15.1 Descriptive Statistics, Measures of Central Tendency, Variability, and Normal Distributions

Descriptive Statistics, Measures of Central Tendency, Variability, and Normal Distributions

Descriptive statistics organize, summarize, and communicate numerical data collected from psychological investigations. Before applying any statistical test, researchers must identify the scale of measurement of their variables, determine the central tendencies and dispersion of their observations, and evaluate the underlying shape of the distribution.

1. S.S. Stevens' Scales of Measurement

In 1946, psychophysicist Stanley Smith Stevens published a landmark taxonomy in Science categorizing measurement into four distinct hierarchical levels: nominal, ordinal, interval, and ratio. Each scale possesses specific mathematical properties that dictate which statistical operations and transformations are permissible.

The Critical Distinction: Interval vs. Ratio Scales

A perennial testing point on the GRE Subject Test in Psychology is the distinction between an arbitrary zero and an absolute zero:

• Arbitrary Zero (Interval): On an interval scale, zero is simply an agreed-upon convention. In Celsius, $0^\circ\text{C}$ represents the freezing point of water, not the physical absence of thermal energy. Consequently, one cannot claim that $40^\circ\text{C}$ is "twice as hot" as $20^\circ\text{C}$. Similarly, an individual with a Wechsler IQ of 140 does not possess "twice the intelligence" of someone with an IQ of 70, nor does an IQ score of 0 represent zero intellectual capacity.
• Absolute Zero (Ratio): On a ratio scale, zero signifies total absence. A participant who takes 400 milliseconds to respond to a visual stimulus took objectively twice as long as a participant responding in 200 milliseconds ($400 / 200 = 2.0$). Ratios of measurement are mathematically valid only when the scale originates from an absolute zero.

2. Measures of Central Tendency

Measures of central tendency identify a single representative numerical value around which observations cluster.

The Mode

The mode is the most frequently occurring score in a distribution.

• Characteristics: A distribution may have a single mode (unimodal), two modes (bimodal), multiple modes (multimodal), or no mode if all scores occur with equal frequency.
• Unique Advantage: The mode is the only measure of central tendency that can be used with nominal data (e.g., identifying the most common psychiatric diagnosis in an inpatient clinic).
• Limitation: It ignores the numerical value of most observations in a quantitative dataset and can fluctuate markedly between samples.

The Median

The median ($Mdn$) is the 50th percentile—the middle value that bisects a rank-ordered distribution such that exactly 50% of scores fall at or below it and 50% fall at or above it.

• Computation:
  • For an odd sample size $N$, the median is the observation at rank $\frac{N + 1}{2}$.
  • For an even sample size $N$, the median is the arithmetic mean of the two central observations at ranks $\frac{N}{2}$ and $\frac{N}{2} + 1$.
• Mathematical Property: The median minimizes the sum of absolute deviations across all observations:

$\sum_{i=1}^N |X_i - Mdn| = \text{minimum}$

• Robustness: The median is resistant to extreme scores (outliers). Whether the highest score in a sample of five reaction times ($210, 215, 220, 225, 950\text{ ms}$) is $230\text{ ms}$ or $950\text{ ms}$, the median remains unchanged at $220\text{ ms}$. Hence, the median is the preferred measure of central tendency for heavily skewed distributions (e.g., household income, housing prices, response latencies).

The Arithmetic Mean

The arithmetic mean (sample mean $\bar{X}$ or population mean $\mu$) is the sum of all scores divided by the total number of observations:

$\bar{X} = \frac{\sum_{i=1}^N X_i}{N}$

• Foundational Algebraic Properties:
  1. Sum of Deviations Equals Zero: The sum of signed deviations of every individual score from the arithmetic mean is always identically zero:

$\sum_{i=1}^N (X_i - \bar{X}) = 0$

1. Least Squares Criterion: The mean minimizes the sum of squared deviations across all scores. No other constant $C$ produces a smaller value for $\sum (X_i - C)^2$:

$\sum_{i=1}^N (X_i - \bar{X})^2 = \text{minimum}$

• Sensitivity to Outliers: Because the mean incorporates the precise numerical value of every score, it is highly sensitive to extreme scores. A single outlier in the tail pulls the mean toward itself, rendering it misleading in skewed distributions.

3. Measures of Variability (Dispersion)

Central tendency captures the midpoint of a dataset, but variability describes how dispersed or tightly clustered the scores are around that center.

Range and Interquartile Range

• Range: The crude distance between the extreme scores:

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

• While intuitive, the range is highly unstable because it is determined entirely by two extreme scores and generally increases as sample size grows.
• Interquartile Range ($IQR$): The range of the middle 50% of rank-ordered observations, defined as the distance between the 75th percentile (third quartile, $Q_3$) and the 25th percentile (first quartile, $Q_1$):

$IQR = Q_3 - Q_1$

• The semi-interquartile range is defined as $\frac{Q_3 - Q_1}{2}$.
• In box-and-whisker plots (boxplots), the box spans the $IQR$, the interior line indicates the median, and whiskers typically extend to $1.5 \times IQR$ beyond the quartiles. Observations outside this boundary are identified as outliers.

Variance and Standard Deviation

To quantify the dispersion of all individual data points around the mean, statisticians compute the Sum of Squared Deviations ($SS$):

$SS = \sum_{i=1}^N (X_i - \bar{X})^2$

Population Variance vs. Sample Variance (Bessel's Correction)

When calculating the variance of an entire population of size $N$, the parameter $\sigma^2$ is computed as:

$\sigma^2 = \frac{\sum_{i=1}^N (X_i - \mu)^2}{N}$

However, when estimating the population variance from a sample of size $N$, dividing by $N$ produces a biased estimator that systematically underestimates the true population variance. Because sample observations cluster more closely around the sample mean $\bar{X}$ than around the true population mean $\mu$, the sum of squared deviations from $\bar{X}$ is smaller than the sum of squared deviations from $\mu$.

To correct for this negative bias, German astronomer Friedrich Bessel introduced Bessel's correction, replacing $N$ with degrees of freedom ($df = N - 1$) in the denominator of the sample variance ($s^2$):

$s^2 = \frac{SS}{N - 1} = \frac{\sum_{i=1}^N (X_i - \bar{X})^2}{N - 1}$

Dividing by $N - 1$ renders $s^2$ an unbiased estimator, meaning that the expected value of the sample variance across repeated random samples equals the population parameter: $E(s^2) = \sigma^2$.

Degrees of Freedom (df = N - 1) Concept:
If N = 4 scores must sum to a Mean of 10 (Total Sum = 40):
Score 1 = 8
Score 2 = 12
Score 3 = 11
Score 4 = MUST BE 9  <-- Exactly 1 score is mathematically constrained!
Only N - 1 = 3 scores are free to vary.

Standard Deviation

Because variance is expressed in squared units of measurement (e.g., $\text{milliseconds}^2$ or $\text{errors}^2$), taking the positive square root returns the metric of dispersion to the original scale of measurement:

$\text{Population Standard Deviation: } \sigma = \sqrt{\frac{\sum (X_i - \mu)^2}{N}}$

$\text{Sample Standard Deviation: } s = \sqrt{\frac{\sum (X_i - \bar{X})^2}{N - 1}}$

4. Distribution Shapes: Skewness and Kurtosis

Distributions in psychological research vary along two primary dimensions of shape: symmetry (skewness) and peakedness/tail weight (kurtosis).

Skewness (Asymmetry)

• Normal / Symmetric Distribution: Skewness is zero. The distribution can be folded in half along the median to produce mirror images. In a perfectly symmetric unimodal distribution:

$\text{Mean} = \text{Median} = \text{Mode}$

• Positive Skew (Right-Skewed): The tail of the distribution extends asymptotically toward the positive (right) end of the horizontal axis. A cluster of low scores is balanced by a small number of extraordinarily high scores.
  • Ordering of Central Tendencies: The mode remains at the peak of the cluster, the median is pulled slightly rightward, and the mean is pulled farthest into the long tail:

$\text{Mode} < \text{Median} < \text{Mean}$

• Psychological Examples: Annual income, reaction times (participants cannot respond faster than physiological limits, but inattentive trials produce extreme positive latencies), number of depressive symptoms endorsed in non-clinical student samples.
• Negative Skew (Left-Skewed): The tail extends toward the negative (left) end of the horizontal axis. A cluster of high scores is accompanied by a small tail of very low scores.
  • Ordering of Central Tendencies: The mean is pulled farthest toward the negative tail, followed by the median, while the mode sits at the high peak:

$\text{Mean} < \text{Median} < \text{Mode}$

• Psychological Examples: Age at death in modern industrialized nations, scores on an exceptionally easy exam that produces a ceiling effect.

Positive Skew (Right-Skewed):          Negative Skew (Left-Skewed):
        Mode                                         Mode
         /\                                           /\
        /  \                                         /  \
       /    \                                       /    \
      /      \                                     /      \
     /  Median\                                   / Median \
    /          \                                 /          \
   /     Mean   \                               /   Mean     \
  /              \________                     ________/              \
  ------------------------>                     ------------------------>
  Mode < Median < Mean                         Mean < Median < Mode
  (Tail points Right)                          (Tail points Left)

Kurtosis (Peakedness and Tail Weight)

Kurtosis reflects the heaviness of a distribution's tails and the sharpness of its central peak relative to a standard normal curve:

• Mesokurtic: A normal distribution with zero excess kurtosis ($K = 0$).
• Leptokurtic ($K > 0$): Characterized by a sharp, elevated central peak and fat, heavy tails. More observations are concentrated near the mean and in the extreme tails than in a normal distribution, with fewer in the intermediate shoulders.
• Platykurtic ($K < 0$): Characterized by a broad, flat central peak and thin, light tails. Observations are dispersed more evenly across the range.

5. The Normal Distribution and Standardized Scores

The normal distribution (Gaussian distribution) is a continuous, symmetrical, mesokurtic bell-shaped distribution mathematically defined by two parameters: its mean ($\mu$) and standard deviation ($\sigma$). Its inflection points, where the curve transitions from convex to concave, occur at exactly $\mu \pm 1\sigma$.

The Empirical Rule (68–95–99.7 Rule)

In any normal distribution, fixed proportions of observations fall within integer standard deviation intervals from the mean:

• $\mu \pm 1\sigma$: Encompasses approximately $68.26\%$ of all scores ($34.13\%$ between the mean and $+1\sigma$, and $34.13\%$ between the mean and $-1\sigma$).
• $\mu \pm 2\sigma$: Encompasses approximately $95.44\%$ of all scores ($47.72\%$ on either side of the mean; exactly $1.96\sigma$ encompasses $95.00\%$).
• $\mu \pm 3\sigma$: Encompasses approximately $99.74\%$ of all scores ($49.87\%$ on either side of the mean).
• Beyond $\pm 3\sigma$: Only $0.26\%$ of observations lie beyond three standard deviations ($0.13\%$ in each extreme tail).

                   Standard Normal Distribution Areas

                                  Mean
                                   │
                        34.13%     │     34.13%
                     ┌─────────────┼─────────────┐
           13.59%    │             │             │    13.59%
        ┌────────────┤             │             ├────────────┐
  2.14% │            │             │             │            │ 2.14%
 ───────┴────────────┴─────────────┴─────────────┴────────────┴───────
 -3σ         -2σ          -1σ      0           +1σ          +2σ       +3σ
 0.13%                                                              0.13%

Standardized z-Scores

A $z$-score represents the signed distance between an individual raw score ($X$) and the population mean ($\mu$) in units of standard deviation ($\sigma$):

$z = \frac{X - \mu}{\sigma} \quad \text{or} \quad z = \frac{X - \bar{X}}{s}$

• Properties of the $z$-Score Distribution:
  • The mean of a $z$-score distribution is always $\mu_z = 0$.
  • The standard deviation of a $z$-score distribution is always $\sigma_z = 1$.
  • Converting raw scores to $z$-scores does not normalize a non-normal distribution; it is a linear transformation that preserves the exact skewness, kurtosis, and relative distances of the raw distribution.

Percentile Equivalence in the Normal Curve

Because the areas under the standard normal curve are mathematically fixed, $z$-scores map directly to percentile ranks:

• $z = -3.0 \rightarrow 0.13\text{th percentile}$
• $z = -2.0 \rightarrow 2.28\text{th percentile}$
• $z = -1.0 \rightarrow 15.87\text{th percentile}$
• $z = 0.0 \rightarrow 50.00\text{th percentile}$ (the median)
• $z = +1.0 \rightarrow 84.13\text{th percentile}$
• $z = +2.0 \rightarrow 97.72\text{th percentile}$
• $z = +3.0 \rightarrow 99.87\text{th percentile}$

Linear Transformations of Standardized Scores

To eliminate negative values and decimal fractions, psychometricians transform raw $z$-scores into alternative standardized metrics using the general linear formula:

$\text{Standardized Score} = \mu_{\text{new}} + (z \times \sigma_{\text{new}})$