mat2020 - unit 1 - measures of central tendency and dispersion

8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion

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©The McGraw-Hill Companies, Inc. 2008 McGraw-Hill/Irwin

What is Statistics

Chapter 1



2

GOALS

Understand why we study statistics.

Explain what is meant by descriptivestatistics and inf erential statistics.

Distinguish between a qualitative variableand a quantitative variable.

Describe how a discrete variable is diff erentf rom a continuous variable.

Distinguish among the nominal, or dinal,interval, and ratio levels of measurement.



3

What is Meant by Statistics?

Statistics is the science of

collecting, or ganizing, presenting,analyzing, and interpreting numerical data to assist in making

more eff ective decisions.



4

Who Uses Statistics?

Statistical techniques are used

extensively by marketing,accounting, quality control,consumers, prof essional sportspeople, hospital administrators,

educators, politicians, physicians,etc...



5

Types of Statistics ± DescriptiveStatistics

Descriptive Statistics - methods of or ganizing,summarizing, and presenting data in an inf ormative way.

EXAMPLE 1: A Gallup poll f ound that 49% of the people in a survey knew the name of the f irst book of the Bible. The statistic 49 describes the number out of every 100persons who knew the answer.

EXAMPLE 2: Accor ding to Consumer Reports, General Electric washing machineowners reported 9 problems per 100 machines during 2001. The statistic 9describes the number of problems out of every 100 machines.

Inf erential Statistics: A decision, estimate,prediction, or generalization about apopulation, based on a sample.



6

Population versus Sample

A population is a collection of all possible individuals, objects, or measurements of interest.

A sample is a portion, or part, of the population of interest



7

Types of Variables

A. Qualitative or Attribute variable - thecharacteristic being studied is nonnumeric.

EXAMPLES: Gender, religious aff iliation, type of automobileowned, state of birth, eye color are examples.

B. Quantitative variable - inf ormation is reported

numerically.EXAMPLES: balance in your checking account, minutes

remaining in class, or number of children in a f amily.



8

Quantitative Variables - Classifications

Quantitative variables can be classif ied as either discreteor continuous.

A. Discrete variables: can only assume certain valuesand there are usually ³gaps´ between values.

EXAMPLE: the number of bedrooms in a house, or the number of hammers sold at the localHome Depot (1,2,3,«,etc).

B. Continuous variable can assume any value within aspecif ied range.

EXAMPLE: The pressure in a tire, the weight of a pork chop, or the height of students in aclass.



9

Summary of Types of Variables



10

Four Levels of Measurement

Nominal level - data that isclassif ied into categories and cannot be arranged in anyparticular or der.

EXAMPLES: eye color, gender,

religious aff iliation.

Or dinal level ± involves dataarranged in some or der, but thediff erences between datavalues cannot be determined or

are meaningless.EXAMPLE: During a taste test of

4 sof t drinks, Mellow Yellowwas ranked number 1, Spritenumber 2, Seven-up number 3, and Orange Crush number 4.

Interval level - similar to the or dinallevel, with the additionalproperty that meaningf ulamounts of diff erences between data values can be determined.

There is no natural zero point.EXAMPLE: Temperature on the

Fahrenheit scale.

Ratio level - the interval level withan inherent zero starting point.Diff erences and ratios are

meaningf ul f or this level of measurement.EXAMPLES: Monthly incomeof sur geons, or distancetraveled by manuf acturer¶srepresentatives per month.



11

Summary of the Characteristics for Levels of Measurement




Describing Data:Frequency Tables, FrequencyDistributions, and Graphic Presentation

Chapter 2



13

GOALS

Or ganize qualitative data into a f requencytable.

Present a f requency table as a bar chart or apie chart.Or ganize quantitative data into a f requencydistribution.Present a f requency distribution f or quantitative data using histograms, f requencypolygons, and cumulative f requencypolygons.



14

Bar Charts



15

Pie Charts



16

Pie Chart Using Excel



17

Frequency Distribution

A Frequencydistribution is a

grouping of data intomutually exclusivecategories showing the number of

observations in eachclass.



18

Frequency Table



19

Relative Class Frequencies

Class f requencies can be converted torelative class frequencies to show the

f raction of the total number of observations in each class.

A relative f requency captures the relationshipbetween a class total and the total number of

observations.



20

Frequency Distribution

Class midpoint: A point that divides a classinto two equal parts. This is the average

of the upper and lower class limits.Class f requency: The number of observations in each class.

Class interval: The class interval is

obtained by subtracting the lower limit of a class f rom the lower limit of the nextclass.



21

EXAMPLE ± Creating a FrequencyDistribution Table

Ms. Kathryn Ball of AutoUS Awants to develop tables,charts, and graphs to showthe typical selling price on

various dealer lots. Thetable on the right reportsonly the price of the 80vehicles sold last month atWhitner Autoplex.



22

EXAMPLE ± Creating a FrequencyDistribution Table



23

Constructing a Frequency Table -Example

Step 1: Decide on the number of classes. A usef ul recipe to determine the number of classes (k ) isthe ³2 to the k rule.´ such that 2k > n.There were 80 vehicles sold. So n = 80. If we try k = 6, which

means we would use 6 classes, then 26 = 64, somewhat lessthan 80. Hence, 6 is not enough classes. If we let k = 7, then 27

128, which is greater than 80. So the recommended number of classes is 7.

Step 2: Determine the class interval or width.The f ormula is: i u (H-L)/ k where i is the class interval, H isthe highest observed value, L is the lowest observed value,and k is the number of classes.($35,925 - $15,546)/7 = $2,911Round up to some convenient number, such as a multiple of 10

or 100. Use a class width of $3,000



24

Step 3: Set the individual class limits

Constructing a Frequency Table -Example



25

Step 4: Tally the vehicleselling prices into the

classes.

Step 5: Count the number of items in each class.

Constructing a Frequency Table



26

Relative Frequency Distribution

To convert a f requency distribution to a rel ativ e f requencydistribution, each of the class f requencies is divided by thetotal number of observations.



27

Graphic Presentation of aFrequency Distribution

The three commonly used graphic f orms

are:Histograms

Frequency polygons

Cumulative f requency distributions



28

Histogram

Histogram f or a f requency distribution based on quantitative data is very similar to the bar chart showing thedistribution of qualitative data. The classes are marked on the horizontal axis and the class f requencies on the verticalaxis. The class f requencies are represented by the heightsof the bars.



29

Histogram Using Excel



30

Frequency Polygon

A frequency polygonalso shows the shapeof a distribution and is

similar to a histogram.

It consists of linesegments connecting the points f ormed bythe intersections of theclass midpoints and theclass f requencies.



31

Cumulative Frequency Distribution




Describing Data:Numerical Measures

Chapter 3



34

GOALS

Calculate the arithmetic mean, weighted mean, median, mode, and geometric mean. Explain the characteristics, uses, advantages, and disadvantages of eachmeasure of location.

Identif y the position of the mean, median, and mode f or both symmetric and skewed distributions. Compute and interpret the range, mean deviation, variance, and standar d deviation. Understand the characteristics, uses, advantages, and disadvantages of each measure of dispersion. Understand Chebyshev¶s theorem and the Empirical Rule as they relate to a

set of observations.



35

Characteristics of the Mean

The arithmetic mean is the most widely used measure of location. It requires the interval

scale. Its major characteristics are: ± All values are used.

± It is unique.

± The sum of the deviations f rom the mean is 0.

± It is calculated by summing the values and dividing by the number of values.



36

Population Mean

For ungrouped data, the population mean is thesum of all the population values divided by the

total number of population values:



37

EXAMPLE ± Population Mean



38

Sample Mean

For ungrouped data, the sample mean is the sum of all the sample values

divided by the number of samplevalues:



39

EXAMPLE ± Sample Mean



40

Properties of the Arithmetic Mean

Every set of interval-level and ratio-level data has a mean.

All the values are included in computing the mean.

A set of data has a unique mean.

The mean is aff ected by unusually lar ge or small data values. The arithmetic mean is the only measure of central tendency

where the sum of the deviations of each value f rom the mean iszero.



41

Weighted Mean

The weighted mean of a set of numbers X 1, X 2, ..., X n, with corresponding weights w 1,w 2, ...,w n, is computed f rom the f ollowing f ormula:



42

EXAMPLE ± Weighted Mean

The Carter Construction Company pays its hourly employees $16.50,$19.00, or $25.00 per hour. There are 26 hourly employees, 14 of which are paid at the $16.50 rate, 10 at the $19.00 rate, and 2 at the$25.00 rate. What is the mean hourly rate paid the 26 employees?



43

The Median

The Median is the midpoint of the valuesaf ter they have been or dered f rom the

smallest to the lar gest. ± There are as many values above the median as

below it in the data array.

± For an even set of values, the median will be the

arithmetic average of the two middle numbers.



45

EXAMPLES - Median

The ages f or a sampleof f ive collegestudents are:

21, 25, 19, 20, 22

Arranging the data in

ascending or der gives:

19, 20, 21, 22, 25.

The heights of f our basketballplayers, in inches, are:

76, 73, 80, 75

Arranging the data in ascending or der gives:

73, 75, 76, 80.

Thus the median is 75.5



46

The Mode

The mode is the value of the observation that appears most f requently.



47

Example - Mode



48

Mean, Median, Mode Using Excel

Table 2±4 in Chapter 2 shows the prices of the 80 vehicles sold last month at Whitner Autoplex in Raytown, Missouri. Determine the mean and the median selling price. The mean and the median selling prices are reported in the f ollowing Excel output. There are 80 vehicles in the study. So thecalculations with a calculator would be tedious and prone to error.



49

Mean, Median, Mode Using Excel



50

The Relative Positions of the Mean,Median and the Mode



51

The Geometric Mean

Usef ul in f inding the average change of percentages, ratios, indexes,or growth rates over time.

It has a wide application in business and economics because we areof ten interested in f inding the percentage changes in sales, salaries,

or economic f igures, such as the GDP, which compound or build on each other.

The geometric mean will always be less than or equal to thearithmetic mean.

The geometric mean of a set of n positive numbers is def ined as thenth root of the product of n values.

The f ormula f or the geometric mean is written:



52

EXAMPLE ± Geometric Mean

Suppose you receive a 5 percent increase in salary this year and a 15 percent increase

next year. The average annual percentincrease is 9.886, not 10.0. Why is this so?We begin by calculating the geometric mean.

098861151051 . ). )( .( GM !!



53

EXAMPLE ± Geometric Mean (2)

The retur n on investment ear ned by Atkinsconstruction Company f or f our successive

years was: 30 percent, 20 percent,-40 percent,and 200 percent. What is the geometric mean rate of retur n on investment?

.. ). )( . )( . )( .( 2941808203602131 44 !!!



54

Dispersion

Why Study Dispersion?

± A measure of location, such as the mean or themedian, only describes the center of the data. It

is valuable f rom that standpoint, but it does nottell us anything about the spread of the data.

± For example, if your nature guide told you that theriver ahead averaged 3 f eet in depth, would youwant to wade across on f oot without additionalinf ormation? Probably not. You would want toknow something about the variation in the depth.

± A second reason f or studying the dispersion in aset of data is to compare the spread in two or

more distributions.



55

Samples of Dispersions



56

Measures of Dispersion

Range

Mean Deviation

Variance and Standar d Deviation



57

EXAMPLE ± Range

The number of cappuccinos sold at the Starbucks location in theOrange Country Airport between 4 and 7 p.m. f or a sample of 5days last year were 20, 40, 50, 60, and 80. Determine the mean deviation f or the number of cappuccinos sold.

Range = Lar gest ± Smallest value= 80 ± 20 = 60



58

EXAMPLE ± Mean Deviation

The number of cappuccinos sold at the Starbucks location in theOrange Country Airport between 4 and 7 p.m. f or a sample of 5days last year were 20, 40, 50, 60, and 80. Determine the mean deviation f or the number of cappuccinos sold.



59

EXAMPLE ± Variance and StandardDeviation

The number of traff ic citations issued during the last f ive months in Beauf ort County, South Carolina, is 38, 26, 13, 41, and 22. Whatis the population variance?



60

EXAMPLE ± Sample Variance

The hourly wages f or a sample of part-time employees atHome Depot are:$12, $20, $16, $18,and $19. What isthe samplevariance?



61

Chebyshev¶s Theorem

The arithmetic mean biweekly amount contributed by the DupreePaint employees to the company¶s prof it-sharing plan is $51.54,and the standar d deviation is $7.51. At least what percent of thecontributions lie within plus 3.5 standar d deviations and minus3.5 standar d deviations of the mean?



62

The Empirical Rule



63

The Arithmetic Mean of Grouped Data

Arithmetic Mean of Grouped Data

where:

is the designation f or the sample mean

is the sum of the class f requencies

is the midpoint of each class

is the sum of the f requency of each class times the midpoint of

the class.



64

Recall in Chapter 2, weconstructed a f requencydistribution f or the vehicleselling prices. The

inf ormation is repeated below. Determine thearithmetic mean vehicleselling price.

The Arithmetic Mean of Grouped Data -Example

Selling Price

($'000)Frequency

15 up to 18 818 up to 21 23

21 up to 24 17

24 up to 27 18

27 up to 30 8

30 up to 33 4

33 up to 36 2

80



65

The Arithmetic Mean of Grouped Data -Example

Selling Price

($'000)Frequency

Mid-Point

(x)fx

15 up to 18 8 16.5 132.0

18 up to 21 23 19.5 448.521 up to 24 17 22.5 382.5

24 up to 27 18 25.5 459.0

27 up to 30 8 28.5 228.0

30 up to 33 4 31.5 126.0

33 up to 36 2 34.5 69.0

80 1,845.0

The Mean Selling Priceis $23,100



66

Standard Deviation of Grouped Data

Variance of Grouped Data

Standard Deviation of Grouped Data



67

Standard Deviation of Grouped Data -Example

Ref er to the f requency distribution f or the Whitner Autoplex data used earlier. Compute the standar d deviation of the vehicle selling prices

Selling Price

($'000)Frequency

Mid-Point

(x)fx fx2

15 up to 18 8 16.5 132.0 2178.0018 up to 21 23 19.5 448.5 8745.75

21 up to 24 17 22.5 382.5 8606.25

24 up to 27 18 25.5 459.0 11704.50

27 up to 30 8 28.5 228.0 6498.00

30 up to 33 4 31.5 126.0 3969.00

33 up to 36 2 34.5 69.0 2380.50

80 1,845.0 44082.00



68

The standar d deviation is the most widely used measure of dispersion.

Alter native ways of describing spread of data includedetermining the locati on of values that divide a set of observations into equal parts.

These measures include quartiles, deciles, and percentiles.

Quartiles, Deciles and Percentiles



69

To f ormalize the computational procedure, let L p ref er to thelocation of a desired percentile. So if we wanted to f ind the 33r d percentile we would use L33 and if we wanted the median, the50th percentile, then L50.

The number of observations is n, so if we want to locate themedian, its position is at (n + 1)/2, or we could write this as

(n + 1)(P /100), where P is the desired percentile.

Percentile Computation



70

Percentiles - Example

Listed below are the commissions ear ned last monthby a sample of 15 brokers at Salomon SmithBar ney¶s Oakland, Calif or nia, off ice. Salomon SmithBar ney is an investment company with off ices

located throughout the United States.

$2,038 $1,758 $1,721 $1,637$2,097 $2,047 $2,205 $1,787$2,287 $1,940 $2,311 $2,054$2,406 $1,471 $1,460

Locate the median, the f irst quartile, and the thir d quartile f or the commissions ear ned.



71

Percentiles ± Example (cont.)

Step 1: Or ganize the data f rom lowest tolar gest value

$1,460 $1,471 $1,637 $1,721

$1,758 $1,787 $1,940 $2,038

$2,047 $2,054 $2,097 $2,205

$2,287 $2,311 $2,406



72

Percentiles ± Example (cont.)

Step 2: Compute the f irst and thir d quartiles.Locate L25 and L75 using:

205,2$

721,1$

lyrespectivearray,in thenobservatio

12thand4ththearequartilesthirdandfirsttheTherefore,

12100

75)115( 4

100

25)115(

75

25

7525

!

!

!!!!

L

L

L L



73

Skewness

In Chapter 3, measures of central location f or a setof observations (the mean, median, and mode) and measures of data dispersion (e.g. range and the

standar d deviation) were introduced Another characteristic of a set of data is the shape.

There are f our shapes commonly observed: ± symmetric,

± positively skewed, ± negatively skewed,

± bimodal.



74

Skewness - Formulas for Computing

The coeff icient of skewness can range f rom -3 up to 3.

± A value near -3, such as -2.57, indicates considerable negative skewness.

± A value such as 1.63 indicates moderate positive skewness.

± A value of 0, which will occur when the mean and median are equal,

indicates the distribution is symmetrical and that there is no skewnesspresent.



75

Commonly Observed Shapes



76

Skewness ± An Example

Following are the ear nings per share f or a sample of 15 sof tware companies f or the year 2005. Theear nings per share are arranged f rom smallest tolar gest.

Compute the mean, median, and standar d deviation.Find the coeff icient of skewness using Pearson¶sestimate. What is your conclusion regar ding theshape of the distribution?



77

Skewness ± An Example UsingPearson¶s Coefficient

017.1

22.5$

)18.3$95.4($3)(3

22.5$115

))95.4$40.16($...)95.4$09.0($

1

95.4$

15

26.74$

222

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!

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!

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7!

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s

edian X sk

n

X X s

n

X X



78

Skewness ± A Minitab Example



79

Describing Relationship between TwoVariables

One graphical technique weuse to show the relationshipbetween variables is called a

scatter diagram. To draw a scatter diagram we

need two variables. We scaleone variable along the

horizontal axis ( X

-axis) of agraph and the other variablealong the vertical axis (Y -axis).



80

Describing Relationship between TwoVariables ± Scatter Diagram Examples



81

In the Introduction to Chapter 2 we presented data f rom AutoUS A. In this case the inf ormation concer ned theprices of 80 vehicles sold last month at the Whitner

Autoplex lot in Raytown, Missouri. The data shown include the selling price of the vehicle as well as theage of the purchaser.

Is there a relationship between the selling price of a

vehicle and the age of the purchaser? Would it bereasonable to conclude that the more expensivevehicles are purchased by older buyers?

Describing Relationship between TwoVariables ± Scatter Diagram Excel Example



82

Describing Relationship between TwoVariables ± Scatter Diagram Excel Example



83

Contingency Tables

A scatter diagram requires that both of thevariables be at least interval scale.

What if we wish to study the relationshipbetween two variables when one or both arenominal or or dinal scale? In this case we tallythe results in a contingency table.



84

Contingency Tables ± An Example

A manuf acturer of preassembled windows produced 50 windowsyester day. This mor ning the quality assurance inspector reviewed each window f or all quality aspects. Each was classif ied asacceptable or unacceptable and by the shif t on which it wasproduced. Thus we reported two variables on a single item. The

two variables are shif t and quality. The results are reported in thef ollowing table.



85

End of Chapter 4



86

Today¶s class was brought to you

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The End

mat2020 - unit 1 - measures of central tendency and dispersion

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