Descriptive Statistics: Central Tendency, Variability, and Parameters

Foundations of Descriptive Statistics and Scale Variables

Scale Variables in Quantitative Research:
- A scale variable consists of true numeric measurements, inherently featuring a broad continuum and wider range of possible values within a dataset.
- In academic and experimental data analyses, statistical tests are conventionally conducted using scale variables as the primary Dependent Variable ( $DV$ ).
Role and Dual Utility of Collected Data:
- Descriptive statistics summarize an entire distribution of values using a concise set of numerical metrics termed "statistics."
- Empirical data gathered for a dependent variable serves two primary analytical functions:
- Descriptive Function: Condensing, organizing, and summarizing the observed scores collected directly from an empirical sample.
- Inferential Function: Extrapolating beyond immediate data to estimate unknown population parameters based upon observed sample statistics.

Measures of Central Tendency

Core Definition and Purpose:
- Measures of central tendency synthesize the overall distribution of data points into a single representative score located near the center of the distribution.
The Mean ( $M$ ):
- Conceptual Definition: The mathematical average representing the center of gravity of all scores in the dataset.
- Mathematical Formula: $M = \frac{\sum X}{N}$
- Component Definitions:
- $X$ : Individual data score/value.
- $\sum X$ : The sum total of all raw scores in the distribution.
- $N$ : Total sample size (number of observed data points).
- $M$ : Arithmetic sample mean.
- Calculation Procedure: Sum every observed numerical value in the dataset and divide the resulting sum by the total count of observations ( $N$ ).
The Median:
- Conceptual Definition: The exact midpoint that divides a rank-ordered dataset into two equal halves.
- Calculation Procedure:
- Arrange all raw scores in ascending rank order from lowest to highest.
- For an odd sample size ( $N$ ), select the single score positioned exactly in the middle.
- For an even sample size ( $N$ ), locate the two middle scores and calculate their arithmetic average.
The Mode:
- Conceptual Definition: The score or value that appears with the greatest frequency within a dataset.
- Modality Properties: A distribution can possess a single mode (unimodal), two distinct modes (bimodal), or several peaks (multimodal).
Comprehensive Worked Example of Central Tendency:
- Raw Dataset ( $N = 8$ ): $5$ , $8$ , $4$ , $7$ , $6$ , $1$ , $9$ , $8$ .
- Mean Calculation: $\sum X = 5 + 8 + 4 + 7 + 6 + 1 + 9 + 8 = 48$ $M = \frac{48}{8} = 6$
- Median Calculation:
- Rank-ordered sequence: $1$ , $4$ , $5$ , $6$ , $7$ , $8$ , $8$ , $9$ .
- The two central values are $6$ and $7$ . $\text{Median} = \frac{6 + 7}{2} = 6.5$
- Mode Calculation: The score occurring with the highest frequency is $8$ (appearing twice).

Central Tendency and Distribution Skewness

Sensitivity of the Mean to Extreme Scores:
- The arithmetic mean incorporates every score in its calculation, making it susceptible to distortion by extreme values or outliers located in the tails of an asymmetrical distribution.
- The mean is systematically pulled in the direction of the elongated tail.
- When a distribution is heavily skewed, the mean loses its representativeness as a central figure, making the median the preferred, robust indicator of central tendency.
- The mode is typically not utilized as the primary index of central tendency for scale variables when distributions contain many distinct numerical values.
Salary Distribution Case Study:
- Observed distribution metrics for professional annual salary (expressed in thousands of dollars):
- Median: $28.58$ (representing $28,580$ dollars).
- Mean: $34.23$ (representing $34,230$ dollars).
- The mean exceeds the median substantially because a small cluster of high earners stretches the distribution tail far to the right.
- In this context, the median of $28.58$ provides a more representative measure of typical earnings than the inflated mean of $34.23$ .

Salary distribution histogram showing positive skewness

Determining Distribution Skewness:
- The direction of skewness is defined by the location of the elongated tail:
- Left-Skewed (Negative Skewness):
  - The elongated tail extends toward lower numerical values on the left.
  - Relative ordering of metrics along the horizontal axis (from left to right): $\text{Mean} < \text{Median} < \text{Mode}$
- Right-Skewed (Positive Skewness):
  - The elongated tail extends toward higher numerical values on the right.
  - Relative ordering of metrics along the horizontal axis (from left to right): $\text{Mode} < \text{Median} < \text{Mean}$

Comparison of left-skewed and right-skewed distributions indicating relative positions of mode, median, and mean

Measures of Variability

Core Concept of Variability:
- Variability quantifies the dispersion, spread, and degree to which individual scores deviate from one another and from the central mean.
The Range:
- Definition: The mathematical difference between the maximum and minimum observed scores in a distribution.
- Formula: $\text{Range} = X_{\text{highest}} - X_{\text{lowest}}$
- The range is expressed as a single scalar value.
Sample Variance ( $SD^2$ ):
- Definition: A metric of total squared distances between every raw score and the sample mean, normalized by the sample size.
- Mathematical Formula: $SD^2 = \frac{\sum (X - M)^2}{N}$
Sample Standard Deviation ( $SD$ ):
- Definition: The standard root-mean-square index measuring the average distance between individual scores and the sample mean.
- Mathematical Formula: $SD = \sqrt{SD^2} = \sqrt{\frac{\sum (X - M)^2}{N}}$
Step-by-Step Calculation of Variance and Standard Deviation:
- Dataset ( $N = 8$ , $M = 6$ ): $5$ , $8$ , $4$ , $7$ , $6$ , $1$ , $9$ , $8$ .
- Calculation Table:

$X$	Deviation Score ( $X - M$ )	Squared Deviation Score ( $(X - M)^2$ )
$5$	$5 - 6 = -1$	$(-1)^2 = 1$
$8$	$8 - 6 = 2$	$(2)^2 = 4$
$4$	$4 - 6 = -2$	$(-2)^2 = 4$
$7$	$7 - 6 = 1$	$(1)^2 = 1$
$6$	$6 - 6 = 0$	$(0)^2 = 0$
$1$	$1 - 6 = -5$	$(-5)^2 = 25$
$9$	$9 - 6 = 3$	$(3)^2 = 9$
$8$	$8 - 6 = 2$	$(2)^2 = 4$

Sum of Raw Scores ( $\sum X$ ): $\sum X = 48$
Sum of Squared Deviations ( $SS = \sum (X - M)^2$ ): $SS = 1 + 4 + 4 + 1 + 0 + 25 + 9 + 4 = 48$
Range Calculation: $\text{Range} = 9 - 1 = 8$
Sample Variance Calculation: $SD^2 = \frac{48}{8} = 6$
Sample Standard Deviation Calculation: $SD = \sqrt{6} \approx 2.45$

Variability and Kurtosis

Nature and Characteristics of Kurtosis:
- Kurtosis characterizes the distribution of data points toward the center relative to the tails, defining the peakedness and tail thickness of a distribution.
- Distributions vary along a continuum from tall, sharply pointed peaks with heavy tails to completely flat, uniform profiles.
Kurtosis Values and Curve Typologies:
- $D = 3$ : Extremely peaked distribution with heavy tails.
- $S = 2$ : Moderately peaked distribution.
- $L = 1.2$ : Mildly peaked distribution.
- $N = 0$ : Standard normal distribution (mesokurtic baseline).
- $C = -0.59376$ : Mildly flattened distribution.
- $W = -1$ : Broad, flattened distribution.
- $U = -1.2$ : Uniform, flat rectangular distribution.

Distribution curves illustrating different degrees of kurtosis

Comparison of distribution shapes illustrating differences in spread and peakedness

Sample Statistics Versus Population Parameters

Epistemological Goals of Empirical Research:
- The goal of empirical science is to deduce generalizable rules and principles that describe existing phenomena and predict future outcomes across an entire target population.
- Exhaustive data collection across an entire population is rarely feasible due to resource, logistical, and boundary constraints.
- Researchers measure a representative subgroup (a sample) to infer characteristics of the larger population, assuming the sample reflects the population in critical aspects.
Distinction Between Statistics and Parameters:
- Sample Statistics: Descriptive values calculated directly from observed sample data.
- Population Parameters: Theoretical or true values describing the complete population, which are rarely measured directly.
Notational Conventions:
- Statistical symbols are italicized in formal research reporting.

Measurement Dimension	Calculated Sample Statistic	Sample Statistic Used to Estimate Population	Actual Population Parameter
Mean	$M$	$M$ or $\mu$	$\mu$
Standard Deviation	$SD$	$s$	$\sigma$
Variance	$SD^2$	$s^2$	$\sigma^2$
Description / Context	Calculated directly from sample data to describe sample characteristics	Calculated with adjustments to estimate true population parameter	Describes population directly; rarely possible to measure directly

Estimating Population Central Tendency from Sample Data

Unbiased Properties of the Sample Mean:
- Although any single sample mean ( $M$ ) fluctuates around the true population mean ( $\mu$ ), repeated random sampling yields a distribution of sample means whose collective average converges upon $\mu$ .
- Consequently, $M$ is an unbiased point estimator for $\mu$ .
Empirical Demonstration Using Cognitive Test Scores:
- Population Baseline: A population of $N = 60$ cognitive test scores ( $X$ ):

Column 1 Scores	Column 2 Scores	Column 3 Scores
$96.67$	$100.05$	$91.38$
$110.68$	$97.80$	$96.14$
$103.15$	$108.34$	$97.42$
$102.72$	$101.35$	$101.16$
$102.57$	$111.00$	$117.27$
$112.89$	$107.44$	$123.48$
$105.13$	$84.44$	$97.63$
$97.22$	$89.75$	$100.29$
$98.52$	$95.63$	$96.87$
$79.88$	$101.29$	$80.11$
$100.00$	$93.71$	$107.08$
$100.89$	$118.57$	$110.52$
$103.69$	$85.39$	$116.20$
$114.54$	$106.80$	$91.22$
$97.38$	$111.41$	$110.68$
$100.42$	$90.26$	$91.85$
$107.64$	$104.76$	$89.73$
$117.73$	$99.20$	$103.58$
$93.19$	$97.59$	$104.15$
$108.95$	$95.35$	$108.14$

True Population Mean: $\mu = 101.4815$
Sampling Procedure: Repeated random extraction of independent samples of size $n = 10$ .
Individual Means Across 11 Random Samples:
- Sample 1: $100.94$
- Sample 2: $101.01$
- Sample 3: $104.44$
- Sample 4: $104.65$
- Sample 5: $99.71$
- Sample 6: $99.44$
- Sample 7: $100.30$
- Sample 8: $99.05$
- Sample 9: $100.18$
- Sample 10: $103.41$
- Sample 11: $103.32$
Mean of the 11 Sample Means: $M_{\text{11 samples}} = 101.4958$
Mean of 21 Sample Means (with additional sampling iterations): $M_{\text{21 samples}} = 101.2845$
Empirical Principle: While each individual sample mean $M$ deviates slightly from $\mu$ , the majority remain close to $\mu$ . After repeated sampling iterations, the mean of the sample means closely matches the true population mean ( $\mu$ ).

Estimating Population Variability from Sample Data

Systematic Bias in Sample Spread:
- A sample distribution mirrors the general shape of its parent population distribution but is smaller and narrower because it contains fewer observations.
- Rare, extreme values found in the tails of the population are less likely to be sampled.
- Consequently, raw sample variability is systematically narrower than the true population variability: $SD < \sigma$ $SD^2 < \sigma^2$
- Directly using sample formulas with $N$ in the denominator systematically underestimates population variability parameters.
Degrees of Freedom Correction ( $N - 1$ ):
- To adjust for this underestimation when inferring population variability, the denominator is reduced from $N$ to $N - 1$ .
- Dividing the sum of squared deviations by a smaller denominator ( $N - 1$ ) scales up the resulting estimate, providing an accurate, unbiased estimation of population variance and standard deviation.
Formula Comparison for Estimating Population Variability:
- 1. Actual Sample Formulas (Purely Descriptive of the Observed Dataset):
- Sample Variance: $SD^2 = \frac{\sum (X - M)^2}{N}$
- Sample Standard Deviation: $SD = \sqrt{\frac{\sum (X - M)^2}{N}}$
- 2. Inferential Sample Formulas (Estimating Population Parameters from a Sample):
- Estimated Population Variance: $s^2 = \frac{\sum (X - M)^2}{N - 1}$
- Estimated Population Standard Deviation: $s = \sqrt{\frac{\sum (X - M)^2}{N - 1}}$
- 3. Population Parameter Formulas (Usable Only When Complete Population Data is Known):
- True Population Variance: $\sigma^2 = \frac{\sum (X - \mu)^2}{N}$
- True Population Standard Deviation: $\sigma = \sqrt{\frac{\sum (X - \mu)^2}{N}}$
Nomenclatural Clarifications:
- Formulas utilizing $N - 1$ in the denominator ( $s^2$ and $s$ ) are frequently labeled "sample variance" and "sample standard deviation" in statistical software and textbooks because sample data is routinely gathered to infer broader population parameters.
- Conversely, formulas dividing by $N$ ( $SD^2$ and $SD$ ) are sometimes referred to as "population variance" and "population standard deviation" formulas because they treat the available dataset as an isolated population rather than as a subset used to estimate a larger population.