Why does Sample Variance divide by n - 1 while Population Variance divides by N?

Population Variance divides sum of squared deviations by N (total population size). Sample Variance uses Bessel's correction, dividing by n - 1 to correct for downward bias when estimating population parameters from a sample.

How is the Interquartile Range (IQR) computed for outlier detection?

IQR = Q3 - Q1 (75th percentile minus 25th percentile). A data point is a statistical outlier if it lies below Q1 - 1.5×IQR or above Q3 + 1.5×IQR.

When is the Median preferred over the Mean as a measure of central tendency?

The Median is preferred for skewed distributions (such as household income or real estate prices) because it is resistant to extreme outlier distortion, whereas the Mean is pulled heavily toward extreme values.

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Descriptive Statistics: Concepts, Formulas & Applications

Master the foundational metrics used to analyze, summarize, and interpret quantitative datasets.

Arithmetic Mean (Average)

The central balancing point of a dataset, calculated by summing all values and dividing by the count.

Formula:μ = (Σ xᵢ) / N (Population) or x̄ = (Σ xᵢ) / n (Sample)

Standard Use: Finding generic benchmarks like average height, test scores, or average stock yields.

Median (Middle Value)

The exact midpoint value of a sorted dataset. Separates the upper 50% from the lower 50%.

Formula: Middle element (Odd Count) or average of two middle elements (Even Count).

Standard Use: Robust indicator of central tendency since it is completely unaffected by extreme outliers (e.g. median household income).

Mode (Most Frequent)

The value or values that appear with the highest frequency in a dataset.

Formula:Mode = MaxFrequency(xᵢ)

Standard Use: Used in retail and operations planning to discover the most common category (e.g. most popular shoe size or product color).

s / σ

Standard Deviation (Population vs. Sample)

Measures the average distance of each data point from the mean. Population SD (σ) represents the entire group, while Sample SD (s) uses Bessel's correction (n - 1) to account for sample bias.

Formula:σ = √[Σ(xᵢ - μ)² / N] (Population) vs. s = √[Σ(xᵢ - x̄)² / (n - 1)] (Sample)

Standard Use: Analyzing product quality variance (Six Sigma), defining grading curves, and evaluating stock return volatility.

s² / σ²

Variance (Population vs. Sample)

The average of squared differences from the Mean, quantifying total spread. Population variance (σ²) divides by N, whereas Sample variance (s²) divides by (n - 1) to offset sample bias.

Formula:σ² = Σ(xᵢ - μ)² / N (Population) vs. s² = Σ(xᵢ - x̄)² / (n - 1) (Sample)

Standard Use: Fundamental baseline metric for Modern Portfolio Theory, financial risk modeling, and ANOVA hypothesis testing.

Quartiles & Interquartile Range (IQR)

Divides a sorted dataset into four equal parts. Q1 is the 25th percentile (lower quartile), Q2 is the median (50th percentile), and Q3 is the 75th percentile (upper quartile).

Formula:IQR = Q3 - Q1 (Interquartile Range)

Standard Use: Building Box-and-Whisker plots, analyzing income distributions, and systematically identifying outliers using the 1.5 × IQR fence method.