Introduction to Probability & Statistics Data Analysis Industrial Engineering

Data Analysis Histograms Industrial Engineering

Experimental Data Suppose we wish to make some estimates on time to fail for a new power supply. 40 units are randomly selected and tested to failure. Failure times are recorded follow:

Histogram Perhaps the most useful method, histograms give the analyst a feel for the distribution from which the data was obtained. Count observations within a set of ranges Average 5 observations per interval class

Histogram Perhaps the most useful method, histograms give the analyst a feel for the distribution from which the data was obtained. Count observations with a set of ranges Average 5 observations per interval class Range for power supply data: 0.5-73.8 Intervals: 0.0 - 10.0 40.1 - 50.0 10.1 - 20.0 50.1 - 60.0 20.1 - 30.0 60.1 - 70.0 30.1 - 40.0 70.1 - 80.0

Histogram Class Interval 0.0 - 10.0 Count = 15

Histogram Class Interval 10.1 - 20.0 Count = 11

Histogram Class Intervals Frequency 0.0 - 10.0 15 10.1 - 20.0 11 0.0 - 10.0 15 10.1 - 20.0 11 20.1 - 30.0 6 30.1 - 40.0 3 40.1 - 50.0 2 50.1 - 60.0 2 60.1 - 70.0 0 70.1 - 80.0 1

Exponential Distribution f x e ( )    Density Cumulative Mean 1/ Variance 1/2 F 1 , x > 0

Histogram; Change Interval Class Intervals Frequency 0.0 - 15.0 21 15.1 - 30.0 10 30.1 - 45.0 4 45.1 - 60.0 3 60.1 - 75.0 1

Histogram; Change Interval Class Intervals Frequency 0.0 - 5.0 8 5.1 - 10.0 7 10.1 - 15.0 6 15.1 - 20.0 4 20.1 - 25.0 3 25.1 - 30.0 3 30.1 - 35.0 2 35.1 - 40.0 1 40.1 - 45.0 1 45.1 - 50.0 1 50.1 - 55.0 1 55.1 - 60.0 1

Histogram; Change Class Mark Class Intervals Frequency -5.0 - 5.0 8 5.1 - 15.0 13 15.1 - 25.0 7 25.1 - 35.0 5 35.1 - 45.0 2 45.1 - 55.0 2 55.1 - 65.0 1 65.1 - 75.0 1

Class Problem The following data represents independent observations on deviations from the desired diameter of ball bearings produced on a new high speed machine.

Class Problem

Class Problem

Histogram Intervals & Class marks can alter the histogram too many intervals leaves too many voids too few intervals doesn’t give a good picture Rule of Thumb # Intervals = n/5 Sturges’ Rule k = [1 + log2n] = [1 + 3.322 log10n]

Class Problem The following represents demand for a particular inventory during a 70 day period. Construct a histogram and hypothesize a distribution.

Relative Histogram Class Freq Rel. 0.0 - 10.0 15 0.375 0.0 - 10.0 15 0.375 10.1 - 20.0 11 0.275 20.1 - 30.0 6 0.150 30.1 - 40.0 3 0.075 40.1 - 50.0 2 0.050 50.1 - 60.0 2 0.050 60.1 - 70.0 0 0.000 70.1 - 80.0 1 0.025

Relative Histogram