1. 1 Statistical Analysis – Descriptive Statistics Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Systems Engineering Program Department of Engineering Management, Information and Systems

2. 2 Basic Concepts Analysis of Location, or Central Tendency Analysis of Variability Analysis of Shape

3. 3 Population the total of all possible values (measurement, counts, etc.) of a particular characteristic for a specific group of objects. Sample a part of a population selected according to some rule or plan. Why sample? - Population does not exist - Sampling and testing is destructive Population vs. Sample

4. 4 Characteristics that distinguish one type of sample from another: the manner in which the sample was obtained the purpose for which the sample was obtained Sampling

5. 5 Simple Random Sample The sample X 1, X 2,...,X n is a random sample if X 1, X 2,..., X n are independent identically distributed random variables. Remark: Each value in the population has an equal and independent chance of being included in the sample. Stratified Random Sample The population is first subdivided into sub-populations for strata, and a simple random sample is drawn from each strata Types of Samples

6. 6 Censored Samples  Type I Censoring - Sample is terminated at a fixed time, t 0. The sample consists of K times to failure plus the information that n-k items survived the fixed time of truncation.  Type II Censoring - Sampling is terminated upon the Kth failure. The sample consists of K times to failure, plus information that n-k items survived the random time of truncation, t k.  Progressive Censoring - Sampling is reduced in stage. Types of Samples - Continued

7. 7 Systematic Random Sample The N items in the population are arranged in some order. Select an item at random from the first K = N/n items, where n is the sample size. Select every K th item thereafter. Types of Samples - Continued

8. 8 Data represents the entire population Statistical analysis is primarily descriptive. Data represents sample from population Statistical analysis - describes the sample - provides information about the population Statistical Analysis Objective

9. 9 Sample (Arithmetic) Mean Sample Midrange Sample Mode Sample Median Sample Percentiles Analysis of Location or Central Tendency

10. 10 Formula: Remarks: Most frequently used statistic Easy to understand May be misleading due to extreme values Sample Mean

11. 11 Definition: Most frequently occurring value in the sample Remarks: A sample may have more than one mode The mode may not be a central value Not well understood, nor frequently used Sample Mode

12. 12 Formula:, if n is odd & K = (n+1)/2, if n is even & K = n/2 where the sample values X 1, X 2,..., Xn are arranged in numerical order Remarks: Not well understood, nor accepted All sample data does not appear to be utilized Not affected by extreme values Sample Median

13. 13 Sample Range Sample Variance Sample Standard Deviation Sample Coefficient of Variation Analysis of Variability

14. 14 Formula: R = X max - X min where X max is the largest value in the sample and X min is the smallest sample value Remarks: Easy to determine Easily understood Determined by extreme values Does not use all sample data Sample Range

15. 15 Sample Variance Sample Standard Deviation s = (sample variance) 1/2 Remarks Most frequently used measure of variability Not well understood Sample Variance & Standard Deviation

16. 16 Remarks Relative measure of variation Used for comparing the variation in two samples of data that are measured in two different units Sample Coefficient of Variation

17. 17 Skewness Kurtosis Analysis of Shape

18. 18 For a unimodal distribution, x r is an indicator of distribution shape < 1, indicates skewed to the left x r = 1, indicates symmetric > 1, indicates skewed to the right Estimate of Skewness

19. 19 The third moment about the mean is related to the asymmetry or skewness of a distribution For a unimodal (i.e., a single peaked) distribution  3 < 0, distribution is skewed to the left  3 = 0, distribution is symmetric  3 > 0, distribution is skewed to the right Measure of skewness relative to degree of spread Measure of Skewness

20. 20 Normal Exponential Comparison of Distribution Skewness

21. 21 Estimate of skewness of a distribution from a random sample where and Estimation of Skewness

22. 22 The fourth moment about the mean is related to the peakedness, called kurtosis, of a distribution Relative measure of Kurtosis where Measurement of Kurtosis

23. 23 Estimate of kurtosis of a distribution (  2 ) from a random sample where and Estimation of Kurtosis

24. 24 Comparison of Kurtosis

25. 25 Presentation of Data

26. 26 40 specimens are cut from a plate for tensile tests. The tensile tests were made, resulting in Tensile Strength, x, as follows: Perform a statistical analysis of the tensile strength data. 40 Specimens

27. 27 40 Specimens The following descriptive statistics were calculated from the data: