Statistics Cental Tendency

8/3/2019 Statistics Cental Tendency

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StatisticsStatistics

Prof. Rushen ChahalDescriptive Statistics -

Measures of Central Tendency

Chapters 3 & 4


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TO CALCULATE THE ARITHMETIC MEAN,THE WEIGHTED MEAN, THE MEDIAN,

THE MODE, AND THE GEOMETRIC MEAN.

TO EXPLAIN THE CHARACTERISTICS,

USE, ADVANTAGES, AND

DISADVANTAGES OF EACH MEASURE OF

CENTRAL TENDENCY.

TO IDENTIFY THE POSITION OF THEARITHMETIC MEAN,MEDIAN, AND MODE

FOR BOTH A SYMMETRICAL AND A

SKEWED DISTRIBUTION.

TODAYS GOALS


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3

Definition: For ungrouped data, the populationmean isthe sum ofall the population values

divided by the total number of population

values. To compute the population mean, use

the following formula.

POPULATION MEAN

mu

Sigma

Population

Size

Individual

value

Q ! XN


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4Parameter: A measurable characteristic ofa

population. For example, the population mean.

A racing team hasa fleet of four cars. The

following are the milescovered by eachcar over

their lives: 23,000, 17,000, 9,000, and 13,000.Find the average milescovered by eachcar.

Since this fleetisthe population, the mean is

(23,000 + 17,000 + 9,000 + 13,000)/4 = 15,500.

EXAMPLE


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Definition: For ungrouped data, the samplemean isthe sum ofall the sample values divided

by the number ofsample values. To compute the

sample mean, use the following formula.

THE SAMPLE MEAN

XXn!

7X-bar Sigma

Sample

Size

Individual

value


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6

Population Mean QSample Mean x

WHATS THE DIFFERENCE?

?


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9Sum of Deviations

Consider the set of values: 3, 8, and 4. The meanis 5. So (3 -5) + (8 - 5) + (4 - 5) = -2 + 3 - 1 = 0.

Symbolically we write:

The mean isalso known asthe expected value

or average.

( )X X ! 0


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Definition: The midpoint ofthe valuesafter theyhave been ordered from the smallestto the

largest, or the largestto the smallest. There are

as many valuesabove the median as below itin

the dataarray.

Note: For an odd set of numbers, the median will

be the middle number in the ordered array.

Note: For an even set of numbers, the medianwill be the arithmeticaverage ofthe two middle

numbers.

THE MEDIAN


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11Compute the median:

The road life for asample of five tiresin milesis:

42,000 51,000 40,000 39,000 48,000

Arranging the datain ascending order gives:

39,000 40,000 42,000 48,000 51,000. Thus

the median is 42,000 miles.

EXAMPLE


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12Compute the median:

The following valuesare years ofservice for a

sample ofsix store managers: 16 12 8 15 7

23.

Arranging in order gives 7 8 12 15 16 23.

Thusthe median is (12 + 15)/2 = 13.5 years.

EXAMPLE


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There isa unique median for each dataset. Itis notaffected by extremely large or small

valuesand istherefore a valuable measure of

central tendency when such values occur.

Itcan be computed for ratio-level, interval-level,

and ordinal-level data.

Itcan be computed for an open-ended frequency

distribution ifthe median does not lie in an

open-ended class.

PROPERTIES OF THE MEDIAN


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14 Definition: The mode isthat value ofthe

observation thatappears most frequently

The exam scores for ten studentsare: 81 93 75

68 87 81 75 81 87. Whatisthe modal exam

score?

Since the score of 81 occursthe most, then the

modal score is 81.

The nextslide showsthe histogram withsixclasses for the water consumption from our

previousclass. Observe thatthe modal classis

the blue box witha midpoint of 15.

THE MODE


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15WATER CONSUMPTION IN 1,000 GALLONS


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16THEWEIGHTED MEANDefinition: The weighted mean ofaset of

numbersX1, X2, ..., Xn, withcorresponding

weightsw1, w2, ... , wn, iscomputed from the

following formula.

Why would anyone wantto weight observations?

Xw X w X w X

w w w

or X

w X

w

wn n

n

w

!

!

1 1 2 2

1 2

...

...

( )


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17EXAMPLE

During a one hour period on a busy Friday

night, fifty soft drinks were sold atthe Kruzin

Cafe. Compute the weighted mean ofthe price

ofthe soft drinks. (Price ($), Number sold):

(0.5, 5), (0.75, 15), (0.9, 15), and (1.10, 15).

The weighted mean is

[0.5v+ 0.75v15 + 0.9v15 + 1.1v15]/[5 +15+15+ 15]= $43.75/50 =

$0.875


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18Definition: The geometric mean (GM)ofaset of

n numbersis defined asthe nth root ofthe

product ofthe n numbers. The formula for the

geometric mean is given by:

One main use ofthe geometric mean isto

average percents.

How do you compute the nth root ofa number?

THE GEOMETRIC MEAN

GM X X X X nn!

1 2 3

...


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19EXAMPLE The profits earned by ABC Construction on

three projects were 6, 3, and 2 percent

respectively. Compute the geometric mean

profitand the arithmetic mean and compare.

The geometric mean is

The arithmetic mean profit =(6 + 3 + 2)/3 =

3.6667.

The geometric mean of 3.3019 givesa moreconservative profit figure than the arithmetic

mean of 3.6667. Thisis because the GM is not

heavily weighted by the profit of 6 percent.

GM! !( )( )( ) . .6 3 23 33019


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20 The other main use ofthe geometric mean to

determine the average percentincrease in sales,

production or other business or economicseries

from one time period to another.

The formula for the geometric mean asappliedto thistype of problem is:

Where did thiscome from?

THE GEOMETRIC MEAN (continued)

GM Value at end of periodValue at beginning of period

n! 1 1


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21 The total enrollmentata large university

increased from 18,246 in 1985 to 22,840 in 1995.

Compute the geometric mean rate ofincrease

over the period.

Here n = 10, so n - 1 = 9 = (number of periods) The geometric mean rate ofincrease is given by

Thatis, the geometric mean rate ofincrease is

2.53%.

EXAMPLE

GM

22,84018,246! !9 1 00253. .


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22 Do you prefer I use the arithmetic mean or the

geometric mean to compute classscore

averages?

EXAMPLE

?


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23 The mean ofasample of data organized in a

frequency distribution iscomputed by the

following formula:

THE MEAN OF GROUPED DATA

X-bar

Sum of

frequencies

Class midpoint

Xvalues -

Sample

size

XXf

f

Xf

n!

!

f- class

frequency


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24A sample oftwenty appliance storesin a large

metropolitan area revealed the following

number of VCRssold last week. Compute the

mean number sold. The formulaand

computation isshown below.

EXAMPLE

XfXf

fXn!

!

= 325/20

= 16.25 VCRs


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25EXAMPLE (continued) The table also gives the necessary computations.


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26SYMMETRIC DISTRIBUTION

Symmetric

Distribution

Mode = Median = Mean

Zero

Skewness


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27RIGHT SKEWED DISTRIBUTION

MODE

MEDIAN

MEAN

Positively skewed

Mean and median

are to the RIGHT ofthe mode.


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28LEFT SKEWED DISTRIBUTION

MODE

MEDIAN

MEAN

Negatively skewed

Mean and median

are to the LEFT of

the mode.


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31TO COMPARE (COMPUTE) VARIOUS MEASURES

OF DISPERSION FOR GROUPED AND

UNGROUPED DATA.

TO EXPLAIN THE CHARACTERISTICS, USES,

ADVANTAGES, AND DISADVANTAGES OF EACHMEASURE OF DISPERSION.

TO EXPLAIN CHEBYSHEVS THEOREM AND THE

EMPIRICAL (NORMAL) RULE.

TO COMPUTE THE COEFFICIENTS OFVARIATION AND SKEWNESS.

DEMONSTRATE COMPUTING STATISTICS WITH

EXCEL.

TODAYS GOALS


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Range: For ungrouped data, the range isthedifference between the highestand lowest values

in aset of data. To compute the range, use the

following formula.

EXAMPLE : A sample of five recentaccountinggraduates revealed the following starting salaries

(in $1000): 17 26 18 20 19. The range isthus

$26,000 - $17,000 = $9,000.

MEASURES OF DISPERSION -

UNGROUPED DATA

RANGE= HIGHEST VALUE - LOWEST VALUE


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33Mean Deviation: The arithmetic mean ofthe

absolute values ofthe deviations from the

arithmetic mean. Itiscomputed by the formula

below:

MEAN DEVIATION

Individual

Value

Arithmetic

Mean

Sample

Size

MD

X X

n!


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35Population Variance: The population variance

for ungrouped dataisthe arithmetic mean ofthe

squared deviations from the population mean. It

iscomputed from the formula below:

POPULATION VARIANCE

Sigma

square

Population

size

Population meanIndividualvalue

W

Q2 2

!( )X

N


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36 The ages ofall the patientsin the isolation ward

of Yellowstone Hospital are 38, 26, 13, 41, and 22years. Whatisthe population variance? The

computationsare given below.

EXAMPLE

Q= 7X)/N

= 140/5 = 28.

W2 = 7(X- Q)2/N

= 534/5= 106.8.


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37ALTERNATIVE FORMULA FOR THEPOPULATION VARIANCE

Verify, using above formula, thatthe populationvariance is 106.8 for the previous example.

Why would you use this formula?

W

22 2

!

X

N

X

N


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Population Standard Deviation: The population

standard deviation (W) isthe square root ofthe

population variance.

For the previous example, the population

standard deviation isW = 10.3344 (square root of

106.8).

Note:If you are given the population standarddeviation, just square that number to get the


THE POPULATION STANDARD

DEVIATION


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39Sample Variance: The formula for the sample

variance for ungrouped datais:

OROR

SAMPLE VARIANCE

2

2

1s

X X

n!

( )

2

22

1s X

X

nn!

( )

Samplevariance

Thissample variance is used to estimate the



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40A sample of five hourly wages for blue-collar

jobsis: 17 26 18 20 19. Find the variance.

EXAMPLE

= 100/5 = 20

s2 = 50/(5 - 1)

= 12.5.

X


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41Sample Standard Deviation: The sample

standard deviation (s) isthe square root ofthesample variance.

For the previous example, the sample standard

s = 3.5355 (square root of 12.5).

Note:If you are given the sample standard

deviation, just square that number to get the

sample variance.

SAMPLE STANDARD DEVIATION


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Chebyshevs theorem: For any set ofobservations (sample or population), the

minimum proportion ofthe valuesthat lie within

kstandard deviations ofthe mean isat least

1 - 1/k2, where kisany constant greater than 1.

Empirical Rule: For any symmetrical, bell-

shaped distribution, approximately 68% ofthe

observations will lie within s 1Wofthe mean (Q);approximately 98% within s 2Wofthe mean (Q);

and approximately 99.7% within s 3Wofthe

mean (Q).

INTERPRETATION AND USES OF THE

STANDARD DEVIATION


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Between:

1. 68.26%

2. 95.44%

3. 99.97%

QW QW QW Q QW QW QW

Bell-Shaped Curve showing the relationship between W and Q.


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46RELATIVE DISPERSION Coefficient of Variation: The ratio ofthe

standard deviation to the arithmetic mean,expressed asa percentage.

For example, ifthe CVfor the yield oftwo

differentstocksare 10 and 25. The stock withthe larger CVhas more variation relative to the

mean yield. Thatis, the yield for thisstockis not

asstable asthe other.

CVs

X! (100%)


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48SYMMETRIC DISTRIBUTION

Symmetric

Distribution

Mode = Median = Mean

Zero

Skewness


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49RIGHT SKEWED DISTRIBUTION

MODE

MEDIAN

MEAN

Positively skewed

Mean and median

are to the RIGHT ofthe mode.


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50LEFT SKEWED DISTRIBUTION

MODE

MEDIAN

MEAN

Negatively skewed

Mean and median

are to the LEFT of

the mode.


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=AVERAGE(A1:A10) ArithmeticMean

=MEDIAN(A1:A10) Median Value

=MODE(A1:A10) Modal Value

=GEOMEAN(A1:A10) GeometricMean=QUARTILE(A1:A10,Q) Quartile Q Value

=MAX(A1:A10)-MIN(A1:A10) Range

=PERCENTILE(A1:A10,P) Percentile P Value

EXCEL FUNCTIONSEXCEL FUNCTIONS


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=AVEDEV(A1:A10) Mean Absolute Deviation (MAD)

=VAR(A1:A10) Sample Variance

=STDEV(A1:A10) Sample Standard Deviation

=VARP(A1:A10) Population Variance

=STDEVP(A1:A10) Population Standard Deviation

=STDEV(A1:A10)/AVERAGE(A1:A10) Coefficeint of

Variation=SKEW(A1:A10) Coefficient of Skewness

(not same as book)

MORE EXCEL FUNCTIONSMORE EXCEL FUNCTIONS

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