1. Random Variable Probability Distribution X=Amt in next bottle X=total of 2 tossed dice X=#G in 4 N(μ=10.2,σ=0.16) B(n=4,p=.5) 2 3 4 5 6 7 8 9 10 11 12 Summary Characteristics Mean Median Mode Std dev Variance skew We must admit that we cannot know exactly what value X will take…. …so that we can do the intelligent thing and talk about something we CAN know, the probability distribution of X. There are summary characteristics of any probability distribution… But knowing these summary measures do not replace our need to know the probability distribution.

2. Class 06: Descriptive Statistics EMBS: 3.1, 3.2, first part of 3.3

3. Characteristics of probability distributions Measures of Location – Mean – Median – Mode Measures of Variability – Standard Deviation – Variance Measure of Shape – skewness Descriptive Statistics (for numerical data) Measures of Location – Sample Mean – Sample Median – Sample Mode Measures of Variability – Sample StDev – Sample Variance Measure of Shape – Sample skewness

4. A positively-skewed pdf Mode is the most likely value P(X median) = 0.5 Mean is the probability- weighted average Skewness > 0

5. http://dept.econ.yorku.ca/~jbsmith/ec2500_1998/lecture9/Lecture9.html

6. A negatively-skewed pdf Skewness < 0

7. An exhibit at MOMA invites visitors to mark their heights on a wall. A normal distribution results:exhibit Well, not quite. The distribution is actually slightly negatively skewed by the confounding presence of children, who are obviously shorter than adults - you can see this in the great number of names well below the central band which are not mirrored by names higher up. Rest assured, however, that the ex- children distribution is itself Gaussian. http://www.thisisthegreenroom.com/2009/bell-curves-in-action/

8. The Normal pdf http://www.comfsm.fm/~dleeling/statistics/no tes06.html Mean = μ median = μ mode = μ Skewness = 0

9. Measures of Variability http://www.google.com/imgres?q=standard+deviation+curve&hl=en&gbv=2&bi w=1226&bih=866&tbm=isch&tbnid=pppxDi8aC37y8M:&imgrefurl=http://www. comfsm.fm/~dleeling/statistics/notes06.html&docid=Hu1RM- siu0MevM&imgurl=http://www.comfsm.fm/~dleeling/statistics/normal_curve_d iff_sx.gif&w=401&h=322&ei=9qAqT8KXAcPptgfC3uX0Dw&zoom=1&iact=hc&vpx =748&vpy=508&dur=1013&hovh=201&hovw=251&tx=142&ty=111&sig=106136 691078404837864&page=1&tbnh=149&tbnw=186&start=0&ndsp=20&ved=1t:4 29,r:13,s:0 σ = 0.7 σ = 1.0 σ = 1.5

10. Skewed pdfs can also have different standard deviations Which pdf has the largest σ?

11. Pdfs Can have different means, but identical standard deviations Which pdf has the largest σ? Which pdf has the largest μ?

13. Characteristics of probability distributions Measures of Location – Mean – Median – Mode Measures of Variability – Standard Deviation – Variance Measure of Shape – skewness Descriptive Statistics (for numerical data) Measures of Location – Sample Mean – Sample Median – Sample Mode Measures of Variability – Sample StDev – Sample Variance Measure of Shape – Sample skewness Probability weighted average 50% point Most likely Expected squared distance from mean Neg if skewed left, 0 if symmetric, pos if skewed right.

14. Characteristics of probability distributions Measures of Location – Mean – Median – Mode Measures of Variability – Standard Deviation – Variance Measure of Shape – skewness Descriptive Statistics (for numerical data) Measures of Location – Sample Mean – Sample Median – Sample Mode Measures of Variability – Sample StDev – Sample Variance Measure of Shape – Sample skewness Probability weighted average 50% point Most likely Expected squared distance from mean Neg if skewed left, 0 if symmetric, pos if skewed right. =average() =median() =mode() =stdev() =var() =skew()

15. Characteristics of probability distributions Measures of Location – Mean – Median – Mode Measures of Variability – Standard Deviation – Variance Measure of Shape – skewness Descriptive Statistics (for numerical data) Measures of Location – Sample Mean – Sample Median – Sample Mode Measures of Variability – Sample StDev – Sample Variance Measure of Shape – Sample skewness Probability weighted average 50% point Most likely Expected squared distance from mean Neg if skewed left, 0 if symmetric, pos if skewed right. GET THEM ALL USING DATA ANALYSIS, DESCRIPTIVE STATISTICS, SUMMARY STATISTCS

16. Characteristics of probability distributions Measures of Location – Mean – Median – Mode Measures of Variability – Standard Deviation – Variance Measure of Shape – skewness Descriptive Statistics (for numerical data) Measures of Location – Sample Mean – Sample Median – Sample Mode Measures of Variability – Sample StDev – Sample Variance Measure of Shape – Sample skewness Probability weighted average 50% point Most likely Expected squared distance from mean Neg if skewed left, 0 if symmetric, pos if skewed right. RANGE COUNT

17. The sample standard deviation

18. Understanding sample standard deviation 010 414 1020 1626 2030 stdev8.25 000 468 10 161412 20 stdev8.257.627.21 00 210 416 1018 20 stdev8.07 It measures variability about the mean. All the data contribute to the measure. It measures variability …. In either direction. XXXXX XXXXX 01234567891011121314151617181920

19. Our Data Section ND IDGender (M=1)HS Stat? HtValue 4 90152645300670 4 90153356101630.06 4 90153607510700.................. 5 90163639910760 5 90164391510720.1 5 90164399500640

20. Data/DataAnalysis/DescriptiveStatistics SummaryStatistics Section ND ID Gender (M=1) HS Stat? Ht Value Mean4.493Mean901589800.3Mean0.609Mean0.217Mean69.351Mean0.185 Standard Error0.061 Standard Error4992.821 Standard Error0.059 Standard Error0.050 Standard Error0.477 Standard Error0.056 Median4 901596170Median1 0 70Median0 Mode4 #N/AMode1 0 71Mode0 Standard Deviation0.504 Standard Deviation41473.487 Standard Deviation0.492 Standard Deviation0.415 Standard Deviation3.959 Standard Deviation0.465 Sample Variance0.254 Sample Variance1720050147 Sample Variance0.242 Sample Variance0.173 Sample Variance15.673 Sample Variance0.216 Kurtosis-2.060Kurtosis0.555Kurtosis-1.847Kurtosis-0.039Kurtosis-0.793Kurtosis6.385 Skewness0.030Skewness-0.581Skewness-0.455Skewness1.401Skewness-0.307Skewness2.706 Range1 228090Range1 1 16Range2 Minimum4 901444465Minimum0 0 60Minimum0 Maximum5 901672555Maximum1 1 76Maximum2 Sum310Sum62209696222Sum42Sum15Sum4785.25Sum12.75 Count69Count69Count69Count69Count69Count69

21. Data/DataAnalysis/DescriptiveStatistics SummaryStatistics Section ND ID Mean4.493Mean901589800.3 Standard Error0.061Standard Error4992.821 Median4 901596170 Mode4 #N/A Standard Deviation0.504 Standard Deviation41473.487 Sample Variance0.254Sample Variance1720050147 Kurtosis-2.060Kurtosis0.555 Skewness0.030Skewness-0.581 Range1 228090 Minimum4 901444465 Maximum5 901672555 Sum310Sum62209696222 Count69Count69

22. Data/DataAnalysis/DescriptiveStatistics SummaryStatistics Gender (M=1) HS Stat? Mean0.609Mean0.217 Standard Error0.059Standard Error0.050 Median1 0 Mode1 0 Standard Deviation0.492Standard Deviation0.415 Sample Variance0.242Sample Variance0.173 Kurtosis-1.847Kurtosis-0.039 Skewness-0.455Skewness1.401 Range1 1 Minimum0 0 Maximum1 1 Sum42Sum15 Count69Count69

23. Data/DataAnalysis/DescriptiveStatistics SummaryStatistics Ht Value Mean69.351Mean0.185 Standard Error0.477Standard Error0.056 Median70Median0 Mode71Mode0 Standard Deviation3.959Standard Deviation0.465 Sample Variance15.673Sample Variance0.216 Kurtosis-0.793Kurtosis6.385 Skewness-0.307Skewness2.706 Range16Range2 Minimum60Minimum0 Maximum76Maximum2 Sum4785.25Sum12.75 Count69Count69

24. Fill Test Data Normal(10.2,0.16)? EXHIBIT 2 LOREX PHARMACEUTICALS Filling Line Test Results with Target = 10.2 9.8910.4110.5310.2010.2310.15 10.17 10.3210.0410.4810.11 10.2910.3510.16 10.1710.19 10.0010.0610.2110.229.7610.22 10.0410.1910.0910.1210.0610.10 10.3510.1710.0210.3610.179.99 10.0510.0710.3210.2410.0410.40 10.1910.2710.1410.0710.4110.76 10.2110.1310.1110.4010.2710.20 9.7910.2410.2010.2910.0010.31 10.5310.1410.3510.2110.2310.16 10.479.849.9610.1010.1110.23 10.2410.3610.3010.2310.1910.17 10.1110.3310.199.9710.00 10.1510.4210.3610.1910.0510.11 10.0610.1610.1710.2910.1210.30 10.1310.2110.1510.2510.3310.64 10.0410.0110.1410.18 10.10 10.2010.2510.0710.4210.5410.23 10.3710.4410.379.859.9110.45 10.2410.4410.4010.4510.2810.17 10.0310.4410.2510.3710.2310.19 10.0110.1310.2410.229.98 10.2010.2910.0310.199.9910.13

25. Fill Test Data Descriptive Statistics Summary Statistics Amount Mean10.198 Standard Error0.014 Median10.190 Mode#N/A Standard Deviation0.163 Sample Variance0.026 Kurtosis0.771 Skewness0.245 Range0.997 Minimum9.758 Maximum10.756 Sum1468.542 Count144

26. Fill Test Data Histogram BinFrequency 9.7581 9.8412 9.9253 10.00810 10.09117 10.17433 10.25736 10.34014 10.42316 10.5067 10.5903 10.6731 More1 1 data point was < 9.758 2 data points were between 9.758 and 9.841 1 was above 10.673 Data Data Analysis Histogram Check chart output

27. Preview of Coming Attractions Class 07 – Find out how to use these counts to test H0: these data came from N(10.2,.16) – Find out how to use the Denmark family counts to test H0: those data came from Binomial(4,.5)