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 ProgramDepartment of Engineering Management, Information and Systems

2

• Basic Concepts

• Analysis of Location, or Central Tendency

• Analysis of Variability

• Analysis of Shape

3

Populationthe total of all possible values (measurement, counts, etc.) of a particular characteristic for aspecific group of objects.

Samplea 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

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

• Simple Random SampleThe sample X1, X2, ... ,Xn is a random sample if X1, X2, ... , Xn 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 SampleThe population is first subdivided into sub-populations for strata, and a simple randomsample is drawn from each strata

Types of Samples

6

•Censored Samples

Type I Censoring - Sample is terminated at a fixed time, t0. 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, tk.

Progressive Censoring - Sampling is reduced in stage.

Types of Samples - Continued

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 Kth item thereafter.

Types of Samples - Continued

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

• Sample (Arithmetic) Mean

• Sample Midrange

• Sample Mode

• Sample Median

• Sample Percentiles

Analysis of Location or Central Tendency

10

• Formula:

• Remarks:

Most frequently used statistic

Easy to understand

May be misleading due to extreme values

n

1iix

n

1x

Sample Mean

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

Formula: , if n is odd & K = (n+1)/2

, if n is even & K = n/2

where the sample values X1, X2, ... , 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

kx

2

xx 1kk 0.5x

Sample Median

13

• Sample Range

• Sample Variance

• Sample Standard Deviation

• Sample Coefficient of Variation

Analysis of Variability

14

• Formula: R = Xmax - Xmin

where Xmax is the largest value in the sampleand Xmin is the smallest sample value

• Remarks:

Easy to determine

Easily understood

Determined by extreme values

Does not use all sample data

Sample Range

15

• Sample Variance

• Sample Standard Deviation

s = (sample variance)1/2

• Remarks

Most frequently used measure of variabilityNot well understood

1nn

xxn

xx1n

1s

2n

1ii

n

1i

2i2n

1ii

2

Sample Variance & Standard Deviation

16

• Remarks

Relative measure of variation

Used for comparing the variation in two samples of data that are measured in two different units

x

sCVs

Sample Coefficient of Variation

17

• Skewness

• Kurtosis

Analysis of Shape

18

For a unimodal distribution, xr is an indicator ofdistribution shape

< 1 , indicates skewed to the left

xr = 1 , indicates symmetric

> 1 , indicates skewed to the right

5.0x

xxr

Estimate of Skewness

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 left3 = 0 , distribution is symmetric3 > 0 , distribution is skewed to the right

• Measure of skewness relative to degree of spread

33 XE

2/3231 )/( 2

2 xE

Measure of Skewness

20

• Normal

•Exponential

01

41

Comparison of Distribution Skewness

21

• Estimate of skewness of a distribution from a random sample

where

and

2/3231 )/(ˆ mm

2n

1ii2 xx

n

1m

3n

1ii3 xx

n

1m

n

1iix

n

1x

Estimation of Skewness

22

•The fourth moment about the mean is related to the peakedness, called kurtosis, of a distribution

• Relative measure of Kurtosis

where

44 xE

2242 /

22 xE

Measurement of Kurtosis

23

• Estimate of kurtosis of a distribution (2) from a random sample

where

and

22422

^

)/(mmb

2n

1ii2 xx

n

1m

4n

1ii4 xx

n

1m

n

1iix

n

1x

Estimation of Kurtosis

24

Comparison of Kurtosis

25

Presentation of Data

26

40 specimens are cut from a plate for tensile tests. The tensile tests were made, resulting in Tensile Strength, x, as follows:

i x i x i x i x1 48.5 11 55.0 21 53.1 31 54.62 54.7 12 55.7 22 49.1 32 49.93 47.8 13 49.9 23 55.6 33 44.54 56.9 14 54.8 24 46.2 34 52.95 54.8 15 49.7 25 52.0 35 54.46 57.9 16 58.9 26 56.6 36 60.27 44.9 17 52.7 27 52.9 37 50.28 53.0 18 57.8 28 52.2 38 57.49 54.7 19 46.8 29 54.1 39 54.8

10 46.7 20 49.2 30 42.3 40 61.2

Perform a statistical analysis of the tensile strength data.

40 Specimens

27

40 Specimens

The following descriptive statistics were calculated from the data:

Descriptive Statistics

Count 40Minimum 42.35Maximum 61.18Range 18.84Sum 2104.82Mean 52.62Median 53.03Sample Variance 19.83Standard Deviation 4.45Kurtosis 2.51Skewness -0.34