Descriptive statistics - Mathematics: Analysis & Approaches IB Study Notes

Descriptive statistics summarizes and organizes data to reveal patterns without making inferences beyond the dataset. Understanding measures of central tendency and dispersion is fundamental.

Measures of Central Tendency

Mean (μ or x̄): The arithmetic average. Formula: $\bar{x} = \frac{\sum x_i}{n}$ where $x_i$ represents each data value and $n$ is the sample size. For grouped data: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ where $f_i$ is frequency and $x_i$ is the midpoint.

Median: The middle value when data is ordered. For $n$ values: position = $\frac{n+1}{2}$. If even, average the two middle values.

Mode: The most frequently occurring value(s). Data can be unimodal, bimodal, or multimodal.

Measures of Dispersion

Range: Maximum value minus minimum value. Simple but sensitive to outliers.

Interquartile Range (IQR): $Q_3 - Q_1$, representing the middle 50% of data. More robust than range.

Variance ($\sigma^2$ or $s^2$): Average squared deviation from mean. Population: $\sigma^2 = \frac{\sum(x_i - \mu)^2}{N}$; Sample: $s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}$

Standard Deviation (σ or s): Square root of variance, in original units. Formula: $s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$

Key insight: Standard deviation measures typical distance from the mean. A smaller σ indicates data clustered closely around the mean.

Quartiles: $Q_1$ (25th percentile), $Q_2$ (median), $Q_3$ (75th percentile) divide ordered data into four equal parts.

Descriptive statistics transforms raw data into meaningful insights across countless applications.

Financial Analysis Example

A fund manager analyzes monthly returns: {2.1%, 3.5%, -1.2%, 4.8%, 2.9%, 3.1%, 5.2%, 1.8%, 2.5%, 3.3%}. The mean return (2.8%) tells investors the average performance, while standard deviation (1.89%) quantifies volatility—crucial for risk assessment. A high σ indicates unpredictable returns; conservative investors prefer low σ portfolios.

Medical Research Application

In clinical trials measuring patient recovery times, the median proves more informative than the mean because outliers (patients with complications) skew averages. If recovery times are {5, 6, 6, 7, 8, 23} days, mean = 9.2 days but median = 6.5 days—the median better represents typical recovery.

Quality Control Analogy

Think of standard deviation as a quality control inspector. Manufacturing bolts with target diameter 10mm: if σ = 0.05mm, production is precise (bolts consistently near 10mm). If σ = 0.5mm, production is erratic (diameters vary wildly).

Educational Assessment

Exam scores with mean = 65% and IQR = 20% reveal that the middle 50% of students scored within a 20-point range. If $Q_1$ = 55% and $Q_3$ = 75%, approximately half the class scored between these values. Outliers below $Q_1 - 1.5×IQR$ or above $Q_3 + 1.5×IQR$ identify struggling or exceptional students.

Real-world principle: Choice of statistic depends on data distribution. Symmetric data → use mean and σ; Skewed data or outliers → use median and IQR.