14.1 Measures of Central Tendency, Dispersion & Resistant Statistics

14.1 Measures of Central Tendency, Dispersion & Resistant Statistics

Measures of Central Tendency: Mean, Median, and Mode

Descriptive statistics begins with quantifying the location or center of a univariate quantitative distribution. Given a dataset containing $n$ observations $\{x_1, x_2, \dots, x_n\}$, three primary metrics characterize central tendency:

1. Arithmetic Mean: The arithmetic mean represents the balance point of the distribution, where the sum of signed deviations equals zero: $\sum_{i=1}^n (x_i - \bar{x}) = 0$. For a sample of size $n$, the sample mean is denoted $\bar{x}$:

$$\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i$$

For an entire finite population of size $N$, the population mean is denoted $\mu$:

$$\mu = \frac{1}{N}\sum_{i=1}^N x_i$$

When values carry differential weights $w_i$, the weighted mean is computed as $\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$.

1. Median: The median, denoted $M$ or $\tilde{x}$, is the 50th percentile ($Q_2$), bisecting an ordered dataset into two subsets of equal cardinality. Sorting the observations such that $x_{(1)} \le x_{(2)} \le \dots \le x_{(n)}$:

• If $n$ is odd, the median is the unique central observation at position $\frac{n+1}{2}$:

$$M = x_{\left(\frac{n+1}{2}\right)}$$

• If $n$ is even, the median is the arithmetic mean of the two central values at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$:

$$M = \frac{x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2} + 1\right)}}{2}$$

1. Mode: The mode represents the observation occurring with the highest frequency. Distributions may be unimodal (one distinct peak), bimodal (two equal peaks), multimodal (three or more peaks), or possess no mode if all observations appear with equal frequency. While mean and median require numerical interval or ratio data, the mode is the sole measure of central tendency applicable to nominal categorical data.

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Measures of Dispersion and Bessel's Correction

Central tendency describes where data cluster, but dispersion quantifies the degree of spread or variability among observations:

1. Range: The simplest spread metric is the difference between extremes:

$$\text{Range} = x_{\max} - x_{\min}$$

Because range depends solely on the two extreme endpoints, it ignores internal distribution structure and is highly sensitive to outliers.

1. Interquartile Range ($IQR$): The distance spanned by the central 50% of the distribution, defined as the difference between the third quartile ($Q_3$, 75th percentile) and the first quartile ($Q_1$, 25th percentile):

$$IQR = Q_3 - Q_1$$

1. Population Variance ($\sigma^2$) and Standard Deviation ($\sigma$): For a known population of size $N$ with mean $\mu$, the parameter variance is the mean squared deviation:

$$\sigma^2 = \frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2, \quad \sigma = \sqrt{\sigma^2}$$

1. Sample Variance ($s^2$) and Sample Standard Deviation ($s$): For a sample of size $n$, the statistic is computed using $n - 1$ in the denominator:

$$s^2 = \frac{1}{n - 1}\sum_{i=1}^n (x_i - \bar{x})^2, \quad s = \sqrt{s^2}$$

The algebraic shortcut formula for manual computation is:

$$s^2 = \frac{1}{n - 1}\left[ \sum_{i=1}^n x_i^2 - \frac{\left(\sum_{i=1}^n x_i\right)^2}{n} \right]$$

The Mathematical Necessity of Bessel's Correction ($n - 1$)

When estimating population variance from a sample, deviations are measured from the sample mean $\bar{x}$ rather than the unobservable true population mean $\mu$. Because $\bar{x}$ minimizes the sum of squared deviations for that specific sample ($\sum (x_i - \bar{x})^2 < \sum (x_i - \mu)^2$ whenever $\bar{x} \neq \mu$), dividing by $n$ systematically underestimates $\sigma^2$.

Analytically, the expected value of the uncorrected sample variance satisfies:

$$\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2\right] = \frac{n-1}{n}\sigma^2$$

Dividing by the degrees of freedom $n - 1$ rather than $n$—an adjustment known as Bessel's correction—counteracts this downward bias:

$$\mathbb{E}[s^2] = \mathbb{E}\left[\frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2\right] = \sigma^2$$

Thus, $s^2$ is an unbiased estimator of $\sigma^2$.

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Statistical Resistance to Extreme Outliers

A descriptive statistic is termed resistant (or robust) if its numerical value remains relatively unaffected by extreme outliers or heavy skewness.

• Non-Resistant Statistics: The arithmetic mean $\bar{x}$, range, sample variance $s^2$, and standard deviation $s$ are non-resistant. In the mean, every observation enters additively into $\sum x_i$, so an extreme value pulls $\bar{x}$ dramatically toward the tail. In variance and standard deviation, deviations $(x_i - \bar{x})$ are squared, magnifying the influence of extreme values exponentially.
• Resistant Statistics: The median $M$ and interquartile range $IQR$ are resistant. The median depends solely on the ordinal rank of data points; changing the highest score in a dataset from $100$ to $10{,}000$ alters neither the median nor the quartiles $Q_1$ and $Q_3$.

Descriptive Selection Guidelines

• Symmetric, mound-shaped distributions: Report the mean $\bar{x}$ and standard deviation $s$. The mean efficiently utilizes every quantitative data value, and standard deviation connects directly to the normal distribution model.
• Skewed distributions or distributions with outliers: Report the five-number summary (median $M$ and $IQR$). These resistant metrics accurately reflect the typical center and core spread without distortion from extreme values.

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Central Tendency and Dispersion Comparison Table

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Linear Transformations of Univariate Data

A linear transformation maps an original random variable $X$ to a new variable $Y$ via the linear function:

$$y_i = a x_i + b$$

where $a$ is a multiplicative scale factor and $b$ is an additive shift constant.

1. Effect of Adding a Constant $b$ ($y = x + b$)

Adding a constant $b$ shifts every data point along the number line by $b$ units without altering the relative spacing between points:

• Measures of Center and Position: Increased by $b$: $$\bar{y} = \bar{x} + b, \quad M_y = M_x + b, \quad Q_{1,y} = Q_{1,x} + b, \quad Q_{3,y} = Q_{3,x} + b$$
• Measures of Spread: Completely unchanged: $$\text{Range}_y = \text{Range}_x, \quad IQR_y = IQR_x, \quad s_y = s_x, \quad s_y^2 = s_x^2$$

2. Effect of Multiplying by a Constant $a$ ($y = a x$)

Multiplying every data point by $a$ stretches or compresses the distribution relative to the origin:

• Measures of Center and Position: Multiplied by $a$: $$\bar{y} = a\bar{x}, \quad M_y = a M_x, \quad \text{Mode}_y = a\text{Mode}_x$$
• Measures of Spread: Multiplied by the absolute value $|a|$: $$\text{Range}_y = |a|\text{Range}_x, \quad IQR_y = |a|IQR_x, \quad s_y = |a|s_x$$ $$\text{Variance: } s_y^2 = a^2 s_x^2$$

Combined Linear Transformation Theorem

For any linear transformation $Y = aX + b$:

$$\bar{Y} = a\bar{X} + b, \quad M_Y = a M_X + b$$ $$s_Y = |a|s_X, \quad IQR_Y = |a|IQR_X, \quad s_Y^2 = a^2 s_X^2$$

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Worked Exemplar: Linear Data Transformation Analysis

Problem: A high school mathematics teacher administers an algebra assessment to a class of 28 students. The raw exam scores exhibit a sample mean of $\bar{x} = 62.0$, a median of $M = 60.0$, a sample standard deviation of $s = 8.0$, and an interquartile range of $IQR = 12.0$. To adjust for exam difficulty, the teacher applies a curved scaling formula: each score is multiplied by $1.2$ and then increased by $10$ points, represented by $y_i = 1.2 x_i + 10$. Calculate the transformed mean $\bar{y}$, transformed median $M_y$, transformed standard deviation $s_y$, transformed variance $s_y^2$, and transformed $IQR_y$.

• Step 1: Identify Transformation Parameters. The linear transformation is $y = ax + b$ with multiplicative factor $a = 1.2$ and additive constant $b = 10$.

• Step 2: Transform Measures of Central Tendency. Both mean and median are linear measures of center; therefore, apply the full transformation $a(\cdot) + b$:

  $$\bar{y} = 1.2\bar{x} + 10 = 1.2(62.0) + 10 = 74.4 + 10 = 84.4$$ $$M_y = 1.2 M + 10 = 1.2(60.0) + 10 = 72.0 + 10 = 82.0$$

• Step 3: Transform Measures of Dispersion. Measures of spread are influenced solely by the multiplicative scale factor $|a| = |1.2| = 1.2$. The additive constant $b = 10$ has zero impact on dispersion:

  $$s_y = |a|s = 1.2(8.0) = 9.6$$ $$IQR_y = |a|IQR = 1.2(12.0) = 14.4$$

• Step 4: Compute Transformed Sample Variance. Variance scales by the square of the multiplicative constant, $a^2$:

  $$s_x^2 = (8.0)^2 = 64.0$$ $$s_y^2 = a^2 s_x^2 = (1.2)^2(64.0) = 1.44(64.0) = 92.16$$

  Alternatively, squaring the transformed standard deviation yields:

  $$s_y^2 = (9.6)^2 = 92.16$$

  Notice that $s_y^2 \neq 1.2(64.0) + 10$; neither the additive constant 10 nor a simple linear multiple applies to variance.