1. 1 Descriptive Statistics Chapter 3 MSIS 111 Prof. Nick Dedeke

2. 2 Objectives Define measures of central tendency, variability, shape and association Define statistical measures Compute statistical measures for ungrouped and grouped data Interpret statistical results

3. 3 Introduction In most competitive sports, one looks for the position of the athletes, e.g. who came in first, second, and so on. In statistics, one is interested in the following measures: - most frequent value in data set - summary of all values in data set - midpoint position of data set - positions of data in data set - distances to midpoint of data set

4. 4 Exercise: Statistical Measure 1 We want to find out which of the following students is the better one using the available data. Kuli1 st 2 nd 1 st 2 nd 1 st 4 th 3 rd 3 rd 2 nd 5 th 1 st Marti 3 rd 2 nd 3 rd 1 st 2 nd 1 st 1 st 1 st 3 rd 2 nd 3 rd

5. 5 Using Statistical Measures Kuli 1 st 2 nd 1 st 2 nd 1 st 4 th 3 rd 3 rd 2 nd 5 th 1 st Marti 3 rd 2 nd 3 rd 2 nd 2 nd 1 st 1 st 1 st 3 rd 2 nd 1 st Mode: Most frequently occurring value of variable Mode for Kuli: Mode for Marti: Mean: Average of the values of a variable Sample mean =  X i n Mean or average score for Kuli Mean or average score for Marti

6. 6 Using Frequency Distributions Analysis of Kuli’s performance Mean =  F i * X i  F i = Mode = Median point = (11+ 1)/2 = 6 th Median value = 2 nd Using cumul. Freq. column = 2 nd XiXi Frequency (F i ) F i * X i Cum. (C F i ) 1 st 444 2 nd 367 3 rd 269 4 th 1410 5 th 1511  25

7. 7 Using Frequency Distributions Analysis of Marti’s performance Mean =  F i * X i  F i = Mode = Median point = (11+ 1)/2 = 6 th Median value = 2 nd Using cumul. Freq. column = 2 nd XiXi Frequency (F i ) F i * X i Cum. (C F i ) 1 st 444 2 nd 488 3 rd 3911 4 th 000 5 th 000  1121

8. 8 New Case: Median measure Analysis of Katie’s performance Mean =  F i * X i  F i = Mode = Median point = (12+ 1)/2 = 6.5 th Median value =(2 nd +3 rd )/2 = 2.5 th Average of the 6 th and 7 th positions. XiXi Frequency (F i ) F i * X i Cum. (C F i ) 1 st 444 2 nd 286 3 rd 51511 4 th 1412  31

9. 9 Percentiles Sometimes we are not analyzing several values from one person, but one value for several persons or objects. For example we have data from the performance of several fund manager’s for year 2006. We want to present the data in the form, XX manager is in the top 10 or tenth percentile or top 25 or 25 th percentile. The method used consists of three steps - organize data in ascending order - calculate location of percentile you want - identify the object in the percentile location from the data set

10. 10 Interpretation: Percentiles If manager is in the tenth percentile of of a group, this means that 90 % of everyone in the data set scored better than the manager. If manager is in the 95 th percentile of of a group, this means that 5 % of everyone in the data set scored higher or better than the manager.

11. 11 Exercise: Percentiles for Known Values First name Fund performance Bill106% Jane109% Sven114% Larry116% Dub121% Anna122% Cole125% Salome129% In which percentile is Sven?

12. 12 Response: Percentiles for Known Values First name Fund performance Bill106% Jane109% Sven114% Larry116% Dub121% Anna122% Cole125% Salome129% In which percentile is Sven? Fi Rel. fi 11/8 1 1 1 1 1 1 1 N=8 Cum fi Percentiles 1/8 12.5 th Percentile 2/8 25 th Percentile 3/8 37.5 th Percentile 4/8 50 th Percentile 5/8 62.5 th Percentile 6/8 75 th Percentile 7/8 87.5 th Percentile 1 100 th Percentile

13. 13 Example: Percentiles for UnKnown Values First name Fund performance Bill106% Jane109% Sven114% Larry116% Dub121% Anna122% Cole125% Salome129% What is the value of the 90 th percentile? Fi Rel. fi 11/8 1 1 1 1 1 1 1 N=8 Cum fi Percentiles 1/8 12.5 th Percentile 2/8 25 th Percentile 3/8 37.5 th Percentile 4/8 50 th Percentile 5/8 62.5 th Percentile 6/8 75 th Percentile 7/8 87.5 th Percentile 1 100 th Percentile

14. 14 Computing Percentile locations 90 th percentile location i = (P/100) * N = 0.9 * 8 = 7.2 th position 90 th percentile is 0.2 or 20% between the 7 th and 8 th The value for the 90 th percentile is computed by averaging the following values = 7 th position’s value + (8 th position’s value - 7 th position value)* Fraction got from computing i 125% + (129% - 125%)*0.2 = 125.8% (~ 126%) 50 th percentile location i = (P/100) * N = 0.5 * 8 = 4 th position

15. 15 Computing Central Tend. Measures Mean=  F i *X i  F i = 1655/15 =110.33 XiXi FiFi F i * X i 552110 601 1003300 1255625 1404560  151655

16. 16 Computing Dispersion Measures Mean (μ) =  F i *X i  F i =1655/15 =110.33 XiXi FiFi F i * X i 552110 601 1003300 1255625 1404560  151655 (X i - μ)(X i - μ) 2 F i * (X i - μ) 2 -55.333061.4096122.818 -50.332533.109 -10.33106.709320.127 14.67215.2091076.045 29.67880.3093521.236 13573.335 Variance (s 2 ) =  F i * (X i - μ) 2 (n –1) =13573.335/(15 –1) =969.52 Standard deviation (s) = 31.137

17. 17 Computing Dispersion Measures 2 XiXi FiFi F i * X i 552110 601 1003300 1255625 1404560  151655 (X i ) 2 F i *(X i ) 2 30256050 3600 1000030000 1562578125 1960078400 196175 Var (s 2 ) =  F i * X i 2 -  (F i *X i ) 2 /n (n –1) = 196175 – (1655 2 /15)/(15 –1) =(196175 – 182601.66)/14 = = 969.52 Standard deviation (s) = 31.137

18. 18 Exercise: Dispersion Measures XiXi FiFi F i * X i 52 61 103 122 141  (X i ) 2 F i *(X i ) 2 Var (s 2 ) =  F i * X i 2 -  (F i *X i ) 2 /n (n –1) Standard deviation (s) =

19. 19 Excel Examples

20. 20 Grouped Data Examples Class intervalFreq (F i ) M [1 – 3) inch162 inches [3 – 5) inch24 inches [5 – 7) inch46 inches [7 – 9) inch38 inches [9 – 11) inch910 inches [11 – 13) inch612 inches  40 F i * MF i * M 2 32 inches64 inches 8 inches32 inches 24 inches144 inches 24 inches192 inches 90 inches900 inches 72 inches864 inches 2502,196 Var (s 2 ) =  F i * M i 2 -  (F i *M i ) 2 /n = 2196 – 1562.5 = 16.24 (n –1) 39 Standard deviation (s) = 4.03 inches

21. 21 Grouped Data Exercise Class intervalFreq (F i ) M [1 – 4) inches4 [4 – 8) inches4 [8 – 12) inches6 [12 – 16) inches12 [16 – 20) inches8 [20 – 24) inches6  40 F i * MF i * M 2 Var (s 2 ) =  F i * M i 2 -  (F i *M i ) 2 /n = (n –1) Standard deviation (s) =