1 statistical indicators and statistical thinking 1 dr. t s g peiris department of mathematics

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Statistical Indicatorsand

Statistical Thinking

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Dr. T S G Peiris

Department of Mathematics

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• Why do we do statistics?

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•What is statistics?

To understand and explain the “variability”

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Why data

• As the global economy becomes increasingly electronic , the application of statistics to business and day to day applications has been increasing fast. 3

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Why data are needed?

• Obtaining appropriate data is essential in conducting business/teaching/etc

• Data contain information needed to make a more informal decision in a particular situation.

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The growth and development of statistics

• Widespread availability of software tools had led to an ever increasing application of statistical methods to business decision making.

• Today managers face problems/burdens, not having to make more and more decisions based on larger and larger volumes of data.

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• Why we want to know about statistics?

Managers/Processionals have access to a large amount of

data.

Lack of knowledge of data and its variability by the managers is difficult to survive due to competition.That is to use available data to make decisions

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• A market researcher needs to assess product characteristics to distinguish one product from another

• An investor wants to determine what firms are likely to have accelerated growth

• A lecturer wants to know the duration of which students access his notes in the website

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• Dean of a Faculty wants to how students performed in an exam

• Students wants compare marks with others

• An engineer wants to know how traffic pass at a busy junction

Farmer wants to about prices of vegetables

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In today’s world…• …we are constantly being bombarded with

statistics and statistical information. For example:– customer surveys – medical news – political polls – economic predictions– marketing information – scanner data

• How can we make sense out of all this data?• How do we differentiate valid from flawed

claims?

• What is statistics?9

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• Statistics is a way to get information from data’

Data

Statistics

Information

Data: Facts,, collected together for reference or information.

Information: Knowledge communicated concerning some particular fact.

Statistics is a tool for creating new understanding from data. 10

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Parameters

Population

Sample

Statistics

Subset

Key statistical concepts(Statistical Thinking)

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Key statistical concepts (Statistical Thinking)

Population• A population is the group of all items of

interest to a statistics practitioner.• Frequently very large; sometimes infinite.

Sample• A sample is a set of data drawn from the

population.• Potentially very large, but less than the

population.. 12

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Parameters

• A descriptive measure of a population.

Statistics• A descriptive measure of a

sample.

Key statistical concepts(Statistical Thinking)

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Statistical inference (Decisions)• Statistical inference is the process of making an

estimate, prediction, or decision about a population based on a sample.

Parameter

Population Sample

Statistic

Inference

What can we infer about a population’s parameters based on a sample’s statistics? 14

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Stats anxiety• A student is anxious about his statistics course, after he had

heard that the course is difficult. The lecturer provides last semester’s final exam marks to the student. What can be discerned from this list of numbers?

Data

Statistics

Information

List of last term’s marks.958970657857

New information derived about the statistics class.

e.g. Class average, proportion of class receiving F’s,most frequent mark,marks distribution, etc.

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Thus statistic

• is a quantity that is calculated from a sample of data.

• It is used to give information about unknown values in the corresponding population.

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Statistical Indicator

• Derived from the sample• A single or composite value to represent a

variability of the populationStudents performance indicatorsUniversity performance indicatorsEducational indicatorsEcological indicatorsHuman Resource IndicatorsSport indicatorsHealth indicatorsClimate indicatorBusiness indicators

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Statistical Thinking

• As today's good decisions are driven by data. Professionals are increasingly required to justify decisions on the basis of data

• Statistical thinking is a science assisting you to make decisions under uncertainties

• To make thinking easier we use statistical indicators

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Indicators to measure central tendency

2030405060708090

100

1 11 21 31 41 51 61 71 81 91

Index Number

Mea

n m

arks

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Indicators to measure central tendency

• Mean = nxxn

i i/

1

Weighted mean =

n

1i i

n

1i ii

w w

xwx

In some cases, the data are weighted according to its importance

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Variation in weighted mean

0.0

10.0

20.0

30.0

1 11 21 31 41 51 61 71 81 91

Index number

Wei

gh

ted

mea

n

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Are both are the same?

What is the indicator to check the association?

Raw

Weig

ht

10090807060504030

90

80

70

60

50

40

30

Scatterplot of Weight vs Raw

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Median=67 - as an indicator

A. means

Perc

ent

10090807060504030

100

80

60

40

20

0

Mean 67.99StDev 12.93N 100

Cumulative distribution of AMNormal

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Selecting Mode as an indicator

MedianMean

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Indicators to Measure dispersion

• (a) Sample variance:

)1()1(

)( 2222

n

xnx

n

xxs ii

(b) Range

)x(Min)x(Max ii

[37,93}

68, 12.9

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Use of Standard Deviation (SD) as an indicator

20406080

100

1 11 21 31 41 51 61 71 81 91

Index number

Mar

ks

Mean Mean + SD Mean - SD

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Quality of information and variation are inversely related

•The larger the variation in the data, the lower the quality of the data (i.e., information): the Devil is in the Deviations

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• Other indicators to measure dispersion are Mean Absolute Deviation (MAD) and Coefficient of Variability (CV)

n

xxMAD

n

1i i

%100*xsd

CV

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Why indicator CV is useful?

Eg. A manufacturer of television tubes has two types of tubes namely A and B. Mean life time tubes A and B are 1495 hrs and 1875 and SD of tubes are 220 hrs and 310 hrs respectively.

• The CV of A = 14.7%

• and CV of B = 16.5%

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Indicators to Measure Percentiles

• First quartile (Q1): The sample 25th percentile (P25)

• Second quartile (Q2): The sample 50th percentile (P50) “median”

• Third quartile (Q3): The sample 75th percentile (P75)

• Inter quartile range (IQR): Q3 – Q1 • Desired Percentile:

100)1(

pnLp

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Indicator to measure association between two variables

2i

2i

ii

XY)yy()xx(

)yy)(x(x

r

Correlation

Coefficient

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Use of statistical indictors for decision making

• Use of Z – Score

3

)( 321 ZZZZ

i

iii S

XXZ

Consumer Price Index – CCPI

is an indicator to measure of the average change in

the prices paid by consumers for a specific basket of

goods and services over time in a country

10000

0101

qP

qPP

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Z Score as a statistical indicator

Sub_1

Frequency

10080604020

20

15

10

5

0

Mean 64.92StDev 20.67N 100

Histogram of Sub_1Normal

Sub_2

Frequency

105907560453015

25

20

15

10

5

0

Mean 62.96StDev 18.23N 100

Histogram of Sub_2Normal

Distribution of raw marks

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Z Score as a statistical indicator

Zsocre_1

Frequency

2.41.60.80.0-0.8-1.6-2.4

20

15

10

5

0

Mean 3.108624E-17StDev 1.000N 100

Histogram of Zsocre_1Normal

Zscore_2

Frequency

210-1-2-3

40

30

20

10

0

Mean -3.77476E-17StDev 1.000N 100

Histogram of Zscore_2Normal

Distribution of new indicator – Z score

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Relationship between raw marks and Z score

Raw_mean

Zsc_

mean

1009080706050403020

1.5

1.0

0.5

0.0

-0.5

-1.0

-1.5

-2.0

Scatterplot of Zsc_mean vs Raw_mean

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Consumer Price Index – CCPI

• It is a statistical indicator to measure of the average change in the prices paid by consumers for a specific basket of goods and services over time in a country.

Colombo Consumer Price Index (CCPI)

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Weights used for Colombo Consumer Price Index (CCPI) – CCPI_N

Items Weight

Food and Beverages 46.71

Cloth and Footwear 3.08

Housing, Water, Electricity 18.29

Furnishing 3.22

Health 4.18

Transport 9.47

Communication 4.42

Recreation & Culture 2.18

Education 5.79

Miscellaneous 2.65

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Grade Point Average _GPA

Credits

Credits sGradePoint GPA Overall

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Use of statistical indictors for decision making

• Body Mass Index Body Mass Index (BMI) is an indicator

calculated from a person's weight and height.

BMI provides a reliable indicator of body fatness for most people and is used to screen for weight categories that may lead to health problems .

HAS THIS UPDATED TO SL?

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BMI - Indicator

• BMI indicates you are

Underweight for BMI less than 18.5,

Normal - 18.5 - 24.9

Overweight - 25.0 - 29.9

Obese > 30.0

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BMI - Indicator

• high blood pressure (hypertension) • high LDL-cholesterol ("bad" cholesterol) • low HDL-cholesterol ("good" cholesterol) • high triglycerides • high blood glucose (sugar) • family history of premature heart disease • physical inactivity • cigarette smoking

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Use of statistical indictors for decision making

• Human Development Index is an indicator to rank countries by level of "human

development", which usually also implies whether a country is a developed, developing or under developing

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HDI - Indicator

• Go to Websites

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Thank you

and

Best Wishes