Quantitative Analysis
Advanced Research Methods &Statistical ApplicationsDr.
Basheer Ahmad Samim15:01 PMStatistical Applications25:04
PMRecommended Readings (Books)Introduction to Statistics, Walpole,
R. E., 3rd Edition (2000)Statistical Methods for Practice and
Research by Ajai S. Gaur and Sanjaya S. Gaur
35:01 PMConstantA characteristic or property that does not
change from individual to individual.45:01 PMVariableA
characteristic or property that varies from individual to
individual.55:01 PMTypes of Variable65:01 PMNominal ScaleVariable
categories are mutually exclusive and exhaustive.Variable
categories have no logical order.Eye Color, Hair Color,
Gender.75:01 PMOrdinal ScaleData categories are mutually exclusive
and exhaustive.Data classifications are ranked or ordered according
to the particular trait they possess.Level of Knowledge about
SPSS85:01 PMInterval ScaleData categories are mutually exclusive
and exhaustive.Data classifications are ranked or ordered according
to the particular trait they possess.Equal differences in the
characteristic are not represented by equal differences in the
measurements.Temperature, Shoe Size and IQ scores95:01 PM10Ratio
ScaleData categories are mutually exclusive and exhaustive.Data
classifications are ranked or ordered according to the particular
trait they possess.Equal differences in the characteristic are
represented by equal differences in the measurements.The zero point
is the essence of the characteristic.Height, Weight, Distance.5:01
PM11Measurement Scales5:01 PM12DataThe information collected for
any kind of investigation.Usually Numerical but can be
Qualitative.5:01 PM1213Primary DataThe initial material collected
during the research process.The information collected directly from
the respondent.Personal Invetigation, Through Investigator, Through
Questionnaire, Through Local Sources, Through Telephone,5:01
PM14Secondary DataThe information collected and processed by the
people other than the researcherGovernment Organizations,
Semi-Government Organizations,
5:01 PM
Data CollectionAny of the following methods may be adopted:(a)
Personal interview(b) Direct observation(c) Mail interview
(internet interview)(d) Telephone interviewWhat are the cons and
pros of each?155:01 PMData managementOffice Editing,Post
Coding,Data entry and Verification.
165:01 PMData organization and AnalysisPreparing data for
analysis,Extracting descriptive measures from the data, Using
advanced statistical techniques to analyze the data and draw
inference there from.175:01 PM18Measures of Central
TendencyArithmetic Mean Quantiles (Median, Quartiles, Deciles,
Percentiles)Mode5:01 PM1819Arithmetic MeanA value obtained by
dividing the sum of all the observations by their number.
If X1, X2, , Xn are n observations of a variable X then
5:01 PM1920Arithmetic MeanThe marks obtained by 8 students
are:
5:01 PM2021QuantilesFor individual observations/discrete
frequency distribution, the ith quartile, jth decile and kth
percentile are located in the array/discrete frequency distribution
by the following relations
5:01 PM2122The weekly TV Watching times (Hours):
QuartilesThe array of the above data is given below:
5:01 PM2223Quartiles
5:01 PM2324
Quartiles5:01 PM2425Quantiles
5:01 PM2526ModeThe mode is a value which occurs most frequently
in a set of data. Or mode is a value that occurs maximum number of
times in a sequence of observations.5:01 PM2627The total automobile
sales (in millions) in the United States for the last 14 years.
ModeMode = 8.2 million5:01 PM2728Measures of DipsersionMeasures
of variation measure the variation present among the values of a
data set, so measures of variation are measures of spread of values
in the data.5:01 PM29Absolute Measures of DispersionRangeQuartile
DeviationMean (Average) DeviationVariance and Standard
Deviation5:01 PM30Relative Measures of DispersionCoefficient of
RangeCoefficient of Quartile DeviationCoefficient of Mean
DeviationCoefficient of Variation (CV)5:01 PM31RangeDifference
between the largest and the smallest observations
5:01 PM32Ignores the way in which data are distributed
Sensitive to outliers
7 8 9 10 11 12Range = 12 - 7 = 57 8 9 10 11 12Range = 12 - 7 = 5
Disadvantages of the
Range1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,51,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120Range
= 5 - 1 = 4Range = 120 - 1 = 1195:01 PMInter-quartile Range
(IQR)Inter-quartile range = 3rd quartile 1st Quartile Q3 - Q1
IQR is independent of outliers
335:01 PMInter-quartile Range34Median(Q2)XmaximumXminimumQ1Q325%
25% 25% 25%12 30 45 57 70Inter-quartile Range (IQR) = 57 30 =
275:01 PM35The Mean (absolute) DeviationX83502-30Mean Deviation is
the average of absolute deviations taken form the mean value.
3036
5:02 PM36VarianceVariance is the average of the squared
deviations taken from the mean value.X
cm(X-Mean)^2X243616616369181124144139169163625660102702
5:02 PM37Comparing Standard DeviationsMean = 15.5 S = 3.338 11
12 13 14 15 16 17 18 19 20 21Data A11 12 13 14 15 16 17 18 19 20
21Mean = 15.5 S = 4.567Data CThe smaller the standard deviation,
the more tightly clustered the scores around meanThe larger the
standard deviation, the more spread out the scores from mean
5:02 PM
11 12 13 14 15 16 17 18 19 20 21Data BMean = 15.5 S =
0.92638Relative Measures of Variation
5:02 PMCoefficient of Variation (CV)Can be used to compare two
or more sets of data measured in different units or same units but
different average size.5:02 PM39
40Use of Coefficient of VariationStock A:Average price last year
= $50Standard deviation = $5
Stock B:Average price last year = $100Standard deviation = $5but
stock B is less variable relative to its price
Both stocks have the same standard deviation5:02 PM41Appropriate
Choice of Measure of VariabilityIf data are symmetric, with no
serious outliers, use range and standard deviation.If data are
skewed, and/or have serious outliers, use IQR.If comparing
variation across two data sets, use coefficient of variation
(C.V)5:02 PM42Five Number SummaryThe five number summary of a data
set consists of the minimum value, the first quartile, the second
quartile, the third quartile and the maximum value written in that
order: Min, Q1, Q2, Q3, Max.
From the three quartiles we can obtain a measure of central
tendency (the median, Q2) and measures of variation of the two
middle quarters of the distribution, Q2-Q1 for the second quarter
and Q3-Q2 for the third quarter.5:02 PM43The weekly TV viewing
times (in hours).
The array of the above data is given below:
Five Number Summary5:02 PM44
Five Number Summary
Minimum value=5.0 Maximum value=66.0
5:02 PM45Box and Whisker DiagramA box and whisker diagram or
box-plot is a graphical mean for displaying the five number summary
of a set of data. In a box-plot the first quartile is placed at the
lower hinge and the third quartile is placed at the upper hinge.
The median is placed in between these two hinges. The two lines
emanating from the box are called whiskers. The box and whisker
diagram was introduced by Professor Jhon W. Tukey.5:02
PM46Construction of Box-PlotStart the box from Q1 and end at
Q3Within the box draw a line to represent Q2Draw lower whisker to
Min. Value up to Q1Draw upper Whisker from Q3 up to Max.
ValueQ1Q3Q25:02 PMMaxValueMinValue47Construction of Box-PlotQ1=22.0
Q3=36.5Q2=30.5Minimum Value=5.0Maximum Value=66.070
60
50
40
30
20
10
05:02 PM48Interpretation of Box-Plot70
60
50
40
30
20
10
0Box-Whisker Plot is useful to identifyMaximum and Minimum
Values in the dataMedian of the data IQR=Q3-Q1, Lengthy box
indicates more variability in the dataShape of the data From
Position of line within boxLine At the center of the
box----SymmetricalLine above center of the box----Negatively
skewedLine below center of the box----Positively SkewedDetection of
Outliers in the data5:02 PM49OutliersAn outlier is the values that
falls well outside the overall pattern of the data. It might be
the result of a measurement or recording error,a member from a
different population,simply an unusual extreme value.
An extreme value needs not to be an outliers; it might, instead,
be an indication of skewness.
5:02 PM50Inner and Outer FencesIf Q1=22.0 Q2=30.5 Q3=36.5
5:02 PM51Identification of the OutliersThe values that lie
within inner fences are normal valuesThe values that lie outside
inner fences but inside outer fences are possible/suspected/mild
outliersThe values that lie outside outer fences are sure
outliers80
70
60
50
40
30
20
10
0Plot each suspected outliers with an asterisk and each sure
outliers with an hollow dot.*Only 66 is a mild outlier5:02 PM52Box
plots are especially suitable for comparing two or more data sets.
In such a situation the box plots are constructed on the same
scale.Uses of Box and Whisker DiagramMaleFemale5:02 PMStandardized
VariableA variable that has mean 0 and Variance 1 is called
standardized variableValues of standardized variable are called
standard scoresValues of standard variable i.e standard scores are
unit-lessConstruction
5:02 PM53X
Z325-1.36241.856164-0.54500.29701190.817410.668212161.08991.1879325404.009
Variable Z has mean 0 and variance 1 so Z is a standard
variable.Standard Score at X=11 is
5:02 PMStandardized Variable55The industry in which sales rep
Mr. Atif works has mean annual sales=$2,500standard deviation=$500.
The industry in which sales rep Mr. Asad works has mean annual
sales=$4,800standard deviation=$600. Last year Mr. Atifs sales were
$4,000 and Mr. Asads sales were $6,000. Performance evaluation by
z-scoresWhich of the representatives would you hire if you have one
sales position to fill?5:02 PM56Performance evaluation by
z-scores
Sales rep. AtifXB= $2,500SB= $500XB= $4,000Sales rep. AsadXP
=$4,800SP = $600XP= $6,000
Mr. Atif is the best choice5:02 PM57A distribution in which the
values equidistant from the centre have equal frequencies is
defined to be symmetrical and any departure from symmetry is called
skewness.
Length of Right Tail = Length of Left TailMean = Median =
ModeSk=0 a) Sk=(Mean-Mode)/SDb) Sk=(Q3-2Q2+Q1)/(Q3-Q1)
5:02 PMMeasures of Skewness58A distribution is positively
skewed, if the observations tend to concentrate more at the lower
end of the possible values of the variable than the upper end. A
positively skewed frequency curve has a longer tail on the right
hand side
Length of Right Tail > Length of Left TailMean > Median
> ModeSK>0Measures of Skewness5:02 PM59A distribution is
negatively skewed, if the observations tend to concentrate more at
the upper end of the possible values of the variable than the lower
end. A negatively skewed frequency curve has a longer tail on the
left side.Length of Right Tail < Length of Left TailMean <
Median < ModeSK< 0
5:02 PMMeasures of Skewness5:02 PM60The Kurtosis is the degree
of peakedness or flatness of a unimodal (single humped)
distribution, When the values of a variable are highly concentrated
around the mode, the peak of the curve becomes relatively high; the
curve is Leptokurtic. When the values of a variable have low
concentration around the mode, the peak of the curve becomes
relatively flat;curve is Platykurtic. A curve, which is neither
very peaked nor very flat-toped, it is taken as a basis for
comparison, is called Mesokurtic/Normal.Measures of Kurtosis615:02
PM
Measures of Kurtosis62Measures of KurtosisIf Coefficient of
Kurtosis > 3 ----------------- Leptokurtic.If Coefficient of
Kurtosis = 3 ----------------- Mesokurtic.If Coefficient of
Kurtosis < 3 ----------------- is Platykurtic.
5:02 PM63
The Empirical Rule
68.26%
95.45%
99.73%5:02 PMSPSS Statistical Package for Social Sciences5:02
PM646772687065687563
2541273243663531155
34263238163038302021
5151620212526273030
31323234353738414366
9.08.28.09.110.311.011.5
10.310.59.89.38.28.28.5
2541273243663531155
34263238163038302021
5151620212526273030
31323234353738414366
