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Math in the Workplace: Overview

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Math in the Work Place

Math in the Workplace - Algebra/Analysis and Probability

MICRON TECHNOLOGY, INC.,

Semiconductor Manufacturing
Quality Control

 

 Job Description: Develop wafer level test strategies and test programs. Provide failure analysis reporting. Monitor device yields, failure rates, and repair rates. Interact with various engineering and product groups to optimize device yields and minimize costs.
   


Problem:

To create a method control chart for quality control, the following numbers were collected while monitoring a fabrication process.

What is the mean of the following set of numbers?

What is the standard deviation (STD)?

3.05
3.02
3.03
2.97
2.98
3.10
2.94
3.06

STD = square root

 S (Xi - M1)2
line
n-1

Where X1 is given numbers
M1 is mean
n is number of values

    solution

     

     

     

     

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    Solution:

    Mean (M1) = sum (S) of readings divided by # of readings 

    M1 =  

    S
    line

    n

    M1 =

    3.05 + 3.02 + 3.03 + 2.97 + 2.98 + 3.10 + 2.94 + 3.06
    line

    8

    M1 =  

    24.15
    line

    		   = 3.02

    8

    Standard Deviation is a statistical measure of the range of variance or deviation from the average. It describes uniformity: the smaller the number, the more uniform the readings; the larger the number, the greater the deviation.

    Deviation = difference between the reading (X) and the average of the readings (M1)

    X - M1

    Standard Deviation (STD) = square root of the mean (M2) of the squares of the difference or deviation of each reading with the mean (M1) of the readings (Xi)

    STD = square root

    		 S (Xi - M1)2
    line
    n-1

    Reading
    Xi    		 Deviation
    Xi - M1    		 Squared Deviation
    (Xi - M1)2 
    3.05

    3.05 - 3.02 = .03

    		 .0009
    3.02

    3.02 - 3.02 = 0

    		 0
    3.03

    3.03 - 3.02 = .01

    		 .0001
    2.97

    2.97- 3.02 = -.05

    		 .0025
    2.98

    2.98 - 3.02 = -.04

    		 .0016
    3.10

    3.10 - 3.02 = .08

    		 .0064
    2.94

    2.94 - 3.02 = -.08

    		 .0064
    3.06

    3.06 - 3.02 = .04

    		 .0016

    M2 =  

    .0009 + 0 + .0001 + .0025 +.0016 + .0064 + .0064 + .0016
    line

    n-1

    M2 =   .0195
    line    		   = .0028
    7

    STD = square rootM2 = .053

    This means that on average, the readings varied only .053 (higher or lower) than the average of the readings.