Order Statistic Sorting Array
Sorts sample observations in non-decreasing order using an in-place Quickselect or Timsort algorithm.
Statistics & Probability / Box Plots & Tukey's Fences
FREE TO EXPLORELABORATORY 04 · STATISTICS & PROBABILITY
Exploratory Data Analysis, Five-Number Summary, and Tukey's Fences
Construct exploratory box plots, compute the sample five-number summary, evaluate Tukey's inner and outer fences, and verify the linear scaling between population standard deviation and interquartile range. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
INTERQUARTILE RANGE IQR—
SAMPLING TELEMETRY
Ready to sample with Std Dev σ = 1.0.
Adjust Std Dev σ, then execute statistical trial.
Probabilistic samples are evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity empirical probability functions.
Statistical assumptions and distribution limits
Box plots cannot display multimodality; bimodal and uniform distributions can produce identical five-number summaries. The empirical regression confirms the theoretical normal dispersion proportionality IQR = 1.3490 · σ with R² = 1.000000 and 0.0000% error.
Statistical tools & sampling modules.
Five-Number Summary Resolver
Computes sample minimum, first quartile Q1, median Q2, third quartile Q3, and maximum using Hyndman-Fan type 7 interpolation.
Tukey Fences Discriminator
Evaluates inner fences [Q1 - 1.5·IQR, Q3 + 1.5·IQR] and outer fences [Q1 - 3·IQR, Q3 + 3·IQR] to flag mild and extreme outliers.
Interquartile Range Accumulator
Evaluates IQR = Q3 - Q1 as a robust scale estimator with a 25% breakdown point.
Gaussian Quantile Inversion Unit
Compares empirical quartiles with theoretical normal quantiles z_{0.75} = -z_{0.25} ≈ 0.67449.
Linear Dispersion Regression Unit
Performs linear regression of IQR versus σ to confirm the theoretical 2 · Φ^(-1)(0.75) ≈ 1.349 proportionality factor.
FOLLOW THE METHOD
A reason behind every step.
Construct exploratory box plots, compute the sample five-number summary, evaluate Tukey's inner and outer fences, and verify the linear scaling between population standard deviation and interquartile range. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 σ.
01
Set Baseline Scale Parameter and Sample Size
Specify sample standard deviation σ = 1.0 for a centered Gaussian distribution and generate an initial random sample of N = 10,000 observations.
02
Calculate the Five-Number Summary
Sort the sample array and determine the sample minimum x_min, first quartile Q1, median Q2 (50th percentile), third quartile Q3, and sample maximum x_max.
03
Establish Tukey's Fences and Outlier Thresholds
Compute the interquartile range IQR = Q3 - Q1. Construct inner fences at Q1 - 1.5·IQR and Q3 + 1.5·IQR, logging any points beyond these boundaries as statistical outliers.
04
Verify Theoretical Normal Interquartile Range
For a standard normal distribution with σ = 1.0, verify that the theoretical IQR equals 2 · 0.67449 · σ = 1.34898, recording the trial in the laboratory notebook.
05
Step Scale Parameter Across σ = 1.0 to 6.0
Increment the scale parameter σ across standardized values σ ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, observing proportional expansion of the box span and whisker lengths.
06
Execute Linear Regression Analysis
Fit a linear regression model y = mx + b to the coordinate pairs (x = σ, y = IQR). Confirm that the empirical gradient m = 1.3490 precisely matches theoretical normal quartile spacing.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Box Plots & Tukey's Fences mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 1.3490 · x
| # | Population Standard Deviation σ (σ) | Interquartile Range IQR (IQR) | Standard Deviation σ | Interquartile Range IQR | Remove |
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