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Lesson 2:

Lesson 2:. Descriptive Statistics. Outline. Population Parameters and Sample Statistics. A population parameter is number calculated from all the population measurements that describes some aspect of the population.

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Lesson 2:

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  1. Lesson 2: Descriptive Statistics

  2. Outline

  3. Population Parameters and Sample Statistics • A population parameter is number calculated from all the population measurements that describes some aspect of the population. • The population mean, denoted , is a population parameter and is the average of the population measurements. • A point estimate is a one-number estimate of the value of a population parameter. • A sample statistic is number calculated using sample measurements that describes some aspect of the sample.

  4. Sample x1, x2, …, xn Sample Mean The Mean Population X1, X2, …, XN m Population Mean

  5. Population Mean • For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values: where µ is the population mean. • N is the total number of observations. • Xis a particular value. •  indicates the operation of adding.

  6. Sample Mean • For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values: Where n is the total number of values in the sample. This sample mean is also referred as arithmetic mean, simple mean, or simply sample average.

  7. EXAMPLE • A sample of five executives received the following bonus last year ($000): 14.0, 15.0, 17.0, 16.0, 15.0

  8. Sample x1, x2, …, xn Sample Proportion xi = 1 if characteristic present, 0 if not Population and Sample Proportions Population X1, X2, …, XN p Population Proportion

  9. EXAMPLE • A sample of five executives received the following bonus last year ($000): 7.0, 15.0, 17.0, 16.0, 15.0 • Changing the first observation from 14.0 to 7.0 will change the sample mean.

  10. Weighted Mean • The weighted meanof a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula:

  11. EXAMPLE • During a one hour period on a hot Saturday afternoon cabana boy Chris served fifty drinks. He sold five drinks for $0.50, fifteen for $0.75, fifteen for $0.90, and fifteen for $1.10. Compute the weighted mean of the price of the drinks sold.

  12. The Median • The Medianis themidpoint of the values after they have been ordered from the smallest to the largest. • There are as many values above the median as below it in the data array. • For an even set of values, the median will be the arithmetic average of the two middle numbers.

  13. EXAMPLE • The ages for a sample of five college students are: 21, 25, 19, 20, 22 • Arranging the data in ascending order gives: 19, 20, 21, 22, 25. Thus the median is 21. • The heights of four basketball players, in inches, are: 76, 73, 80, 75 • Arranging the data in ascending order gives: 73, 75, 76, 80. Thus the median is 75.5

  14. The Mode • The mode is the value of the observation that appears most frequently. • EXAMPLE: The exam scores for ten students are: 81, 93, 84, 75, 68, 87, 81, 75, 81, 87. Because the score of 81 occurs the most often, it is the mode.

  15. Properties of Mean and Median

  16. Measures of dispersion • Range • Mean Deviation • Variance and standard deviation • Coefficient of variation

  17. Range • The range is the difference between the largest and the smallest value. • Only two values are used in its calculation. • It is influenced by an extreme value. • It is easy to compute and understand.

  18. Mean Deviation • TheMean Deviationis the arithmetic mean of the absolute values of the deviations from the arithmetic mean. • All values are used in the calculation. • It is not influenced too much by large or small values. • The absolute values are difficult to manipulate. Mean deviation is also known as Mean Absolute Deviation (MAD).

  19. EXAMPLE: Range and Mean Deviation • The weights of a sample of crates containing books for the bookstore (in pounds ) are: 103, 97, 101, 106, 103 Find the range and the mean deviation. Range = 106 – 97 = 9

  20. EXAMPLE: Range and Mean Deviation The first step is to find the mean weight. The mean deviation is:

  21. Population Variance • The population variance is the arithmetic mean of the squared deviations from the population mean. • All values are used in the calculation. • More likely to be influenced by extreme values than mean deviation. • The units are awkward, the square of the original units.

  22. Sample x1, x2, …, xn s2 Sample Variance The Variance Population X1, X2, …, XN s2 Population Variance Note in the sample variance formula the sum of deviation is divided by (n-1) instead of n in order to yield an unbiased estimator of the population variance.

  23. EXAMPLE: Population variance • The ages of the Dunn family are: 2, 18, 34, 42 What is the population variance?

  24. EXAMPLE: Population Standard Deviation • The population standard deviation (σ) is the square root of the population variance. • In the last example, the population variance is 236. Hence, the population standard deviation is 15.36, found by

  25. EXAMPLE: Sample variance The hourly wages earned by a sample of five students are: $7, $5, $11, $8, $6. Find the variance.

  26. EXAMPLE: Sample Standard Deviation • The sample standard deviation is the square root of the sample variance. • In the last example, the sample variance is 5.29. Hence, the sample standard deviation is 2.30

  27. Sample Variance For Grouped Data • The formula for the sample variance for grouped data is:

  28. EXAMPLE: Sample Variance For Grouped Data • During a one hour period on a hot Saturday afternoon cabana boy Chris served fifty drinks. He sold five drinks for $0.50, fifteen for $0.75, fifteen for $0.90, and fifteen for $1.10. Compute the variance of the price of the drinks.

  29. Interpretation and Uses of the Standard Deviation • Chebyshev’s theorem: For any set of observations, the minimum proportion of the values that lie within k standard deviations of the mean is at least: where k is any constant greater than 1.

  30. Chebyshev’s theorem Chebyshev’s theorem:For any set of observations, the minimum proportion of the values that lie within k standard deviations of the mean is at least 1- 1/k2

  31. Interpretation and Uses of the Standard Deviation • Empirical Rule: For any symmetrical, bell-shaped distribution: • About 68% of the observations will lie within 1s the mean, • About 95% of the observations will lie within 2s of the mean • Virtually all the observations will be within 3s of the mean Empirical rule is also known as normal rule.

  32. Bell-shaped Curve showing the relationship between σ and μ m-2s m-1s m m+1s m+2s m+ 3s m-3s

  33. Why are we concern about dispersion? • Dispersion is used as a measure of risk. • Consider two assets of the same expected (mean) returns. • -2%, 0%,+2% • -4%, 0%,+4% • The dispersion of returns of the second asset is larger then the first. Thus, the second asset is more risky. • Thus, the knowledge of dispersion is essential for investment decision. And so is the knowledge of expected (mean) returns.

  34. Relative Dispersion • The coefficient of variation is the ratio of the standard deviation to the arithmetic mean, expressed as a percentage:

  35. Sharpe Ratio and Relative Dispersion • Sharpe Ratio is often used to measure the performance of investment strategies, with an adjustment for risk. • If X is the return of an investment strategy in excess of the market portfolio, the inverse of the CV is the Sharpe Ratio. • An investment strategy of a higher Sharpe Ratio is preferred. http://www.stanford.edu/~wfsharpe/art/sr/sr.htm

  36. Skewness • Skewness is the measurement of the lack of symmetry of the distribution. • The coefficient of skewness can range from 3.00 up to 3.00. • A value of 0 indicates a symmetric distribution. • It is computed as follows: Or

  37. Why are we concerned about skewness? • Skewness measures the degree of asymmetry in risk. • Upside risk • Downside risk • Consider the distribution of asset returns: • Right skewed implies higher upside risk than downside risk. • Left skewed implies higher downside risk than upside risk.

  38. Symmetric Distribution zero skewness: mode =median= mean Density Distribution (the height may be interpreted as relative frequency) The area under the density distribution is 1. The sum of relative frequency is 1. Thus median always splits the density distribution into two equal areas.

  39. Right Skewed Distribution Positively skewed: (Skew to the right) Mean and Median are to the right of the Mode. Mode<Median<Mean

  40. Left Skewed Distribution Negatively Skewed: (skew to the left) Mean and Median are to the left of the Mode. Mean<Median<Mode

  41. Working with mean and Standard Deviation

  42. Working with mean and Standard Deviation • (2) = (1) – mean(1): • Mean(2)=0; Stdev(2)=Stdev(1) • (3) = (1) + mean(1) • Mean(3)=Mean(1); Stdev(3)<Stdev(1). • (4) = (1)*2; (5) = (1)*3 • Mean(4)=mean(1)*2; mean(5)=mean(1)*3 • Stdev(4)=stdev(1)*2; stdev(5)=stdev(1)*3

  43. Working with mean and Standard Deviation • (6)=(1) multiplied by some frequency • Mean(6)=Mean(1); Stdev(6)=Stdev(1). • (9) = (7)+(8) • Mean(9)=mean(7)+mean(8) • (10) = (7) *(8) • Mean(10)=mean(7)*mean(8)

  44. Lesson 2: Descriptive Statistics - END -

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