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Shanghai Jiao Tong University 1
ME250: Statistics & Probability
ME 250: Design & Manufacturing I
School of Mechanical Engineering
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Outline
Introduction Tolerances Reliability Statistical Process Control Regression Summary
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Introduction Uncertainty in Design & Manufacturing Learning Objectives
Understand the elementary aspects of statistics and probability.
How to represent data. Discrete and continuous distributions. The normal distribution and how to use tables of the
cumulative distribution. Probability of single and multiple events. Obtaining trends from noisy data by regression. Applications in statistical process control and geometric
dimensioning and tolerances. Taylor’s tool life equation and how to use it.
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Example 1 Suppose a company produces 1000 engines per year and 5
engines fail on average per year. The failures can be eliminated by a 100% inspection program. If it costs $10,000 to replace an engine but $100 to inspect each one, then the costs are:
Inspect: 1000 * $100 = $100,000 Replace: 5 * $10,000 = $50,000
Obviously, it is cheaper not to inspect. If 10 or more engines fail, then the inspection program is cheaper.
Therefore, it is very important for the company to maintain good statistical data on its manufacturing process.
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Tolerances
Tolerances refer to how close a dimension or a surface finish of a manufactured part is to the desired or “nominal” dimension.
The explicit specification of tolerances is critical for determining the manufacturing process(es) required to make a part and therefore its final cost.
If the surface finish is not important, it may be possible to use cheaper or faster process than if the surface finish is critical.
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Reliability
Reliability is the likelihood that an object will perform its function or a process will yield satisfactory results.
Often, the reliability of system must be predicted from an estimate of the reliability of its components.
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Statistical Process Control
Statistical Process Control is used to monitor processes in a way such that normal variability of the operation is accounted for, but long term degradation is observed.
The process is sampled at given intervals and the statistics of the samples compared. The cost of the inspection must be balanced versus the dangers of either missing a change in the process or misinterpreting a statistical fluctuation for a change in the process.
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Regression
Regression allows us to determine a relationship between two or more variables, even when there is uncertainly in the measurements and the relationship may not be immediately obvious.
We will use regression to establish the Taylor’s Tool Life Equation.
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Terminology
If the parameters of a model of a system or a process are known, given, or established from past history, then we have a probability problem and we can deduce the behavior of the system from the model. However, if the parameters are unknown and have to be estimated from the available data, then we have a statistical problem.
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Representing Statistical Data
Histogram Group the data into bins or intervals and show
the size of each bin (interval) graphically. Cumulative distribution
Show graphically how much of the data is below a given value
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Example 1 Consider the following grade
points from a class with 28 students:
70, 73, 74, 76, 76, 76, 76, 77, 77, 77, 78, 78,79, 79,83, 83, 84, 85, 86, 87, 89, 89, 90, 90, 92, 92, 93, 97
where the numbers have been arranged in numerical order. Grouping the results together in six intervals yields
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Discrete Distributions
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Variance
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The solution of the example
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Exercise in class (1)
Given the following number: 51,53,54,56,56,56,56,57,57,57,58,59,63,63,64,65,
66,67,69,69,70,72,72,72,73,76
(a) Plot a histogram of this data using six intervals.
(b) Plot the cumulative distribution also.
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Exercise in class (1)
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Discrete Distributions
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Distributions
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Continuous Distributions
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Continuous Distributions
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Continuous Distributions
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Uniform Distributions
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Normal Distributions
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Normal Distributions
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Normal Distributions
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The standard normal distributions
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The standard normal distributions
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The standard normal distributions
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The standard normal distributions
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Example 2
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Example 3
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Example 4a
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Example 4b
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Example 4c
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Example 4d
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Exercise in class (2)
The length of a batch of 500 steel rods are approximately normally distributed with mean 11 cm and standard deviation 1 cm. Estimate the number of rods which are longer than 10 cm.
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Exercise in class (2)
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Probability: Weibull Distributions
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Weibull Distributions
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Example 5a
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Example 5b
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Example 5c
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Example 5d
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Exercise in class (3)
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Exercise in class (3)
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Combining Statistics
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Combining Statistics
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Combining Statistics
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Combining Statistics
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Example 6a
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Example 6b
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Exercise in class (4)
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Exercise in class (4)
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Exercise in class (5)
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Exercise in class (5)
Do = Di+2W = 3.734+2*0.139 = 4.012 to = 0.028+2*0.004 = 0.036 Do = 4.012+-0.036
To = 0.02857
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Probability
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Probability
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Probability
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Probability
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Example 7
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Example 8
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Example 9
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Exercise in class (6)
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Exercise in class (7)
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Exercise in class (7)
0.99*(1-(1-0.90)2)*0.99 = 0.99*0.99*0.99 = 0.97
0.99*(1-(1-0.90)n)*0.99 > 0.98; n>3.991 N=4
N 接近无穷大， (1-0.90)n=0 0.99*1*0.99=0.9801<0.99
Therefore it is impossible.
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Summary
Uncertainty is a natural part of any human enterprise.
Statistics and probability is used in many aspects of the design and manufacturing cycle.
