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Statistics

Statistics. Statistics. Branch of Mathematics that deals with the collection and analysis of data Descriptive Statistics: used to analyze and describe data

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Statistics

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  1. Statistics

  2. Statistics • Branch of Mathematics that deals with the collection and analysis of data • Descriptive Statistics: used to analyze and describe data • Inferential Statistics: used to use the information to make statements regarding the relationships between variables or the expectations about future events.

  3. Measures of Central Tendency

  4. Measures of Central Tendency • Arithmetic Mean • Median • Mode • Geometric Mean

  5. Arithmetic Mean • Other names • Average • Mean

  6. Arithmetic Mean • The calculation is identical, just the notation varies slightly

  7. Summation Notation • Notice that the first form uses less vertical space on the page • This makes accountants very happy • The first can also be easier to fit into a line of text

  8. Example • Ten second year BBA students wrote the CSC exam last month • Their scores were: 71, 72, 88, 69, 77, 63, 91, 81, 83, 75

  9. Calculating the Mean • Arithmetic mean • sum the observations and divide by the number of observations • Example: 5%, 7%, -2%, 12%, 8%

  10. Problem with the Arithmetic Mean • Arithmetic mean is incorrect for variables that are related multiplicatively, like rates of growth, rates of return and rates of change • $1,000 at 6% for 5 years should be $1,338.23

  11. Geometric Mean • The Geometric Mean should be used for rates of change, like rates of return

  12. Geometric Mean • The Geometric Mean should be used for rates of change, like rates of return Means: The product of these factors from 1 to N

  13. Geometric vs. Arithmetic Mean • The more variable the underlying data, the greater the error using the Arithmetic mean • The Geometric Mean is often easier to calculate: • Stock prices: 1992: $20; 1999: $40, R = 10.41%

  14. Geometric vs. Arithmetic Mean • For analysis of past performance, use the Geometric mean • The past returns have averaged 5.898% • To use the past returns to estimate the future expected return, use the Arithmetic mean • The expected return is 6%

  15. Median and Mode • Median: Midpoint • If odd number of observations: Middle observation • If even number of observations: Average of middle 2 observations • Mode: Most frequent

  16. Example • Our CSC mark data was (sorted): 63, 69, 71, 72, 75, 77, 81, 83, 88, 91 • The median is 76 • There is no mode

  17. Example • The Deviation is the difference between each observation and the mean • The sign indicates whether the observation is above (+) or below (-) the mean

  18. Example • The average deviation is always zero • If it isn’t, you must have made a mistake!

  19. Measures of Dispersion

  20. Measures of Dispersion • So far, we have look at measures of central tendency • What about measuring the tendency of the data to vary from these centre?

  21. Measures of Dispersion • Range • Highest - Lowest • Variance • Standard Deviation

  22. Example • The range is 91-63=28 • The range can be extremely sensitive to outlier observations • Suppose one of these students had a very bad day and scored 8. • The range would now be 91-8=83

  23. Mean Absolute Deviation • The Mean Absolute Deviation is a measure of average dispersion that is not used very much • It has some undesirable mathematical properties beyond the level of this course

  24. Mean Squared Deviation • The Mean Squared Deviation is very commonly used • The MSD in this example is 694/10=69.4 • The more common name of the MSD is the VARIANCE

  25. Variance • Variance measures the amount of dispersion from the mean. • For Populations: For Samples:

  26. Standard Deviation • Standard Deviation measures the amount of dispersion from the mean. • For Populations: For Samples:

  27. Standard Deviation Example • Using the previous example • The data is sample data

  28. Interpreting the Std. Dev. • You have heard of the Bell Shaped or Normal Distribution • The properties of the Normal Distribution are well known and give us the EMPIRICAL RULE

  29. Normal Distribution

  30. Empirical Rule For approximately Normally Distributed data: • Within 1s of the mean: approx.. 2/3s • Within 2s of the mean: approx. 95% (19/20) • Within 3s of the mean: virtually all

  31. Quartiles, Percentiles, etc. • The Median splits the data in half • Quartiles split the data into quarters • Deciles split the data into tenths • Percentiles split the data into one-hundredths

  32. Rank Measures • “That was a top-half performance” • “WTG Special fund has been a top quartile performer for the past 5 years” • “Our programme accepts only students proven to be top decile performers” • “I was in the 92nd percentile on the GMAT”

  33. Using Excel • Full Descriptive Statistics • Tools • Data Analysis • Descriptive Statistics

  34. Measures of Association

  35. Bivariate Statistics • So far, we have been dealing with statistics of individual variables • We also have statistics that relate pairs of variables

  36. Interactions Sometimes two variables appear related: • smoking and lung cancers • height and weight • years of education and income • engine size and gas mileage • GMAT scores and MBA GPA • house size and price

  37. Interactions • Some of these variables would appear to positively related & others negatively • If these were related, we would expect to be able to derive a linear relationship: y = a + bx • where, b is the slope, and • a is the intercept

  38. Linear Relationships • We will be deriving linear relationships from bivariate (two-variable) data • Our symbols will be:

  39. Example • Consider the following example comparing the returns of Consolidated Moose Pasture stock (CMP) and the TSE 300 Index • The next slide shows 25 monthly returns

  40. Example Data

  41. Example • From the data, it appears that a positive relationship may exist • Most of the time when the TSE is up, CMP is up • Likewise, when the TSE is down, CMP is down most of the time • Sometimes, they move in opposite directions • Let’s graph this data

  42. Graph Of Data

  43. Example Summary Statistics • The data do appear to be positively related • Let’s derive some summary statistics about these data:

  44. Observations • Both have means of zero and standard deviations just under 3 • However, each data point does not have simply one deviation from the mean, it deviates from both means • Consider Points A, B, C and D on the next graph

  45. Graph of Data

  46. Implications • When points in the upper right and lower left quadrants dominate, then the sums of the products of the deviations will be positive • When points in the lower right and upper left quadrants dominate, then the sums of the products of the deviations will be negative

  47. An Important Observation • The sums of the products of the deviations will give us the appropriate sign of the slope of our relationship

  48. Covariance(Showing the formula only to demonstrate a concept)

  49. Covariance

  50. Covariance • In the same units as Variance (if both variables are in the same unit), i.e. units squared • Very important element of measuring portfolio risk in finance

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