GPT Copy of discussion 3
.pdf
School
University of California, Riverside**We aren't endorsed by this school
Course
STATISTICS 8
Subject
Statistics
Date
Dec 26, 2024
Pages
2
Uploaded by gobrandonwang
Key Statistical ConceptsChebyshev’s Theorem●Definition: For any distribution shape, at least (1−1/k2)×100%(1 - 1/k^2) \times 100\% ofthe observations lie within μ±kσ\mu \pm k\sigma, where k>1k > 1.●Key Features:○Applicable to any distribution.○Provides the minimum percentage of observations within a range.○Keyword:At least.Empirical Rule●Definition: For symmetric, mound-shaped distributions:○Approximately 68% of observations fall within μ±1σ\mu \pm 1\sigma.○Approximately 95% of observations fall within μ±2σ\mu \pm 2\sigma.○Approximately 99.7% of observations fall within μ±3σ\mu \pm 3\sigma.●Key Features:○Only applies to symmetric, bell-shaped distributions.○Provides approximate percentages for specific ranges.○Keyword:Approximately.Mean●Formula: Mean=Sum of all data pointsNumber of data points\text{Mean} =\frac{\text{Sum of all data points}}{\text{Number of data points}}Sample Variance (S2S^2)●Formula: S2=∑Xi2−(∑Xi)2nn−1S^2 = \frac{\sum X_i^2 - \frac{(\sum X_i)^2}{n}}{n - 1}○∑Xi2\sum X_i^2: Sum of the squared values.○(∑Xi)2(\sum X_i)^2: Square of the sum of all values.○nn: Total number of data points.Sample Standard Deviation●Formula: S=S2S = \sqrt{S^2}
○The positive square root of the sample variance.Z-Score●Definition: The number of standard deviations an observation is from the mean.●Formula:○Sample: Z=Xi−XˉSZ = \frac{X_i - \bar{X}}{S}○Population: Z=Xi−μσZ = \frac{X_i - \mu}{\sigma}●Outlier Rule: An observation is considered an outlier if∣Z∣≥3|Z| \geq 3.