1. Measures of Central Tendency: Mean, Median, and Mode

Central tendency quantifies the center point or typical value of a probability distribution or dataset:

• Arithmetic Mean (x̄): The sum of all values divided by count n: x̄ = (∑ xᵢ) / n. The mean minimizes the sum of squared deviations ∑ (xᵢ - c)², but is sensitive to extreme outliers.
• Median (M): The 50th percentile dividing ordered data into two equal halves. The median minimizes the sum of absolute deviations ∑ |xᵢ - c|, making it robust against heavily skewed data (such as national income or housing prices).
• Mode: The most frequently occurring score in the dataset. Unimodal distributions have one mode; bimodal distributions exhibit two peaks.

2. Dispersion & Why Bessel's Correction (n - 1) Is Essential

When computing the variance of an entire population of size N, we divide by N:

However, when estimating variance from a sample of size n, using the sample mean x̄ instead of true population mean μ systematically underestimates the true variance because sample deviations are calculated relative to their own center. Friedrich Bessel proved that dividing by (n - 1) yields an unbiased estimator whose expected value equals the true population variance:

3. Five-Number Summary, IQR, and Tukey's Outlier Rule

Introduced by John Tukey in 1977, the box-and-whisker plot summarizes a distribution using five key values: Minimum, First Quartile (Q₁), Median (Q₂), Third Quartile (Q₃), and Maximum.

The Interquartile Range (IQR = Q₃ - Q₁) encompasses the middle 50% of the data. Data points falling beyond the standard Tukey fences are flagged as statistical outliers:

1. Measures of Central Tendency: Mean, Median, and Mode

Central tendency quantifies the center point or typical value of a probability distribution or dataset:

• Arithmetic Mean (x̄): The sum of all values divided by count n: x̄ = (∑ xᵢ) / n. The mean minimizes the sum of squared deviations ∑ (xᵢ - c)², but is sensitive to extreme outliers.
• Median (M): The 50th percentile dividing ordered data into two equal halves. The median minimizes the sum of absolute deviations ∑ |xᵢ - c|, making it robust against heavily skewed data (such as national income or housing prices).
• Mode: The most frequently occurring score in the dataset. Unimodal distributions have one mode; bimodal distributions exhibit two peaks.

2. Dispersion & Why Bessel's Correction (n - 1) Is Essential

When computing the variance of an entire population of size N, we divide by N:

However, when estimating variance from a sample of size n, using the sample mean x̄ instead of true population mean μ systematically underestimates the true variance because sample deviations are calculated relative to their own center. Friedrich Bessel proved that dividing by (n - 1) yields an unbiased estimator whose expected value equals the true population variance:

3. Five-Number Summary, IQR, and Tukey's Outlier Rule

Introduced by John Tukey in 1977, the box-and-whisker plot summarizes a distribution using five key values: Minimum, First Quartile (Q₁), Median (Q₂), Third Quartile (Q₃), and Maximum.

The Interquartile Range (IQR = Q₃ - Q₁) encompasses the middle 50% of the data. Data points falling beyond the standard Tukey fences are flagged as statistical outliers: