscores vary in importance

where represents the scores and the corresponding weights

w xx x w

w

Formula

1 1 2 2

1 2

...

...n n

n

w x w x w xx

w w w

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Example Weighted Mean

• The final grade computation for a freshman statistics course is based on weighted components.

Tests 20% each

Final 40%

If the Test scores are 83%, 73%, 82% , and a final exam score of 91% … What is the final grade?

20 83 20 73 20 82 40 91

20 20 20 40wx

840084

100wx

A straight percentage grade based on all tests being 100 pNote oints

3298225 82

400. %

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Weighted Mean common error

The weights do not need to sum to 100

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Example Weighted Mean

• Two algebra classes had the following average test scores. Period 2 had a mean score of 40 with 24 students and period 8 had a mean score of 34 with 16 students. What is the mean of the two classes combined?

24 140 346

24 16wx

150437 6 points

40.wx

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Example Weighted Mean

• Your first semester grades at college are as follows:

Subject Grade Credits

Biology A 3

Calculus B 4

College Writing B 3

Archery C 1

Chemistry A 3

What is your GPA?

3 4 3 1 3

3 4 3

4 3 3 2 4

1 3GPA

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3 35714

.GPA

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Advantages - Disadvantages

Table 2-13

Best Measure of Center

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Extreme Values and Measures of Center

Consider the following data set of salaries at a small shipping company.

30,000 30,000 30,000 30,000 30,000 40,000 45,000 125,000

What is the mean? The median? The mode?

What is the mean? The median? The mode?

36000045 000 30 000 mode 30 000

8$ , $ , $ ,x x

Now change the $125,000 to $250,000

485 00060 625 30 000 mode 30 000

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,$ , $ , $ ,x x

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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

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Skewness

Mode = Mean = Median

SYMMETRIC

Figure 2-13 (b)

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Skewness

Mode = Mean = Median

SKEWED LEFT(negatively)

SYMMETRIC

Mean Mode Median

Figure 2-13 (b)

Figure 2-13 (a)

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Skewness

Mode = Mean = Median

SKEWED LEFT(negatively)

SYMMETRIC

Mean Mode Median

SKEWED RIGHT(positively)

Mean Mode Median

Figure 2-13 (b)

Figure 2-13 (a)

Figure 2-13 (c)

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Review of Concepts

• The mean of a data set is the balance point of the values. Think of the mean as a give and take

• The median is the middle value of an ordered data set.

• The mode is the data value that occurs most frequently. There may be no mode, one mode, or more than one mode.

• If a distribution has few values or it is skewed, then the measures of center may not actually be near the center of the distribution. You must make appropriate decisions as to use which measure of center is most appropriate.

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Page 68 #20, 21, 24

Page 107 # 2, 3

Handout data analysis of mean