5.4 Descriptive Statistics: Central Tendency, Dispersion & Frequency Distributions

5.4 Descriptive Statistics: Central Tendency, Dispersion & Frequency Distributions

Descriptive statistics synthesizes raw quantitative data into standardized summary metrics that capture central location, spread, and distributional shape. On the Bocconi Admission Test, statistics problems test algebraic relationships, data transformation theorems, and comparative distribution properties under strict time constraints, rather than demanding extensive tabular arithmetic. Candidates must master the distinct behavior of means versus medians in skewed distributions, the shortcut variance formula, and the linear transformation invariance rules that dictate how scaling and shifting affect statistical parameters.

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1. Measures of Central Tendency

Measures of central tendency locate the single representative value around which numerical observations cluster.

Arithmetic Mean

The arithmetic mean (sample mean $\bar{x}$, population mean $\mu$) of $n$ observations $x_1, x_2, \dots, x_n$ is the sum of all values divided by $n$: $\mu = \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$

• Zero-Sum Deviation Property: The sum of signed deviations from the mean is always identically zero: $\sum_{i=1}^n (x_i - \bar{x}) = 0$
• Least Squares Property: The sum of squared deviations $\sum_{i=1}^n (x_i - c)^2$ is strictly minimized when $c = \bar{x}$.

Weighted Arithmetic Mean

When observations carry different relative importance or frequencies $w_i$ (or $f_i$): $\bar{x}_w = \frac{\sum_{i=1}^k w_i x_i}{\sum_{i=1}^k w_i}$

• Combined Subgroup Means: If $m$ cohorts have sample sizes $n_1, n_2, \dots, n_m$ and means $\bar{x}_1, \bar{x}_2, \dots, \bar{x}_m$, the combined overall mean is: $\bar{x}_{\text{comb}} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2 + \dots + n_m \bar{x}_m}{n_1 + n_2 + \dots + n_m}$ Exam Warning: The unweighted simple average $\frac{\bar{x}_1 + \bar{x}_2}{2}$ is valid only if $n_1 = n_2$.

Median

The median (denoted $\text{Med}$ or $Q_2$) is the value separating the ordered data into two equal halves (50% $\le \text{Med}$ and 50% $\ge \text{Med}$).

Algorithm for Discrete Datasets:

1. Sort the $n$ observations in non-decreasing order: $x_{(1)} \le x_{(2)} \le \dots \le x_{(n)}$.
2. If $n$ is odd: The median is the unique middle observation at rank $\frac{n + 1}{2}$: $\text{Med} = x_{\left(\frac{n + 1}{2}\right)}$
3. If $n$ is even: The median is the arithmetic mean of the two central observations at ranks $\frac{n}{2}$ and $\frac{n}{2} + 1$: $\text{Med} = \frac{x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2} + 1\right)}}{2}$

• Robustness Against Outliers: The median is resistant to extreme values. Replacing the largest observation with an arbitrarily large number does not alter the median, whereas it inflates the arithmetic mean significantly.

Mode

The mode ($\text{Mod}$) is the observation occurring with the highest frequency.

• Unimodal: A single value attains the maximum frequency.
• Bimodal / Multimodal: Two or more distinct values tie for the highest frequency.
• No Mode: All values appear with equal frequency.

Distribution Symmetry and Skewness

The relative positioning of mean, median, and mode reflects the directional skewness of a distribution:

• Symmetric Distribution (Bell-shaped): $\text{Mean} \approx \text{Median} \approx \text{Mode}$
• Positively Skewed (Right-Skewed, long right tail): Extreme high outliers pull the mean upward: $\text{Mode} < \text{Median} < \text{Mean}$
• Negatively Skewed (Left-Skewed, long left tail): Extreme low values pull the mean downward: $\text{Mean} < \text{Median} < \text{Mode}$

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2. Measures of Dispersion (Variability)

Measures of dispersion quantify the extent to which data points spread out from the central value.

Range

$R = x_{\max} - x_{\min}$ While fast to calculate, the range depends exclusively on two extreme values and ignores all intermediate distribution structure.

Population Variance ($\sigma^2$)

The variance measures the average squared deviation from the mean: $\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2$

The Computational Shortcut for Variance

Calculating deviations $(x_i - \mu)$ for every data point is slow and error-prone. Instead, apply the algebraic shortcut identity: $\sigma^2 = \frac{\sum_{i=1}^n x_i^2}{n} - \mu^2 = \overline{x^2} - (\bar{x})^2$ Verbal Rule: 'The mean of the squares minus the square of the mean.' This formula computes variance in a fraction of the time on the Bocconi test.

Standard Deviation ($\sigma$)

The standard deviation is the non-negative square root of the variance: $\sigma = \sqrt{\sigma^2} = \sqrt{\frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2}$ Because it shares the original units of measurement of $x$ (unlike variance, which is in squared units), standard deviation is the universal benchmark for risk and dispersion.

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3. Linear Transformations of Statistical Data

When every observation in a dataset undergoes a linear transformation $y_i = a \cdot x_i + b$ (where $a, b \in \mathbb{R}$):

Mathematical Proof of Dispersion Shift Invariance

Why does additive shift $b$ vanish from variance? $\sigma_y^2 = \frac{1}{n} \sum_{i=1}^n (y_i - \bar{y})^2 = \frac{1}{n} \sum_{i=1}^n [(a x_i + b) - (a \bar{x} + b)]^2 = \frac{1}{n} \sum_{i=1}^n [a(x_i - \bar{x})]^2 = a^2 \left[\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2\right] = a^2 \sigma_x^2$ The constant $b$ cancels identically inside the deviation brackets. Adding a constant translates all points equally without altering relative distances.

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4. Frequency Distributions & Grouped Data

When data is presented in frequency tables:

• Absolute Frequency ($f_i$): Count of occurrences of value $x_i$, with $\sum f_i = n$.
• Relative Frequency ($p_i$): Proportion $p_i = \frac{f_i}{n}$, satisfying $\sum p_i = 1$.
• Cumulative Frequency ($F_k$): Running sum $F_k = \sum_{i=1}^k f_i$.

Joint, Marginal and Conditional Frequencies (Two-Way Tables)

Bocconi's statistics syllabus asks candidates to "derive absolute and relative frequencies, conjoint and conditioned frequency distributions given in tables or diagrams." A two-way (contingency) table cross-classifies the same units by two characteristics at once, and each of the three frequency types answers a different question.

Take 200 Bocconi applicants classified by intended area and by whether they took a quantitative secondary-school track:

• Joint (conjoint) frequency $f_{ij}$ — the count in a single interior cell. The joint relative frequency divides by the grand total: $p_{ij} = f_{ij} / n$. Economics applicants from a quantitative track: $72 / 200 = 36.0%$ of all applicants.
• Marginal frequency — a row or column total, which recovers the one-variable distribution. The marginal relative frequency of Management is $80 / 200 = 40.0%$.
• Conditional frequency — a cell divided by the total of the group you are conditioning on, so the row (or column) sums to $100%$. Among Management applicants, the conditional relative frequency of a quantitative track is $36 / 80 = 45.0%$. Among quantitative-track applicants, the conditional relative frequency of Management is $36 / 114 \approx 31.6%$.

The Direction Trap: those last two numbers, $45.0%$ and $31.6%$, describe the same 36 people and answer different questions. "What percentage of Management applicants took a quantitative track?" conditions on the row; "what percentage of quantitative-track applicants chose Management?" conditions on the column. Identify the denominator before touching the arithmetic — reversing it is the most common error on two-way table items, and it is the same inversion fallacy that appears in conditional probability in Section 5.3.

Independence check. Two characteristics are statistically independent when every conditional distribution equals the corresponding marginal distribution. Here the marginal share of quantitative-track applicants is $114 / 200 = 57.0%$, but the conditional share within Economics & Finance is $72 / 100 = 72.0%$ and within International Politics only $6 / 20 = 30.0%$. The conditional distributions differ sharply from the marginal, so area of interest and school track are not independent in this cohort.

Grouped Data Approximations

When observations are grouped into continuous intervals $[L_i, U_i)$:

1. Compute the class midpoint: $m_i = \frac{L_i + U_i}{2}$.
2. Approximate the grouped mean: $\bar{x} \approx \frac{\sum f_i m_i}{\sum f_i}$.
3. Approximate the grouped variance: $\sigma^2 \approx \frac{\sum f_i m_i^2}{\sum f_i} - \bar{x}^2$.

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5. Quartiles, Percentiles & Outlier Detection

Quartiles

• First Quartile ($Q_1$ / 25th Percentile): Median of the lower half of data; 25% of observations fall below $Q_1$.
• Second Quartile ($Q_2$ / 50th Percentile): The overall median $\text{Med}$.
• Third Quartile ($Q_3$ / 75th Percentile): Median of the upper half of data; 75% of observations fall below $Q_3$.

Interquartile Range (IQR)

$\text{IQR} = Q_3 - Q_1$ The IQR spans the middle 50% of observations and provides an outlier-resistant measure of dispersion.

Tukey's 1.5 $\times$ IQR Outlier Detection Rule

In boxplot analysis, fences identify anomalous observations:

• Lower Outlier Fence: $\text{LF} = Q_1 - 1.5 \cdot \text{IQR}$
• Upper Outlier Fence: $\text{UF} = Q_3 + 1.5 \cdot \text{IQR}$ Any data point $x < \text{LF}$ or $x > \text{UF}$ is classified as an outlier.

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6. Common Bocconi Traps & Fast Shortcuts

1. Averaging Averages Without Weights: Never compute $\frac{\bar{x}_1 + \bar{x}_2}{2}$ when combining classes of different sizes. Always multiply each mean by its cohort size, sum the totals, and divide by the combined total $n$.
2. The Negative Standard Deviation Trap: When transforming $y = -3x + 10$, candidate answers frequently state $\sigma_y = -3\sigma_x$. Standard deviation is always non-negative: $\sigma_y = |-3|\sigma_x = 3\sigma_x$.
3. Assuming Shift Alters Spread: Adding a flat bonus of 10 points to every student's score increases the mean by 10 points, but changes the variance and standard deviation by exactly zero.
4. Failure to Use the Computational Shortcut: On discrete sets like ${1, 3, 5, 7, 9}$, do not compute $(-4)^2 + (-2)^2 + \dots$. Directly evaluate $\frac{1 + 9 + 25 + 49 + 81}{5} - 5^2 = \frac{165}{5} - 25 = 33 - 25 = 8$.

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7. Worked Step-by-Step Examples

Example 1: Multi-Cohort Combined Average

In a business analytics department, Section 1 has 25 students with an average exam grade of 80, Section 2 has 35 students with an average of 88, and Section 3 has 40 students with an average of 75. What is the combined weighted average of all 100 students?

1. Calculate Total Score Points per Section:
   • Section 1: $25 \times 80 = 2000$
   • Section 2: $35 \times 88 = 3080$
   • Section 3: $40 \times 75 = 3000$
2. Sum All Points and Total Sample Size: $\text{Total Points} = 2000 + 3080 + 3000 = 8080$ $N = 25 + 35 + 40 = 100$
3. Compute Weighted Average: $\bar{x}_{\text{comb}} = \frac{8080}{100} = 80.8$

Example 2: Fast Variance Evaluation Using Shortcut Formula

Compute the population variance $\sigma^2$ and standard deviation $\sigma$ for the dataset $S = {2, 4, 6, 8, 10}$.

1. Compute Arithmetic Mean: $\mu = \frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6$
2. Compute Sum of Squares: $\sum x_i^2 = 2^2 + 4^2 + 6^2 + 8^2 + 10^2 = 4 + 16 + 36 + 64 + 100 = 220$
3. Apply Computational Variance Identity: $\sigma^2 = \frac{\sum x_i^2}{n} - \mu^2 = \frac{220}{5} - 6^2 = 44 - 36 = 8$
4. Compute Standard Deviation: $\sigma = \sqrt{8} = 2\sqrt{2} \approx 2.828$

Example 3: Comprehensive Linear Transformation

A distribution of raw employee productivity scores $x$ has mean $\mu = 50$, median $\text{Med} = 48$, variance $\sigma^2 = 16$, and standard deviation $\sigma = 4$. If scores are converted via $y = -2x + 15$, find the new statistical metrics.

1. Transformed Mean: $\mu_y = -2(50) + 15 = -100 + 15 = -85$.
2. Transformed Median: $\text{Med}_y = -2(48) + 15 = -96 + 15 = -81$.
3. Transformed Variance: $\sigma_y^2 = (-2)^2 \cdot \sigma_x^2 = 4 \times 16 = 64$.
4. Transformed Standard Deviation: $\sigma_y = |-2| \cdot \sigma_x = 2 \times 4 = 8$.