Quantitative Techniques – Class I

1. Quantitative Techniques – Class I Making Data Simple

2. What is Statistics? • Study of the collection, organization, analysis, interpretation and presentation of data • Theoretical statistics – the mathematical basis of the process of statistical analysis. Most research and work on finding new tools, fixing errors in existing tools and improving them • Applied Statistics – using the existing tools of statistics on certain data to improve our understanding of the problem in hand • What type of assumptions? • What is Normal? • Simple concept like average (mean) and standard deviation

3. Concepts • Population and Sample • Mean • Median • Standard Deviation • Risk Adjusted Returns • Basic Probability • Sample Selection • Measurement Bias • Spreadsheets and Relational Data • Charting your Data – Pie Charts, Bar Graphs, Histograms

4. Types of Data • Nominal Data – Groups • Ordinal Data – Has some meaningful order • Interval Data – Ordinal data, but same intervals • Ratio Data • Data can be continuous or discrete

5. Descriptive Statistics • Mean • Median • Mode • Percentile • Interquartile Range • Standard Deviation

6. Charts • Bar Charts – Perfect for Discrete Data with Only few categories • Stacked Charts – When Comparing Similar Bars, over time • Pie Chart – When the proportions are important, not the actual values • Box Plot – Median, shown with the Interquartile Range, and Extended Line showing the Minimum and Maximum Values (Range) • Histograms – Display continuous data, similar to par charts, but can be used for more categories, since it shows a trend • Scatterplots – Show the relation between two variables • Line Graphs

7. Probability - Definitions • Sample Space – Like population, the entire range of values possible • Event – The actual realization of the values • Union – The likelihood of either of multiple events occurring • Intersection – The likelihood of both events occurring • Complement – Everything in the sample that is not occuring • Mutual Exclusivity – If one event occurs, then the other cannot • Independence – When the events are not related to each other – that is, the probability of one, does not affect the other • Permutations – The number of ways to arrange some objects • Combinations – Permutation, when order is not important

8. Rules (Axioms) of Probability • An “event” E will occur or not occur • P(E) is a number that equals the probability that E will occur. • By convention, 0 < P(E) < 1. • E' = the event that E does not occur • P(E') = the probability that E does not occur.

9. Essential Results for Probability • If P(E) = 0, then E cannot (will not) occur • If P(E) = 1, then E must (will) occur • E and E' are exhaustive – either E or E' will occur. • Something will occur, P(E) + P(E') = 1 • Only one thing can occur. If E occurs, then E' will not occur – E and E' are exclusive.

10. Joint Events • Pairs (or groups) of events: A and B One or the other occurs: A or B ≡ A  B Both events occur A and B ≡ A  B • Independent events: Occurrence of A does not affect the probability of B • An addition rule: P(A  B) = P(A)+P(B)-P(A  B) • The product rule for independent events: P(A  B) = P(A)P(B)

11. Independent Events • Events are independent if the occurrence of one does not affect probabilities related to the other. • Events A and B are independent if P(A|B) = P(A). I.e., conditioning on B does not affect the probability of A.

12. Expected Value • Toss a coin • If you get head, you make Rs. 10 • If you get tail, you make Rs. 2 • What is your expected value? • Remember, probability of Head = 0.5 (50%) • Probability of Tail = 0.5 (50%) • EV = 0.5 x 10 + 0.5 x 2 = Rs. 6 • If someone says that you can take this bet for Rs. 5 then you should always take it