1. SESSION 17 & 18 Last Update 16 th March 2011 Measures of Dispersion Measures of Variability

2. Lecturer:Florian Boehlandt University:University of Stellenbosch Business School Domain:http://www.hedge-fund- analysis.net/pages/vega.php

3. Grouped Data – Investment B Intervalsxff(<)xfActual -25 to < -15-2022-40 -15 to < -5-1057-50 -5 to < 505120 5 to < 151041640 15 to < 2520622120 25 to < 353032590 Total25160 Total / 212.5 Mean6.4007.072 O me 5 f(<)12 f me 4 Median6.2504.700 O mo 15 fmfm 6 f m-1 4 f m+1 3 Mode19.000multimodal

4. Learning Objectives 1.Measures of relative standing: Median, Quartiles, Deciles and Percentiles 2.Measures of dispersion: Range 3.Measures of variability: Variance and Standard Deviation

5. Percentiles The Pth percentile is the value for which P percent are less than that value and (100 – p)% are greater than that value. Some special percentiles commonly used include the median and the quartiles. Percentiles are measures of relative standing.

6. Terminology 50 th Percentile25 th, 50 th, 75 th,100 th Percentile 20 th, 40 th,…, 100 th Percentile 10 th, 20 th,…, 100 th Percentile ½  1 Median¼  4 Quartiles 1/5  5 Quintiles 1/10  10 Deciles Q2Q2 Q 1, Q 2, Q 3,Q 4, LpLp

7. Location of a Percentile The location L of a percentile is a function of the required percentile P and the sample size n: L p = (n + 1) * (P / 100) As with the median, all observations must be placed in ascending or descending order first.

8. Calculation of Percentile 1.Place all observations in order 2.Calculate the location of the percentile 3.Since the location will often be a fraction (e.g. n/2), the distance between the two observations in question must be multiplied with the fractional part of the location 4.The result of 3. is added to the preceding observation to yield the percentile

9. Percentile: An example The following denotes the number of hours spent on the internet: 0 0 5 7 8 9 12 14 22 23 The values are already placed in order. The sample size is n = 10. We wish to determine L 25, L 50 and L 75 (this is analogous to the quartiles Q 1, Q 2 and Q 3 )

10. Solution – Step 1 Use the formula to calculate the location for each percentile / quartile ObsDataQuartileLp 10252.75=(10 + 1) * (25 / 100) 20505.50=( + 1) * (50 / 100) 35758.25=( + 1) * (75 / 100) 47n 5810 69 712 814 922 1023

11. Solution – Step 2 Determine the fractional part of the location ObsDataQuartileLpFraction 10252.750.75=2.75 - 2 20505.500.50=5.5 - 5 35758.250.25=8.25 - 8 47n 5810 69 712 814 922 1023

12. Solution – Step 3 ObsDataQuartileLpFractionLowerUpper 10252.750.7505 20505.50.5089 35758.250.251422 47n 5810 69 712 814 922 1023 Determine the next lower and next higher observation associated with the location. For 2.75, the two observations are 2  0 and 3  5.

13. Solution – Step 4 In order to determine the quartile associated with a given location, you need to calculate the following: Solution = Lower + (Upper – Lower) * Fraction ObsDataQuartileLpFractionLowerUpperSolution 10252.750.75053.75=0 + (5 - 0) * 0.75 20505.50.50898.50=8 + (9 - 8) * 0.5 35758.250.25142216.00=14 + (22 - 14) * 0.25 47n 5810 69 712 814 922 1023

14. Exercises You may use shortcuts if you want! 1.Determine the first, second and third quartiles: 5 8 2 9 5 3 7 4 2 7 4 10 4 3 5 2.Determine the third and eighth deciles (30 th and 80 th percentile): 10.5 14.7 15.3 17.7 15.9 12.2 10.0 14.1 13.9 18.5 13.9 15.1 15.7

15. Range The range is the difference between the minimum and maximum observation. It is a measure of dispersion. The interquartile range is the difference between the third and the first quartile: Interquartile Range = Q 3 – Q 1

16. Variance The variance expresses the sum of the squared deviation of every single observation from the sample / population mean. All differences are squared so that positive and negative deviations from the mean are not cancelled out. The variance in a measure of variability.

17. Population and Sample Variance We need to differentiate between population variance and sample variance. From the calculation of the mean, the sample variance has one less degrees of freedom (n-1) in calculating the variance. For the hypothetically infinite population of size N this is not the case.

18. Formulas SamplePopulation Sample size Total population size Observation Sample Mean Population Mean Sample StatisticPopulation Parameter

19. Calculation of Variance 1.Calculate the average: Sum of observations / number of observations 2.Subtract the average from every obervation 3.Square the difference 4.Sum the squared differences 5.Divide the result from 4. by either N (population) or n-1 (sample)

20. Variance: An example The following denotes the number of hours spent on the internet for a sample of n = 10 adults: 0 7 12 5 33 14 8 0 9 22 Calculate the variance.

21. Solution – Step 1 Use the mean to calculate the differences between the mean and every observation ObsDataDifference 10-8=(0 - 8) 27=(7 - 8) 3124=(12 - 8) 45-3=(5 - 8) 53-5=(3 - 8) 6146=(14 - 8) 780=(8 - 8) 80-8=(0 - 8) 991=(9 - 8) 102214=(22 - 8) Total80 n10 n-1 Average8

22. Solution – Step 2 Square all differences. Next, Sum the differences and divide the sum by n – 1 (sample only) ObsDataDifferenceSqr Diff 10-864=(-8)^2 271=(-1)^2 312416=(4)^2 45-39=(-3)^2 53-525=(-5)^2 614636=(6)^2 7800=(0)^2 80-864=(-8)^2 9911=(1)^2 102214196=(14)^2 Total80412 n10 n-19 Average845.778 In case of the sample, the sumsq is divided by n-1, in the case of the population it is divided by N

23. Interpretation Variance The variance may be difficult to interpret. Remember that all differences are squared to avoid positive and negative differences from cancelling out. The statistic may be standardized by taking the square root of the variance. This statistic is called the standard deviation. However, the variances from two datasets may still be referred to when determining the more volatile dataset.

24. Example – Standard Deviation The population standard deviation: Similarly, the sample standard deviation: Thus, for the internet usage example:

25. Solution – Step 3 ObsDataDifferenceSqr Diff 10-864 271 312416 45-39 53-525 614636 7800 80-864 9911 102214196 Total80412 n10 n-19 Average845.778 Sqrt6.766 Interpretation: On average, observations of internet usage within the sample of ten people deviates by 6.766 h from the sample mean.

26. Exercises 1.Calculate the variance and standard deviation for the following data: 2 8 9 4 1 7 5 4 2.Calculate the variance and standard deviation for the following data: 7 -5 -3 8 4 -4 1 -5 9 3