Quantitative Methods. Varsha Varde. Course Coverage. Essential Basics for Business Executives Data Classification & Presentation Tools Preliminary Analysis & Interpretation of Data Correlation Model Regression Model Time Series Model Forecasting Uncertainty and Probability

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Quantitative Methods

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Course Coverage • Essential Basics for Business Executives • Data Classification & Presentation Tools • Preliminary Analysis & Interpretation of Data • Correlation Model • Regression Model • Time Series Model • Forecasting • Uncertainty and Probability • Sampling Techniques • Estimation and Testing of Hypothesis varsha Varde

Preliminary Analysis of Data Central Tendency of the Data at Hand: • Need to Size Up the Data At A Glance • Find A Single Number to Summarize the Huge Mass of Data Meaningfully: Average • Tools: Mode Median Arithmetic Mean Weighted Average varsha Varde

Mode, Median, and Mean • Mode: Most Frequently Occurring Score • Median: That Value of the Variable Above Which Exactly Half of the Observations Lie • Arithmetic Mean: Ratio of Sum of the Values of A Variable to the Total Number of Values • Mode by Mere Observation, Median needs Counting, Mean requires Computation varsha Varde

Example: Number of Sales Orders Booked by 50 Sales Execs April 2006 0, 0, 0, 0, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 12, 14, 15, 16, 17, 19, 21, 24, 28, 30, 34, 43 • Mode:9 (Occurs 5 Times) Orders • Median:8 (24 Obs. Above & 24 Below) Total Number of Sales Orders: 491 Total Number of Sales Execs : 50 • Arithmetic Mean: 491 / 50 = 9.82 Orders varsha Varde

This Group This Group of Participants: Mode of age is years Median is years, Arithmetic Mean is years varsha Varde

Arithmetic Mean - Example • Arithmetic Mean: 65 / 4 = 16.25 % • Query: But, Are All Products of Equal Importance to the Company? • For Instance, What Are the Sales Volumes of Each Product? Are They Identical? • If Not, Arithmetic Mean Can Mislead. varsha Varde

A Comparison • Mode: Easiest, At A Glance, Crude • Median: Disregards Magnitude of Obs., Only Counts Number of Observations • Arithmetic Mean: Outliers Vitiate It. • Weighted Av. Useful for Averaging Ratios • Symmetrical Distn: Mode=Median=Mean • +ly Skewed Distribution: Mode < Mean • -ly Skewed Distribution: Mode > Mean varsha Varde

Preliminary Analysis of Data Measure of Dispersion in the Data: • ‘Average’ is Insufficient to Summarize Huge Data Spread over a Wide Range • Need to Obtain another Number to Know How Widely the Numbers are Spread • Tools: Range & Mean Deviation Variance & Standard Deviation Coefficient of Variation varsha Varde

Range and Mean Deviation • Range: Difference Between the Smallest and the Largest Observation • Mean Deviation: Arithmetic Mean of the Deviations of the Observations from an Average, Usually the Mean. varsha Varde

Computing Mean Deviation • Select a Measure of Average, say, Mean. • Compute the Difference Between Each Value of the Variable and the Mean. • Multiply the Difference by the Concerned Frequency. • Sum Up the Products. • Divide by the Sum of All Frequencies. • Mean Deviation is the Weighted Average. varsha Varde

Mean Deviation • Sum of the Products: 318.12 • Sum of All Frequencies: 50 • Mean Deviation: 318.12 / 50 = 6.36 • Let Us Compute With a Simpler Example varsha Varde

Arithmetic Mean • Arithmetic Mean is the Average of the Observed Downtimes. • Arithmetic Mean= Total Observed Downtime/ total number of days • Arithmetic Mean= 2000 / 100 = 20 Minutes • Average Machine Downtime is 20 Minutes. varsha Varde

Mean Deviation • Definition: Mean Deviation is mean of Deviations (Disregard negative Sign) of the Observed Values from the Average. • In this Example, Mean Deviation is the Weighted Average(weights as frequencies) of the Deviations of the Observed Downtimes from the Average Downtime. • Mean Deviation = 1000 / 100 = 10 Minutes varsha Varde

Variance • Definition: Variance is the average of the Squares of the Deviations of the Observed Values from the mean. varsha Varde

Standard Deviation • Definition: Standard Deviation is the Average Amount by which the Values Differ from the Mean, Ignoring the Sign of Difference. • Formula: Positive Square Root of the Variance. varsha Varde

Variance & Standard Deviation • Variance = 14500 / 100 = 145 Mts Square • Standard Deviation = Sq. Root of 145 = 12.04 Minutes • Exercise: This Group of 65: Compute the Variance & Standard Deviation of age varsha Varde

Simpler Formula for Variance • Logical Definition: Variance is the Average of the Squares of the Deviations of the Observed Values from the mean. • Simpler Formula: Variance is the Mean of the Squares of Values Minus the Square of the Mean of Values.. varsha Varde

Variance (by Simpler Formula) • Mean of the Squares of Values = 54500/100 = 545 • Square of the Mean of Values=20x20=400 • Variance = Mean of Squares of Values Minus Square of Mean of Values = 545 – 400 = 145 • Standard Deviation = Sq.Root 145 = 12.04 varsha Varde

Significance of Std. Deviation In a Normal Frequency Distribution • 68 % of Values Lie in the Span of Mean Plus / Minus One Standard Deviation. • 95 % of Values Lie in the Span of Mean Plus / Minus Two Standard Deviation. • 99 % of Values Lie in the Span of Mean Plus / Minus Three Standard Deviation. Roughly Valid for Marginally Skewed Distns. varsha Varde

Interpretation from Mean & Std Dev Machine Downtime Data • Mean = 20 and Standard Deviation = 12 • Span of One Std. Dev. = 20–12 to 20+12 = 8 to 32: 60% Values • Span of Two Std. Dev. = 20–24 to 20+24 = -4 to 44: 95% Values • Span of Three Std. Dev. = 20–36 to 20+36 = -16 to 56: 100% Values varsha Varde

Interpretation from Mean & Std Dev Sales Orders Data • Mean = 9.82 & Standard Deviation = 6.36 • Round Off To: Mean 10 and Std. Dev 6 • Span of One Std. Dev. = 10–6 to 10+6 = 4 to 16: 31 Values (62%) • Span of Two Std. Dev. = 10–12 to 10+12 = -2 to 22: 45 Values (90%) • Span of Three Std. Dev. = 10–18 to 10+18 = -8 to 28: 47 Values (94%) varsha Varde

BIENAYME_CHEBYSHEV RULE • For any distribution percentage of observations lying within +/- k standard deviation of the mean is at least ( 1- 1/k square ) x100 for k>1 • For k=2, at least (1-1/4)100 =75% of observations are contained within 2 standard deviations of the mean varsha Varde

Coefficient of Variation • Std. Deviation and Dispersion have Units of Measurement. • To Compare Dispersion in Many Sets of Data (Absenteeism, Production, Profit), We Must Eliminate Unit of Measurement. • Otherwise it’s Apple vs. Orange vs. Mango • Coefficient of Variation is the Ratio of Standard Deviation to Arithmetic Mean. • CoV is Free of Unit of Measurement. varsha Varde

Coefficient of Variation • In Our Machine Downtime Example, Coefficient of Variation is 12.04 / 20 = 0.6 or 60% • In Our Sales Orders Example, Coefficient of Variation is 6.36 / 9.82 = 0.65 or 65% • The series for which CV is greater is said to be more variable or less consistent , less uniform, less stable or less homogeneous.

Coefficient of Variation • In Our Machine Downtime Example, Coefficient of Variation is 12.04 / 20 = 0.6 • In Our Sales Orders Example, Coefficient of Variation is 6.36 / 9.82 = 0.65 • The series for which CV is greater is said to be more variable or less consistent , less uniform, less stable or less homogeneous.

Example • Mean and SD of dividends on equity stocks of TOMCO & Tinplate for the past six years is as follows • Tomco:Mean=15.42%,SD=4.01% • Tinplate:Mean=13.83%, SD=3.19% • CV:Tomco=26.01%,Tinplate=23.01% • Since CV of dividend of Tinplates is less it implies that return on stocks of Tinplate is more stable • For investor seeking stable returns it is better to invest in scrips of Tinplate

Exercise • List Ratios Commonly used in Cricket. • Study Individual Scores of Indian Batsmen at the Last One Day Cricket Match. • Are they Nominal, Ordinal or Cardinal Numbers? Discrete or Continuous? • Find Median & Arithmetic Mean. • Compute Range, Mean Deviation, Variance, Standard Deviation & CoV. .. varsha Varde

Rule of thumb • Not too few to lose information content and not too many to lose pattern • The number of classes chosen is usually between 6 and15. • Subject to above the number of classes may be equal to the square root of the number of data points. • The more data one has the larger is the number of classes. varsha Varde

Rule of thumb • Every item of data should be included in one and only one class • Adjacent classes should not have interval in between • Classes should not overlap • Class intervals should be of the same width to the extent possible varsha Varde

Illustration Frequency and relative frequency distributions (Histograms): Example Weight Loss Data 20.5 19.5 15.6 24.1 9.9 15.4 12.7 5.4 17.0 28.6 16.9 7.8 23.3 11.8 18.4 13.4 14.3 19.2 9.2 16.8 8.8 22.1 20.8 12.6 15.9 • Objective: Provide a useful summary of the available information varsha Varde

Illustration • Method: Construct a statistical graph called a “histogram” (or frequency distribution) Weight Loss Data class boundaries - tally class rel . freq, f freq, f/n 1 5.0-9.0 3 3/25 (.12) 2 9.0-13.0 5 5/25 (.20) 3 13.0-17.0 7 7/25 (.28) 4 17.0-21.0 6 6/25 (.24) 5 21.0-25.0 3 3/25 (.12) 6 25.0-29.0 1 1/25 (.04) Totals 25 1.00 Let • k = # of classes • max = largest measurement • min = smallest measurement • n = sample size • w = class width varsha Varde

Formulas • k = Square Root of n • w =(max− min)/k • Square Root of 25 = 5. But we used k=6 • w = (28.6−5.4)/6 w = 4.0 varsha Varde