descriptive statistics (1)

5/26/2018 Descriptive Statistics (1)

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WHAT IS STATISTICS

Statistics is defined as the science ofcollecting, organizing, presenting,

analyzing, and interpreting data toassist in making more effectivedecisions.

ORCollection of numerical information iscalled statistics.

Dr. Iftikhar Hussain Adil


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WHAT IS STATISTICS

Broadly defined, it is the science,technology and art of extractinginformation from observational data,

with an emphasis on solving real worldproblems.

It is a logic and methodology for the

measurement of uncertainty and forexamination of the consequences of thatuncertainty in the planning andinterpretation of experimentation and

observation.


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TYPES OF STATISTICS


StatisticalMethods

Descriptive

Statistics

Inferential

Statistics


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TYPES OF STATISTICS

DESCRIPTIVE STATISTICSMethods of organizing, summarizing,and presenting data in an informativeway.

INFERENTIAL STATISTICS

The methods used to determine

something about a population on thebasis of a sample.


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DESCRIPTIVE STATISTICS



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Inferential Statistics

Aim to draw conclusions about anadditional population outside of yourdatasets/sample is known to beinferential statistics.


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Population versus Sample

A population is the complete set of allitems that interests an investigator.Population size N, can be very large or

even infinite.

e.g. All the registered voters of Pakistan

All the students at NUST

Sample is an observed subset of thepopulation values with sample sizegiven by n


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Sampling Techniques

Simple Random Sampling

Systematic Sampling

Stratified Sampling Possible strata: (Male and female strata, Resident

and non-resident strata, White, Black, Hispanic, and Asianstrata, Protestant, Catholic, Jewish, Muslim, etc., strata)

Clustered Sampling Sample of Convenience


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Parameter and Statistic

A parameter is a specific characteristic of apopulation. A statistic is a specificcharacteristic of a sample.

e.g. NBS surveyed its students to determine theaverage daily expense. From a sample of 80students the average expense was computedRs.133.

What is population?

What is sample?

What is parameter?

What is statistic?

Is Rs.133 a parameter or statistic?


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Types of Variables

Variable: A characteristic of an item orindividual that will be analyzed byusing statistics.

e.g. Gender, Party affiliation of registeredvoters, HH income of citizens who live inspecific geographic area, Publishing

category (hard cover, trade paper book,mass marked paper book, text book) ofa book. No of televisions in a householdetc.


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Example (Types of variables)

Reg # GenderAge FA/FSC or

equivalent

Family

Members

1 M 18.2 67 4

2 F 19 70 3

3 M 20 80 5

4 F 19.4 85 6

5 F 20.6 73 3

6 M 21 76 4

7 F 20.3 67 58 F 19.8 89 4


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Types of Variables

Categorical Variables

A categorical variable is a variable that can takeon one of a limited, and usually fixed, number

of possible values. Categorical variables areoften used to represent categorical data.

The values of these variables are selected from anestablished list of categories.

Male/ Female, Pass/ Fail, SA,A,D,SD Numerical variables

The values of these variables involve a counted ormeasured valued


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Types of Variables

Discrete Variables: The vales of thesevariables counts.

e.g. Number of people living in a HHContinuous Variables: These variables

have continuous values and any valuecan theoretically occur limited only by

the precision of the measuringprocess. E.g time to complete a work,air pressure in tyre.


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Levels of Measurement

Levels of measurement often dictatethe calculations that can be done tosummarize and present the data. Italso determines the statistical testthat should be performed.

e.g. Balls in a bag are of different colors

like brown, yellow, blue, green,orange or red etc.


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Types of Levels of Measurement

Ratio Level Data: When a scaleconsist of not only of equidistantpoints but also has a meaningful zeropoint, then we refer it as ratio scale.

Ratio scales are more sophisticated ofscales since it incorporates all the

characteristics of nominal, ordinal andinterval scales. E.g. income data


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Properties of Ratio Level

Equal differences in the characteristic arerepresented by equal differences in thenumbers assigned to the classifications.

Can be added or subtracted i.e.X1+X2 or X1-X2is possible

Can be multiplied or divided

X1*X2 orX1/X2 is possible

Can be ordered

X1X2 Meaningful zero point


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Interval Scale: An interval scale satisfies x2-x1or x2x1or x1x2but not the ratio.

e.g. 100Ois not twice as warm as 50o

(no zero point, no ratio but x2x1or x1x2)

Ordinal Scale: When item are classifiedaccording to more or less characteristics, thescale used is referred as ordinal scale. Thisscale is common in marketing, satisfaction andattitudinal research.E.g. Excellent, v good,good, fair, poor ( No zero point, no equal gap,no ratio but just comparison)


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Nominal Scale: a discrete classificationof data, in which data are neithermeasured nor ordered but subjectsare merely allocated to distinctcategories: for example Male female,married unmarried widowed or

separated (No ratio, No zero point,No equal gap and no comparison)


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Example

A sample of customers in a specialty icecream store was asked a series ofquestions.

What is your favorite flavor of ice cream.

How many times do you eat ice cream

Do you have children under the age of ten

living in your home Have you tried our latest ice cream

flavor?


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Self Review 1-1

Chicago-based Market Facts asked a sample of1,960 consumers to try a newly developedchicken dinner by Boston Market. Of the 1,960

sampled, 1,176 said they would purchase thedinner if it is marketed.

(a) What could Market Facts report to BostonMarket regarding acceptance of the chickendinner in the population?

(b) Is this an example of descriptive statisticsor inferential statistics? Explain.


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DESCRIPTIVE STATISTICSFREQUENCY DISTRIBUTION

A grouping of data into mutuallyexclusive classes showing the numberof observations in each. The raw data

are more easily interpreted iforganized into a frequency distribution.

How to find maximum of data

How to find minimum of data Where is the cluster of data

What is the typical price of vehicleDr. Iftikhar Hussain Adil


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Step 1: Decide on the number ofclasses.

Step 2: Determine the class interval'or width.

Step 3: Set the individual class limits

Step 4: Tally the vehicle selling pricesinto the classes.



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Step 5: Count the number of items in each

class.

class frequency The number ofobservations in each class.

class midpoint

class interval

Relative frequency



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Self Review 2.2

Barry Bonds of the San Francisco Giantsestablished a new single season home runrecord by hitting 73 home runs during the

2001 Major League Baseball season. Thelongest of these home runs traveled 488 feetand the shortest 320 feet. You need toconstruct a frequency distribution of thesehome run lengths.

(a) How many classes would you use?

(b) What class interval would you suggest?

(c) What actual classes would you suggest?


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Exercise Page 31

1. A set of data consists of 38 observations. How manyclasses would you recommend for the frequencydistribution?

2. A set of data consists of 45 observations between $0

and $29. What size would you recommend for the classinterval?

3. A set of data consists of 230 observations between$235 and $567. What class interval would yourecommend?

4. A set of data contains 53 observations. The lowestvalue is 42 and the largest is 129. The data are to beorganized into a frequency distribution.

a. How many classes would you suggest?

b. What would you suggest as the lower limit of the first

class?


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5. Wachesaw Manufacturing, Inc. produced the followingnumber of units the last 16 days. 27, 27, 27, 28, 27,25, 25, 28, 26, 28, 26, 28, 31, 30, 26,26

The information is to be organized into a frequencydistribution.

a. How many classes would you recommend?

b. What class interval would you suggest?

c. What lower limit would you recommend for the first

class? d. Organize the information into a frequency distribution

and determine the relative frequency distribution.

e. Comment on the shape of the distribution.


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HISTOGRAM

A graph in which the classes aremarked on the horizontal axis and theclass frequencies on the vertical axis.

The class frequencies are representedby the heights of the bars, and thebars are drawn adjacent to each

other.


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HISTOGRAM


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Frequency Polygon

It consists of line segmentsconnecting the points formed by theintersections of the class midpoints

and the class frequencies.

cumulative frequency distribution

cumulative frequency polygon


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Frequency Polygon


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Frequency Polygon


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Cumulative Frequency Polygon


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Pareto Diagram

A pareto diagram is a bar chart thatdisplays the frequency of defectcauses

Line Graphs


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Bar Charts

A bar chart can be used to depict any of

the levels of measurement-nominal,

ordinal, interval, or ratio.

The level of education is an ordinalscale variable and is reported on thehorizontal axis


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Difference b/w Histogram andBar Chart

In a histogram, the horizontal axis refersto the ratio scale variable-vehicle sellingprice. This is a continuous variable; hence

there is no space between the bars.Another difference between a bar chartand a histogram is the vertical scale. In ahistogram the vertical axis is the

frequency or number of observations. In abar chart the vertical scale refers to anamount.


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Measures of Location

Measures of Variability

Measure of Relative Position

Measure of Shape



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POPULATION MEAN:

For raw data, that is, data that has notbeen grouped in a frequencydistribution, the population mean isthe sum of all the values in thepopulation divided by the number of

values in the population.Or



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The Sample Mean:

For raw data, that is, ungrouped data,the mean is the sum of all thesampled values divided by the totalnumber of sampled values

or


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Examples: To obtain grade A, Ben mustachieve an average of at least 80 percent infive tests. If his average marks for the first

four tests is 78, what is the lowest marks hecan get in his fifth test and still obtain grade A?

The speeds to the nearest mile per hr, of 120vehicles passing a check point were recordedand grouped into the table below. Estimate themean of this distribution.

Speedmph

21-25 26-30 31-35 36-45 46-60

No of

vehicles

22 48 25 16 9


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Measures of LocationProperties of Mean

1. Every set of interval- or ratio-leveldata has a mean.

2. All the values are included incomputing the mean.

3. The mean is unique.

4. The sum of the deviations of eachvalue from the mean will always bezero.


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The Weighted Mean

The weighted mean is a special caseof the arithmetic mean. It occurswhen there are several observationsof the same value.


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Example: A candidate obtained thefollowing results at NBS

Quizzes Mid Assignments Final

92% 95% 90% 65%

The regulations states that quizzeshaving weight of 15%, assignments10%, mid 25% and final 50%.What isthe candidates final percentage?


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The Median:

The midpoint of the values after theyhave been ordered from the smallestto the largest, or the largest to thesmallest.


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Properties of Median

The median is unique.

It is not affected by extremely largeor small values.

It can be computed for ratio-level,interval-level, and ordinal-level data.


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MODE:The value of the observationthat appears most frequently.


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Properties of Mode

It is Robust measure.

In several data sets there is no modeor more than one mode


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Geometric Mean

The geometric mean is useful infinding the average of percentages,ratios, indexes, or growth rates.


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Why Study Dispersion1. The average is not representative because of

the large spread.

2. A second reason for studying the dispersion ina set of data is to compare the spread in twoor more distributions.

A small value for a measure of dispersion

indicates that the data are clustered closely,say, around the arithmetic mean. The mean istherefore considered representative of thedata. Conversely, a large measure ofdispersion indicates that the mean is notreliable.


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Range

The range is based on the largest andthe smallest values in the data set. Itis the difference of largest andsmallest value.

Range = Largest value - Smallest value


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MEAN DEVIATION

The arithmetic mean of the absolutevalues of the deviations from thearithmetic mean.

Ad a ta e a d D a ba k


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Advantages and Drawbackof Mean Deviation

it uses all the values in thecomputation.

It is easy to understand.

It uses absolute values and it isdifficult to work with absolute values

so this measure is not frequentlyused.


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VARIANCE:The arithmetic mean of thesquared deviations from the mean.

STANDARD DEVIATION: The squareroot of the variance.

Population Variance:

Sample Variance:


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CHEBYSHEV'S THEOREM

For any set of observations (sampleor population), the proportion of thevalues that lie within k standard

deviations of the mean is at least

(1 1/k2)

where k is any constant greater than

1.


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EMPIRICAL RULE

For a symmetrical, bell-shapedfrequency distribution, approximately 68percent of the observations will lie

within plus and minus one standarddeviation of the mean; about 95 percentof the observations will lie within plusand minus two standard deviations of

the mean; and practically all (99.7percent) will lie within plus and minusthree standard deviations of the mean.

Q a tiles Deciles and


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Quartiles, Deciles, andPercentiles

a percentile(or centile) is the valueof a variable below which a certainpercent of observations fall

Lp=(n+1)*P/100

91, 75, 61, 101,43,104


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Box Plots

Abox plot is a graphical display,based on quartiles, that helps uspicture a set of data.

To construct a box plot, we needonly five statistics: the minimumvalue, Q1(the first quartile), the

median, Q3(the third quartile), andthe maximum value.


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Outlier: An outlier is a value that isinconsistent with the rest of the data.

Inter Quartile Range:

The inter quartile range is thedistance between the first and thenthird quartile.


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Skewness

Symmetric: In a symmetric set ofobservations the mean and median are equaland the data values are evenly spread around

these values. The data values below the meanand median are a mirror image of those above.

Positively Skewed:A set of values isskewed to the right or positively skewed if

there is a single peak and the values extendmuch further to the right of the peak than tothe left of the peak. In this case the mean islarger than the median.


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Skewness

Negatively Skewed:In a negativelyskewed distribution there is a singlepeak but the observations extend

further to the left, in the negativedirection, than to the right. In negativelyskewed distribution the mean is smallerthan the median.

Bimodal:A bimodal distribution will havetwo or more peaks. This is often thecase when the values are from two

populations.

How to Access Skewness with


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How to Access Skewness withthe help of Boxplot

Symmetric

The distance from Min to Q2= Q2toMax

The distance from Min to Q1= Q3toMax

The distance from Q1to Q2= Q2to Q3



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Right Skewed

The distance from Q2to Max > Min toQ2

The distance from Q3to Max > Min to

Q1

The distance Q2to Q3> Q1to Q2



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Left Skewed

The distance from Min to Q2> Q2toMax

The distance from Min to Q1> Q3toMax

The distance Q1to Q2> Q2to Q3


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Skewness


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Measures of Skewness


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Univariate Vs Bivariate

Scatter Diagram

we use to show the relationshipbetween variables is called a scatterdiagram.

CONTINGENCY TABLE

A table used to classify observationsaccording to two identifiablecharacteristics.


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Stem and Leaf Plot


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Stem and leaf


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descriptive statistics (1)

Documents

byusing statistics

beinferential statistics

descriptive statisticsdr

sample sizegiven

nust sample

types of statisticsdr

anadditional population

types of variablesvariable