Design of Experiments (DoE) is a systematic method for determining the relationship between factors affecting a process and the output of that process. It allows researchers to understand how different input variables impact the final result with minimal experimental runs.
Factorial Design
Fractional Factorial Design
Response Surface Methodology
Experimental Units
Randomization
Replication
Two-Factor Factorial Design Example: Consider a manufacturing process with two factors:
Temperature (A): Low (1) and High (2)
Pressure (B): Low (1) and High (2)
Experimental Matrix:
|
Run |
Temperature (A) |
Pressure (B) |
Yield |
|
1 |
Low (1) |
Low (1) |
85.2 |
|
2 |
Low (1) |
High (2) |
92.7 |
|
3 |
High (2) |
Low (1) |
88.5 |
|
4 |
High (2) |
High (2) |
95.3 |
Statistical Quality Control (SQC) is a methodology used to monitor, control, and improve product quality through statistical methods, focusing on reducing variability in manufacturing and service processes.
Control Charts
Process Capability Analysis
Acceptance Sampling
Six Sigma Methodology
Process Mean Calculation:
Collect subgroup samples
Calculate subgroup means
Determine control limits
Sample Calculation:
Mean (X̄) = Σ(Sample Means) / Number of Samples
Upper Control Limit (UCL) = X̄ + 3(σ/√n)
Lower Control Limit (LCL) = X̄ - 3(σ/√n)
Bootstrap is a resampling technique used to estimate the sampling distribution of a statistic by repeatedly sampling with replacement from the original dataset.
Confidence Interval Estimation
Hypothesis Testing
Variance Estimation
Model Validation
import numpy as np
def bootstrap_mean(data, num_iterations=1000):
bootstrap_means = []
for _ in range(num_iterations):
sample = np.random.choice(data, size=len(data), replace=True)
bootstrap_means.append(np.mean(sample))
return np.mean(bootstrap_means), np.std(bootstrap_means)
# Example usage
data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
mean_estimate, std_estimate = bootstrap_mean(data)
Bayesian statistics is a mathematical approach to data analysis based on Bayes' theorem, which updates probability estimates as more information becomes available.
Prior Probability
Likelihood
Posterior Probability
Bayesian Inference
Markov Chain Monte Carlo (MCMC)
P(A|B) = [P(B|A) * P(A)] / P(B)
Where:
P(A|B): Probability of A given B
P(B|A): Probability of B given A
P(A): Prior probability of A
P(B): Probability of B
Clear Problem Statement
Methodology Description
Statistical Analysis
Key Findings
Practical Implications
Future Work Recommendations
Recap of Key Statistical Techniques
Practical Applications in Engineering
Critical Thinking in Data Analysis
Connecting Theoretical Concepts to Real-World Problems