Engineering Data Analysis Simulations

A collection of interactive 3D visualizations and simulations to help you master concepts in engineering data analysis.

Introduction to Data Analysis - Theory & Concepts

Overview of statistics in engineering, data collection methods, observational studies versus experiments, and sampling techniques.

Engineering Data Analysis

Sampling Methods & Bias Explorer

Population Size (

N

)120

Sample Size (

n

)15

Sampling Methodology

Population Map (

N

circles)

Introduction to Data Analysis - Theory & Concepts - Engineering Data Analysis Bias Precision

Overview of statistics in engineering, data collection methods, observational studies versus experiments, and sampling techniques.

Engineering Data Analysis • Topic 1

Bias vs. Precision Target Visualizer

Bias (Systematic Error)Low

Centred (Accurate)Highly Biased

Precision (Random Error)High (Tight Cluster)

Broad SpreadTight Cluster

Concept Summary

• Accuracy (Low Bias): The average of the measurements is very close to the true value (the bullseye).

• Precision: The degree to which repeated measurements show the same results (tightness of the grouping), regardless of whether they are correct.

Descriptive Statistics - Theory & Concepts

Measures of central tendency, dispersion, position, skewness, and kurtosis, including grouped data.

Engineering Data Analysis

Descriptive Statistics Explorer

Add Data Point

Dataset (5)

Mean (Average)

\bar{x}

12.00

The sum of all values divided by the sample size.

Median

\tilde{x}

12.00

The middle value when the data is sorted in order.

Mode

\text{Mo}

None

The most frequently occurring value(s) in the dataset.

Range

R

15.00

Sample Std. Dev.

s

5.87

Descriptive Statistics - Theory & Concepts - Engineering Data Analysis Box Whisker

Measures of central tendency, dispersion, position, skewness, and kurtosis, including grouped data.

Engineering Data Analysis • Topic 2

Interactive Box & Whisker Plot

Data Values

Value x₁20

Value x₂35

Value x₃40

Value x₄50

Value x₅55

Value x₆60

Value x₇75

Value x₈95

Median (Q2)52.5

Q137.5

Q367.5

IQR30.0

• Outliers are values beyond fences:

[\text{Q1} - 1.5\text{IQR}, \text{Q3} + 1.5\text{IQR}]

.No outliers.

Descriptive Statistics - Theory & Concepts - Skewness Kurtosis

Measures of central tendency, dispersion, position, skewness, and kurtosis, including grouped data.

Engineering Data Analysis

Distribution Shape: Skewness & Kurtosis

Skewness (

\gamma_1

): 0.0Symmetric (Zero Skew)

Negative SkewSymmetric (0)Positive Skew

Kurtosis (

\beta_2

): 3.0Mesokurtic (Normal)

Platykurtic (Flat)Mesokurtic (3)Leptokurtic (Peaked)

Statistical Moments

• Skewness measures the asymmetry of the PDF around the mean. A positive skew has a tail extending towards more positive values.

• Kurtosismeasures the "tailedness" of the distribution. Fatter tails and a sharper peak characterize high kurtosis (Leptokurtic).

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Probability Fundamentals - Theory & Concepts - Engineering Data Analysis Venn Diagram

Basic probability theory, sample spaces, events, counting rules, and probability rules.

Engineering Data Analysis • Topic 3

Venn Diagram & Set Operations

Set Operation Presets

Set Mathematical Notation

\varnothing \quad \text{(Empty Set)}

Set Operations Guide

• Click on any region of the Venn diagram (including Set A, Set B, the central intersection lens, or the surrounding Universal space) to toggle highlights and construct custom set formulas dynamically.

Probability Fundamentals - Theory & Concepts

Basic probability theory, sample spaces, events, counting rules, and probability rules.

Engineering Data Analysis

Probability Playground: Law of Large Numbers

Heads

0 flips (0%)

Tails

0 flips (0%)

Heads: 0.0%Theoretical: 50%Tails: 0.0%

Total Flips (

n

)

Simulation Speed

SlowFast

Note: As trials $n \to \infty$ increase, the experimental relative frequencies approach the theoretical probability $P(\text{Heads}) = 0.5$ . This is the cornerstone of empirical probability modeling in engineering.

Conditional Probability - Theory & Concepts

Understanding how probabilities change when new information is available, including Bayes' Theorem and Independence.

Engineering Data Analysis

Bayes' Theorem & Diagnostic Testing Explorer

Prior / Base Rate

P(D)

(Defective)2.0%

The prior probability of a random component being defective.

Sensitivity / True Positive

P(T|D)

95.0%

The probability that the test is positive given that a defect is present.

False Alarm / False Positive

P(T|G)

10.0%

The probability that the test flag is positive when NO defect is present.

Posterior Probability Calculation

16.2%

$P(D|T)$ — Probability that a component is actually defective given a positive test flag.

Total Positives

P(T)

11.7%

True Positives

P(T|D)P(D)

1.90%

False Positives

P(T|G)P(G)

9.80%

Conditional Probability - Theory & Concepts - Engineering Data Analysis Diagnostic Testing

Understanding how probabilities change when new information is available, including Bayes' Theorem and Independence.

Engineering Data Analysis • Topic 4

Bayes' Theorem in Diagnostic Testing

Prevalence

P(D)

10%

Sensitivity

P(+\mid D)

90%

Specificity

P(-\mid D^c)

90%

Population Grid (N = 100)

True Positive (TP: 9)

False Negative (FN: 1)

False Positive (FP: 9)

True Negative (TN: 81)

PPV (Post-test prob of disease | positive test)

50.0%

P(D\mid +) = \frac{P(+\mid D)P(D)}{P(+\mid D)P(D) + P(+\mid D^c)P(D^c)}

NPV (Post-test prob of healthy | negative test)

98.8%

P(D^c\mid -) = \frac{P(-\mid D^c)P(D^c)}{P(-\mid D^c)P(D^c) + P(-\mid D)P(D)}

Discrete Probability Distributions - Theory & Concepts

Expected value, Binomial, Poisson, Negative Binomial, Geometric, and Hypergeometric distributions.

Engineering Data Analysis

Discrete Probability Distributions Explorer

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Number of Trials (

n

)10

Success Probability (

p

)0.50

Theoretical Properties

Mean (

\mu

):5.00

Variance (

\sigma^2

):2.50

Discrete Probability Distributions - Theory & Concepts - Engineering Data Analysis Hypergeometric Geometric

Expected value, Binomial, Poisson, Negative Binomial, Geometric, and Hypergeometric distributions.

Engineering Data Analysis • Topic 5

Discrete Probability Distributions Sandbox

Distribution Model

Success Prob (

p

)0.30

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Mean (Expected Value)3.333

Variance7.778

Continuous Probability Distributions - Theory & Concepts

Probability density functions, Normal, Uniform, Exponential, Gamma, Weibull, and Lognormal distributions.

Engineering Data Analysis

Continuous Normal Distribution Explorer

Mean (

\mu

)0.0

Std. Dev (

\sigma

)1.0

Upper Bound (

x

)0.0

Calculated Probability (CDF)

P(X \leq 0.00)

0.5000

Adjust the slider values to observe the shift (

\mu

), spread (

\sigma

), and shaded area representing cumulative density (

P(X \le x)

Continuous Probability Distributions - Theory & Concepts - Engineering Data Analysis Exponential Uniform

Probability density functions, Normal, Uniform, Exponential, Gamma, Weibull, and Lognormal distributions.

Engineering Data Analysis • Topic 6

Continuous Probability Distributions Sandbox

Distribution Type

Rate Parameter (

\lambda

)0.50

Threshold (

x

)3.0

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Cumulative Probability

P(X ≤ 3.0) = 77.69%

P(X \le x) = 1 - e^{-\lambda x} = 1 - e^{-0.50 \times 3.0}

Joint Probability Distributions - Theory & Concepts

Joint probability mass/density functions, marginal and conditional distributions, covariance, and correlation.

Engineering Data Analysis

Discrete Joint Probability Explorer

$X \setminus Y$	$Y = 1$	$Y = 2$	$Y = 3$	Marginal $g(x)$
$X = 10$				0.150
$X = 20$				0.600
$X = 30$				0.250
Marginal $h(y)$	0.200	0.500	0.300	Sum: 1.000

Mean of X (

\mu_X

)

21.00

Mean of Y (

\mu_Y

)

2.10

Covariance

\text{Cov}(X,Y)

2.9000

If $\text{Cov}(X,Y) > 0$ , X and Y vary together. If $\text{Cov}(X,Y) = 0$ , they may be independent.

Joint Probability Distributions - Theory & Concepts - Engineering Data Analysis Bivariate Normal

Joint probability mass/density functions, marginal and conditional distributions, covariance, and correlation.

Engineering Data Analysis • Topic 7

Bivariate Normal Distribution Contours

Std Dev X (

\sigma_X

)2.0

Std Dev Y (

\sigma_Y

)2.0

Correlation (

\rho

)0.50

Covariance Matrix (

\mathbf{\Sigma}

)

\mathbf{\Sigma} = \begin{bmatrix} 4.00 & 2.00 \\ 2.00 & 4.00 \end{bmatrix}

Contours at 1σ, 2σ, and 3σ

Sampling Distributions - Theory & Concepts

How sample statistics behave, the Central Limit Theorem, t-distribution, Chi-square, and F-distribution.

Engineering Data Analysis

Central Limit Theorem & Sampling Distribution

Population Distribution

Sample Size (

n

)30

Number of random items in each sample.

Sampling Statistics

Total Samples:0

Mean of Means (

\mu_{\bar{x}}

):0.000

Std Error (

\sigma_{\bar{x}}

):0.000

Generate samples to construct the sampling distribution.

Click "+1 Sample" or "Run Auto". As sample size $n \ge 30$ grows, the distribution of sample means approaches normality regardless of the population shape.

Sampling Distributions - Theory & Concepts - Engineering Data Analysis Probability Distribution Shapes

How sample statistics behave, the Central Limit Theorem, t-distribution, Chi-square, and F-distribution.

Engineering Data Analysis • Topic 8

Probability Distribution Shapes

Select Distribution

Degrees of Freedom (

\nu

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• Observe how as the degrees of freedom $\nu$ increases, the tails of the t-distribution become lighter, and the curve converges directly to the Standard Normal distribution $N(0, 1)$ .

Estimation - Theory & Concepts

Point estimation, confidence intervals for means, proportions, and variances, and prediction/tolerance intervals.

Engineering Data Analysis

Confidence Interval & Parameter Estimation

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95% Confidence Interval

z_{\alpha/2} = z_{0.025} = 1.96

[2416.84, 2583.16]

\bar{x} \pm E = 2500.0 \pm 83.16

(Margin of Error)

Sample Mean (

\bar{x}

)2500

Standard Deviation (

s

)300

Sample Size (

n

)50

Standard Error: $\sigma_{\bar{x}} = s / \sqrt{n} = 42.426$

Confidence Level (

1 - \alpha

)

Estimation - Theory & Concepts - Engineering Data Analysis Sample Size Calculator

Point estimation, confidence intervals for means, proportions, and variances, and prediction/tolerance intervals.

Engineering Data Analysis • Topic 9

Sample Size Calculator

Estimation Target

Confidence Level

Margin of Error (

E

)0.050

Std Dev (

\sigma

)0.25

Required Sample Sizen = 97

Governing Formula

n = \left(\frac{Z_{\alpha/2} \cdot \sigma}{E}\right)^2

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Tests of Hypotheses - Theory & Concepts

Null and alternative hypotheses, Type I/II errors, P-values, and tests for means, proportions, variances, and Goodness-of-Fit.

Engineering Data Analysis

Hypothesis Testing Simulator

Test Type

Significance Level (

\alpha

)

Test Statistic (Z)1.96

Conclusion

Fail to Reject H₀

The test statistic falls in the acceptance region. There is insufficient evidence to reject H₀.

Tests of Hypotheses - Theory & Concepts - Engineering Data Analysis P Value Significance

Null and alternative hypotheses, Type I/II errors, P-values, and tests for means, proportions, variances, and Goodness-of-Fit.

Engineering Data Analysis • Topic 10

p-Value vs. Significance Level (α) Visualizer

Hypothesis Direction

Significance Level (

\alpha

)0.050

p-Value0.035

Conclusion

Reject Null Hypothesis (H₀)

Since the p-value (0.035) is $\le$ significance level $\alpha$ (0.050), the result is statistically significant.

Regression and Correlation - Theory & Concepts

Simple and multiple linear regression, correlation coefficients, least squares method, and residual analysis.

Engineering Data Analysis

Linear Regression Sandbox

Click on the grid to plot points

Slope (m)0.000

Intercept (b)0.000

Correlation (r)0.0000

R² (Coeff. of Det.)0.0000

Regression Equation

y = 0.00x + 0.00

The line of best fit is calculated using Ordinary Least Squares (OLS) which minimizes the sum of squared residuals:

\sum (y_i - \hat{y}_i)^2

Regression and Correlation - Theory & Concepts - Engineering Data Analysis Residual Outlier

Simple and multiple linear regression, correlation coefficients, least squares method, and residual analysis.

Engineering Data Analysis • Topic 11

Residuals & Leverage Outliers sandbox

Outlier Scenario

Slope (m)1.129

Intercept (b)2.69

R² score0.991

Outlier ResidualN/A

Analysis of Variance - Theory & Concepts

One-way ANOVA, Randomized Complete Block Design (RCBD), and post-hoc tests for comparing multiple means.

Engineering Data Analysis

One-Way ANOVA Simulator

Adjust Group Means & Variance

Group 1 Mean (

\mu_1

)50

Group 2 Mean (

\mu_2

)50

Group 3 Mean (

\mu_3

)50

Within-Group Variance (

\sigma^2

)10

F-Test Result

F = 0.06

Critical F-value (

\alpha=0.05

): 3.35

Fail to Reject Null (No significant difference detected)

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Analysis of Variance - Theory & Concepts - Engineering Data Analysis A N O V A Overlap

One-way ANOVA, Randomized Complete Block Design (RCBD), and post-hoc tests for comparing multiple means.

Engineering Data Analysis • Topic 12

ANOVA Variance & Overlap Visualizer

Between-Group Diff (

\delta

)3.0

Within-Group Noise (

\sigma

)1.2

F-ratio Estimation

F ≈ 62.5

Reject H₀. The groups are separated enough from each other compared to within-group noise.

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• Between-group variance measures how separated the distribution peaks are.

• Within-group variance measures the spread or standard deviation of each distribution.

• ANOVA compares these variances via the F-ratio. Larger F-ratios occur when peaks are far apart and noise is low.

Statistical Quality Control - Theory & Concepts

Control charts for variables and attributes, process capability indices, and natural tolerance limits.

Engineering Data Analysis

Control Chart ( $\bar{X}$ ) Simulator

Process Parameters

Target Mean (

\mu

)100

Std Dev (

\sigma

Subgroup Size (

n

Mean Shift (Assignable Cause)+0

Applies a step shift to the mean starting at subgroup 11.

UCL0.00

CL (

\bar{\bar{X}}

)0.00

LCL0.00

Statistical Quality Control - Theory & Concepts - Engineering Data Analysis Process Capability

Control charts for variables and attributes, process capability indices, and natural tolerance limits.

Engineering Data Analysis • Topic 13

Process Capability Index (Cₚ & Cₚₖ) Visualizer

Process Mean (

\mu

)100.0

Std Dev (

\sigma

)3.50

Lower Limit (LSL)90

Upper Limit (USL)110

Cp Index0.95

Cpk Index0.95

EvaluationNot Capable (Produces Defects)

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• Cₚ (Process Capability): Measures potential capability if the process was perfectly centered. Formula: $C_p = \frac{\text{USL} - \text{LSL}}{6\sigma}$

• Cₚₖ (Actual Capability): Accounts for the centering of the mean relative to limits. Formula: $C_{pk} = \min\left(\frac{\text{USL} - \mu}{3\sigma}, \frac{\mu - \text{LSL}}{3\sigma}\right)$