Abstract
The Gaussian distribution (normal or bell-shaped distribution) is a widely used statistical distribution and it is generally used as the foundation for statistical quality control. Simply measuring the time-zero values of a parameter (resistor values, mechanical tolerances, children heights, class grades on a test, etc.) can result in a distribution of values which can be described by a normal distribution.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Mean can be estimated: \( {x}_{50}={\sum}_{i=1}^N{x}_i/N \),when N is the sample size.
- 2.
Standard deviation can be estimated: \( \sigma ={\left[{\sum}_{i=1}^N{\left({x}_i-{x}_{50}\right)}^2/\left(N-1\right)\right]}^{1/2} \).
- 3.
A more precise value is 15.87 %.
- 4.
One gm-f equals 9.8 × 10−3 N.
- 5.
A cumulative probability of exactly F = 1 cannot be plotted. Therefore, in order to ensure that all 25 data points can be plotted, then an unbiased estimate of the cum F is needed. In reliability physics and engineering, Eq. (6.4) is generally used.
Bibliography
Ash, C.: The Probability Tutoring Book, IEEE Press, (1993).
Bowker, A. and G. Lieberman: Engineering Statistics, Prentice-Hall Publishing, (1972). Dixon, W. and F. Massey: Introduction to Statistical Analysis, McGraw-Hill Book Co., (1957). Fowler, J., L. Cohen and P. Jarvis: Practical Statistics for Field Biology, John Wiley & Sons, (1998).
Larsen, R.: Engineering with EXCEL, 2nd Ed., Pearson/Prentice Hall Publishing, (2005). Miller, I. and J. Freund: Probability and Statistics for Engineers, Prentice Hall Publishing, (1977).
Author information
Authors and Affiliations
Problems
Problems
-
1.
O-rings (from a manufacturing line) were randomly selected for diameter measurements. The 25 measurements are shown in the below table (all measurements are in mm). Find the Normal Distribution parameters: median diameter size (x50) and the standard deviation σ.
181.4 | 173.0 | 172.2 | 173.5 | 180.5 |
|---|---|---|---|---|
187.8 | 178.6 | 170.7 | 179.5 | 186.5 |
171.1 | 180.0 | 183.4 | 177.3 | 187.0 |
176.7 | 186.1 | 182.5 | 174.2 | 188.7 |
184.0 | 185.6 | 190.0 | 175.4 | 189.5 |
Answers: x50 = 180.7 mm σ = 6.6 mm
-
2.
For the O-ring manufacturing process in Problem 1 (x50 = 180.7 mm, σ = 6.6 mm), find the capability parameters: Cp and Cpk. Assume that the upper spec limit is 215 mm and the lower spec limit is 155 mm.
Answers: Cp = 1.52 Cpk = 1.30
-
3.
The breakdown-strength distribution for capacitor dielectrics had a median value of (Ebd)50 = 10.50 MV/cm and a σ = 1.8 MV/cm.
-
(a)
Find the fraction of caps with a breakdown ≤ 8 MV/cm.
-
(b)
Find the fraction of caps with a breakdown ≥ 12 MV/cm.
Answers: (a) 0.082 (b) 0.202
-
(a)
-
4.
The rupture strength distribution of water pipes had a median value of (Rupture-Stress)50 = 900 MPa and a σ = 120 MPa.
-
(a)
Find the fraction of pipes with a rupture stress of ≤ 600 MPa.
-
(b)
Find the fraction of pipes with a rupture stress of ≥ 1,300 MPa.
Answers: (a) 6.21 × 10−3 = 6,210 ppm (b) 4.29 × 10−4 = 429 ppm
-
(a)
-
5.
Resistors have a resistance value distribution with a median value of (R)50 = 189 Ω and a σ = 3.5 Ω.
-
(a)
Find the fraction of resistors with a resistance value of ≤ 160 Ω.
-
(b)
Find the fraction of resistors with a resistance value of ≥ 200 Ω.
-
(a)
Answers: (a) 5.55 × 10−17 = 0.555 × 10−10 ppm (b) 8.37 × 10−4 = 837 ppm
-
6.
A group of patients had a heart rate distribution with a median value (HR)50 = 60 beats/min and a σ = 2 beats/min.
-
(a)
Find the fraction of patients with a heart rate of ≤ 50 beats/min.
-
(b)
Find the fraction of patients with a heart rate of ≥ 70 beats/min.
Answers: (a) 2.87 × 10−7 = 0.287 ppm (b) 2.87 × 10−7 = 0.287 ppm
-
(a)
-
 7.
Using the breakdown-strength distribution, defined in Problem 3, what are the process capability parameters: Cp and Cpk? Assume an upper-level limit of 12 MV/cm and a lower-level limit of 8 MV/cm.
Answers: Cp = 0.37 Cpk = 0.28
-
 8.
For the rupture strength distribution, defined in Problem 4, what are the process capability parameters: Cp and Cpk? Assume an upper-level limit of 1,300 MPa and a lower-level limit of 600 MPa.
Answers: Cp = 0.97 Cpk = 0.83
-
 9.
For the resistor distribution, defined in Problem 5, what are the process capability parameters: Cp and Cpk? Assume an upper-level limit of 200 Ω and a lower-level limit of 160 Ω.
Answers: Cp = 1.90 Cpk = 1.05
-
10.
For the heart rate distribution, defined in Problem 6, what are the capability parameters: Cp and Cpk for this group of patients? Assume an upper-level limit of 70 beats/min and a lower-level limit of 50 beats/min.
Answers: Cp = 1.67 Cpk = 1.67
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
McPherson, J.W. (2019). Gaussian Statistics: An Overview. In: Reliability Physics and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-93683-3_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-93683-3_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-93682-6
Online ISBN: 978-3-319-93683-3
eBook Packages: EngineeringEngineering (R0)