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Lesson Directive // Statistics & DataREF_CORE

Measures of Central Tendency

σ\sigma==∑(xi−μ)2\sum (x_i - \mu)^2NN

Hover over a variable in the formula above, or see glossary below:

σ\sigma
Standard Deviation
Same unit as data
μ\mu
Mean (Average)
Same unit as data
xix_i
Individual Data Point
Same unit as data
NN
Number of Data Points
Count

The **mean** (average) \mu = \frac{\sum x_i}{N} finds the central value. The **median** is the middle value when sorted. The **mode** is the most frequent value. Each measure has different strengths depending on the data distribution.

INSIGHT: Use median instead of mean when data has extreme outliers.

Standard Deviation & Spread

Standard deviation \sigma measures how far data points typically stray from the mean. A dataset of sensor readings {100, 101, 99, 100} is much tighter than {50, 150, 80, 120} even if both have the same mean.

INSIGHT: σ tells you the "typical distance" from the average.

The Normal Distribution

In a normal (bell curve) distribution, approximately 68% of data falls within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. This is the famous empirical rule used in quality control, science, and AI model evaluation.

INSIGHT: 68-95-99.7 rule: most data lies within 3 standard deviations of the mean.
Detailed Theory & ReferencesEXT_DOC

Descriptive Statistics and Distributions

Statistics is the science of collecting, analysing, interpreting, and presenting data. In engineering and science, it is essential for understanding measurement uncertainty, signal processing, and model validation.

Measures of Central Tendency

  • Mean: μ=1N∑i=1Nxi\mu = \frac{1}{N} \sum_{i=1}^{N} x_i
  • Median: The middle value when data is sorted in ascending order.
  • Mode: The most frequently occurring value.

Measures of Spread

  • Variance: σ2=1N∑i=1N(xi−μ)2\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2
  • Standard Deviation: σ=σ2\sigma = \sqrt{\sigma^2}
  • Range: xmax⁡−xmin⁡x_{\max} - x_{\min}

Note: Population variance uses 1/N1/N. Sample variance uses 1/(N−1)1/(N-1) (Bessel's correction).

The Normal Distribution

f(x)=1σ2πe−12(x−μσ)2f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}

The Empirical Rule (68-95-99.7):

Interval% of Data
μ±1σ\mu \pm 1\sigma≈ 68.27%
μ±2σ\mu \pm 2\sigma≈ 95.45%
μ±3σ\mu \pm 3\sigma≈ 99.73%

Correlation vs. Causation

Two variables may be statistically correlated without one causing the other. Establishing causation requires controlled experiments.

References

AI NOTICE

AI Assistance Disclaimer: This module uses AI-assisted educational models and interactive visual representations to help explain scientific and mathematical concepts. For formal research or academic evaluation, please verify formulas and data against standard primary reference materials.