WELCOME

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STATISTICS STATISTICS Class IXClass IX

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NATIONAL ANTHEMNATIONAL ANTHEM

Jana-gana-mana-adhinayaka,jaya heJana-gana-mana-adhinayaka,jaya heBharata-bhagya-vidhata.Bharata-bhagya-vidhata.Punjab-Sindh-Gujarat-MarathaPunjab-Sindh-Gujarat-MarathaDravida-Utkala-BangaDravida-Utkala-BangaVindhya-Himachala-Yamuna-GangaVindhya-Himachala-Yamuna-GangaUchchala –Jaladhi-taranga.Uchchala –Jaladhi-taranga.Tava shubha asisa jage,Tava shubha asisa jage,Tava subha asisa mage,Tava subha asisa mage,Gahe tava jaya gatha,Gahe tava jaya gatha,Jana-gana-mangala-dayaka jaya heJana-gana-mangala-dayaka jaya heBharata-bhagya-vidhata.Bharata-bhagya-vidhata.Jaya he, jaya he, jaya he,Jaya he, jaya he, jaya he,Jaya jaya jaya , jaya he!Jaya jaya jaya , jaya he!

  

PLEDGEPLEDGEIndia is my country. All Indians are my brothers and sisters. I love my India is my country. All Indians are my brothers and sisters. I love my country, and I am proud of its rich and varied heritage. I shall always strive country, and I am proud of its rich and varied heritage. I shall always strive to be worthy of it. I shall give respect to my parents, teachers and all elders to be worthy of it. I shall give respect to my parents, teachers and all elders and treat everyone with courtesy. I pledge my devotion to my country and and treat everyone with courtesy. I pledge my devotion to my country and my people. In their well-being and prosperity alone lies my happinessmy people. In their well-being and prosperity alone lies my happiness

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INDEXINDEX

REVIEW ON OWNREVIEW ON OWN

• Summarizing Qualitative DataSummarizing Qualitative Data• Summarizing Quantitative DataSummarizing Quantitative Data• Measures of Location and Measures of Location and

Variability/DispersionVariability/Dispersion

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Summarizing Qualitative DataSummarizing Qualitative Data

Frequency Distribution Relative Frequency Relative Frequency Percent Frequency DistributionPercent Frequency Distribution Bar GraphBar Graph Pie ChartPie Chart

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Frequency DistributionFrequency Distribution

A A frequency distributionfrequency distribution is a tabular summary is a tabular summary of data showing the frequency (or number) of of data showing the frequency (or number) of items in each of several nonoverlapping items in each of several nonoverlapping classes.classes.

The objective is to provide insights about the The objective is to provide insights about the data that cannot be quickly obtained by data that cannot be quickly obtained by looking only at the original data.looking only at the original data.

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Example: Marada InnExample: Marada InnGuests staying at Marada Inn were asked to rate the Guests staying at Marada Inn were asked to rate the

quality of their accommodations as being quality of their accommodations as being excellentexcellent, , above averageabove average, , averageaverage, , below averagebelow average, or , or poorpoor. .

The Theratings provided by a sample of 20 guests are shownratings provided by a sample of 20 guests are shownbelow.below.

Below Average Average Below Average Average Above AverageAbove AverageAbove Average Above Average Above Average Above Average Above Average Above Average Above Average Below Average Above Average Below Average Below Average Below Average Average Poor Average Poor PoorPoorAbove Average ExcellentAbove Average Excellent Above Average Above Average Average Above AverageAverage Above Average AverageAverageAbove Average Average Above Average Average

How many rated How many rated worseworse than average? than average?

How many rated How many rated betterbetter than average? than average?

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Frequency DistributionFrequency Distribution

Rating Rating Frequency Frequency

PoorPoor 2 2

Below AverageBelow Average 3 3

AverageAverage 5 5

Above AverageAbove Average 9 9

ExcellentExcellent 1 1

TotalTotal 20 20

Example: Marada InnExample: Marada Inn

How many rated How many rated worseworse than average? than average?

How many rated How many rated betterbetter than average? than average?

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Frequency DistributionFrequency Distribution

Rating Rating Frequency Frequency

PoorPoor 2 2

Below AverageBelow Average 3 3

AverageAverage 5 5

Above AverageAbove Average 9 9

ExcellentExcellent 1 1

TotalTotal 20 20

Example: Marada InnExample: Marada Inn

The GM of Marada Inn has a goal that no more than The GM of Marada Inn has a goal that no more than 10% of all guests will rate their stay as 10% of all guests will rate their stay as worseworse than than

average.average.

How is the inn doing?How is the inn doing?

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Relative Frequency andRelative Frequency andPercent Frequency DistributionsPercent Frequency Distributions

The The relative frequencyrelative frequency of a class is the fraction of a class is the fraction or proportion of the total number of data items or proportion of the total number of data items belonging to the class.belonging to the class.

A A relative frequency distributionrelative frequency distribution is a tabular is a tabular summary of a set of data showing the relative summary of a set of data showing the relative frequency for each class.frequency for each class.

The The percent frequencypercent frequency of a class is the relative of a class is the relative frequency multiplied by 100.frequency multiplied by 100.

AA percent frequency distributionpercent frequency distribution is a tabular is a tabular summary of a set of data showing the percent summary of a set of data showing the percent frequency for each class.frequency for each class.

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Example: Marada InnExample: Marada Inn

Relative Frequency and Percent Frequency Relative Frequency and Percent Frequency DistributionsDistributions

RelativeRelative PercentPercent

RatingRating FrequencyFrequency FrequencyFrequency

PoorPoor .10 .10 1010

Below AverageBelow Average .15 .15 1515

AverageAverage .25 .25 2525

Above AverageAbove Average .45 .45 4545

ExcellentExcellent .05 .05 5 5

TotalTotal 1.00 1.00 100 100

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Bar GraphBar Graph

A A bar graphbar graph is a graphical device for depicting is a graphical device for depicting qualitative data that have been summarized in a qualitative data that have been summarized in a frequency, relative frequency, or percent frequency, relative frequency, or percent frequency distribution.frequency distribution.

On the horizontal axis we specify the labels that On the horizontal axis we specify the labels that are used for each of the classes.are used for each of the classes.

A A frequencyfrequency, , relative frequencyrelative frequency, or , or percent percent frequencyfrequency scale can be used for the vertical scale can be used for the vertical axis.axis.

Using a bar of fixed width drawn above each Using a bar of fixed width drawn above each class label, we extend the height appropriately.class label, we extend the height appropriately.

The The bars are separatedbars are separated to emphasize the fact to emphasize the fact that each class is a separate category.that each class is a separate category.

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Two DefinitionsTwo Definitions

HISTOGRAMHISTOGRAM• Quantitative dataQuantitative data

BAR GRAPHBAR GRAPH• Qualitative (non-numerical) dataQualitative (non-numerical) data• See examples that followSee examples that follow

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Example: Marada InnExample: Marada Inn

Bar GraphBar Graph

11

22

33

44

55

66

77

88

99

PoorPoor BelowAverageBelow

AverageAverageAverage Above

AverageAbove

AverageExcellentExcellent

Fre

qu

en

cy

Fre

qu

en

cy

RatingRating

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0% 0%

25%

50%

25%

0%

20%

40%

60%

A B C D F

ECO 6416 Grade Distribution

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3D BAR GRAPH3D BAR GRAPH

100%

0% 0% 0% 0%

0%

50%

100%

A B C D F

ECO 6416 Grade Distribution

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Pie ChartPie Chart

The The pie chartpie chart is a commonly used graphical is a commonly used graphical device for presenting relative frequency device for presenting relative frequency distributions for qualitative data.distributions for qualitative data.

First draw a circle; then use the relative First draw a circle; then use the relative frequencies to subdivide the circle into sectors frequencies to subdivide the circle into sectors that correspond to the relative frequency for that correspond to the relative frequency for each class.each class.

Since there are 360 degrees in a circle, a class Since there are 360 degrees in a circle, a class with a relative frequency of .25 would with a relative frequency of .25 would consume .25(360) = consume .25(360) =

90 degrees of the circle.90 degrees of the circle.

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Pie ChartPie Chart

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Example: Marada InnExample: Marada Inn

Pie ChartPie Chart

Average 25%Average 25%

BelowAverage 15%

BelowAverage 15%

Poor 10%Poor 10%

AboveAverage 45%

AboveAverage 45%

Exc. 5%Exc. 5%

Quality Ratings Quality Ratings

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Summarizing Quantitative DataSummarizing Quantitative Data

Frequency DistributionFrequency Distribution Relative Frequency and Percent Frequency Relative Frequency and Percent Frequency

DistributionsDistributions HistogramHistogram

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91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76104 74 62 68 97 105 77 65 80 10985 97 88 68 83 68 71 69 67 7462 82 98 101 79 105 79 69 62 73

91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76104 74 62 68 97 105 77 65 80 10985 97 88 68 83 68 71 69 67 7462 82 98 101 79 105 79 69 62 73

Example: Hudson Auto RepairExample: Hudson Auto Repair

The manager of Hudson Auto would like to get The manager of Hudson Auto would like to get aa

better picture of the distribution of costs for better picture of the distribution of costs for engineengine

tune-up parts. A sample of 50 customer invoices tune-up parts. A sample of 50 customer invoices hashas

been taken and the costs of parts, rounded to been taken and the costs of parts, rounded to thethe

nearest dollar, are listed below.nearest dollar, are listed below.

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Example: Frequency Distribution Example: Frequency Distribution TableTable

This is what a frequency distribution table looks likeThis is what a frequency distribution table looks like

CumulativeCumulative

RelativeRelative Cumulative Cumulative PercentPercent

Cost ($)Cost ($) Frequency Frequency FrequencyFrequency Frequency Frequency FrequencyFrequency

50-5950-59 2 2 .04 .04 2 2 4 4

60-69 60-69 13 13 .26 15 .26 15 30 30

70-7970-79 16 16 .32 .32 31 31 62 62

80-8980-89 7 7 .14 .14 38 38 76 76

90-9990-99 7 7 .14 .14 45 45 90 90

100-109100-109 55 .10.10 50 50 100 100

Totals 50Totals 50 1.00 1.00

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Frequency DistributionFrequency Distribution

Guidelines for Selecting Number of ClassesGuidelines for Selecting Number of Classes• Use between 5 and 20 classes.Use between 5 and 20 classes.• Data sets with a larger number of elements Data sets with a larger number of elements

usually require a larger number of classes.usually require a larger number of classes.• Smaller data sets usually require fewer Smaller data sets usually require fewer

classes.classes. Guidelines for Selecting Width of ClassesGuidelines for Selecting Width of Classes

• USE CLASSES OF EQUAL WIDTH• Approximate Class Width =Approximate Class Width =

Largest Data Value Smallest Data ValueNumber of Classes

Largest Data Value Smallest Data ValueNumber of Classes

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Example: Hudson Auto RepairExample: Hudson Auto Repair Frequency DistributionFrequency Distribution

If we choose six classes: If we choose six classes:

Approximate Class Width = (109 - 52)/6 = 9.5 Approximate Class Width = (109 - 52)/6 = 9.5 1010

Cost ($)Cost ($) FrequencyFrequency

50-5950-59 2 2

60-69 60-69 13 13

70-7970-79 16 16

80-8980-89 7 7

90-9990-99 7 7

100-109100-109 5 5

Total 50Total 50Would it be wrong if 50-59,60-79,80-89,90-99,100-Would it be wrong if 50-59,60-79,80-89,90-99,100-109?109?

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Relative Frequency and Percent Frequency Relative Frequency and Percent Frequency DistributionsDistributions

Relative PercentRelative Percent

Cost ($)Cost ($) FrequencyFrequency FrequencyFrequency

50-5950-59 .04.04 4 4

60-6960-69 .26.26 2626

70-7970-79 .32.32 3232

80-8980-89 .14.14 1414

90-9990-99 .14.14 1414

100-109100-109 .10.10 1010

Total 1.00Total 1.00 100 100

Example: Hudson Auto RepairExample: Hudson Auto Repair

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HistogramHistogram

Another common graphical presentation of Another common graphical presentation of quantitative data is a quantitative data is a histogramhistogram..

The variable of interest is placed on the The variable of interest is placed on the horizontal axis and the frequency, relative horizontal axis and the frequency, relative frequency, or percent frequency is placed on frequency, or percent frequency is placed on the vertical axis.the vertical axis.

A rectangle is drawn above each class interval A rectangle is drawn above each class interval with its height corresponding to the interval’s with its height corresponding to the interval’s frequency, relative frequency, or percent frequency, relative frequency, or percent frequency.frequency.

Unlike a bar graph, a histogram has Unlike a bar graph, a histogram has no natural no natural separation between rectanglesseparation between rectangles of adjacent of adjacent classes.classes.

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Example: Hudson Auto RepairExample: Hudson Auto Repair

HistogramHistogram

Cost ($)Cost ($)

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44

66

88

1010

1212

1414

1616

1818

Fre

qu

en

cy

Fre

qu

en

cy

50 60 70 80 90 100 11050 60 70 80 90 100 110

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NBA Salaries, few yrs ago

0

510

15

20

2530

35

40

Salaries ($100,000)

Nu

me

r o

f P

laye

rs

2 4 6 8 10 12 20 30 4014 16 18 22 24 26 28 32 34 36 38 42

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Measures of LocationMeasures of Location

MeanMean MedianMedian ModeMode xx

%%

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Example: Apartment RentsExample: Apartment Rents

Given below is a sample of monthly rent Given below is a sample of monthly rent values ($)values ($)

for one-bedroom apartments. The data is a for one-bedroom apartments. The data is a sample of 70sample of 70

apartments in a particular city. The data are apartments in a particular city. The data are presentedpresented

in ascending order. in ascending order.

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

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MeanMean

The The meanmean of a data set is the average of all of a data set is the average of all the data values.the data values.

If the data are from a sample, the mean is If the data are from a sample, the mean is denoted by denoted by

..

If the data are from a population, the mean is If the data are from a population, the mean is denoted by (mu).denoted by (mu).

xxnixxni

xNi x

Ni

xx

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Example: Apartment RentsExample: Apartment Rents

MeanMean

xxni

34 35670

490 80,

.xxni

34 35670

490 80,

.

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

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MedianMedian

The The medianmedian of a data set is the value in the of a data set is the value in the middle when the data items are arranged in middle when the data items are arranged in ascending order.ascending order.

If there is an odd number of items, the median If there is an odd number of items, the median is the value of the middle item.is the value of the middle item.

If there is an even number of items, the If there is an even number of items, the median is the average of the values for the median is the average of the values for the middle two items.middle two items.

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Example: Apartment RentsExample: Apartment Rents

MedianMedian

Since 70 is even and ½ of 70 = 35Since 70 is even and ½ of 70 = 35, average , average 35th and 36th data values:35th and 36th data values:

Median = (475 + 475)/2 = 475Median = (475 + 475)/2 = 475

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

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Example: Apartment RentsExample: Apartment Rents

ALTERNATIVE METHOD - MedianALTERNATIVE METHOD - Median

Median = 50% percentileMedian = 50% percentile

ii = ( = (pp/100)/100)nn = (50/100)70 = 35, average 35th = (50/100)70 = 35, average 35th and 36th data values: and 36th data values: ((see later slidesee later slide))

Median = (475 + 475)/2 = 475Median = (475 + 475)/2 = 475

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ModeMode

The The modemode of a data set is the value that occurs of a data set is the value that occurs with greatest frequency.with greatest frequency.

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Example: Apartment RentsExample: Apartment Rents ModeMode 450 occurred most frequently (7 times)450 occurred most frequently (7 times)

Mode = 450Mode = 450

Mean = $491Mean = $491 Median = $475 Median = $475Mode = $450Mode = $450

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615