Download presentation
Presentation is loading. Please wait.
Published byWalter Ball Modified over 9 years ago
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:
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.