Preview
Measures of Central Tendency and Dispersion
IB DP Mathematics: Applications and Interpretation | SL & HL | Statistics and Probability — Topic 4.3
📋 Learning Intentions
- Calculate the mean, median, and mode from raw data, frequency tables, and grouped frequency tables.
- Find the interquartile range (IQR) and standard deviation using the GDC and interpret these values in context.
- Identify which measure of central tendency and which measure of dispersion are most appropriate for a given data set.
- Interpret the standard deviation and IQR as measures of spread, explaining what a larger or smaller value means in context.
- Determine the effect of outliers on the mean, median, and standard deviation.
- Use the GDC (TI-84 Plus CE) to compute all summary statistics from raw data and frequency tables efficiently.
Real-World Introduction
A school reports that the "average" student scores 72% on a mathematics exam. But does this tell the whole story? If one student scores 20% and another scores 98%, the average of 59% tells us almost nothing about the typical student's performance.
Choosing the right summary statistic — and understanding how spread out the data is — is the foundation of all statistical analysis. Epidemiologists use median household income (not mean) to describe wealth distribution because a few billionaires would distort the mean. Sports analysts use standard deviation to measure consistency of athletic performance.
🌍 Why This Matters The United Nations Human Development Index uses median income rather than mean income to compare living standards across countries, precisely because extreme values distort the mean. Understanding when to use each measure is a skill tested directly in IB exams and used daily in data-driven professions.
Core Concepts & Definitions
Before computing any summary statistic, it is essential to understand what each one measures and what it is sensitive to. Measures of central tendency describe where the "centre" of a data set lies. Measures of dispersion describe how spread out the values are around that centre.
Concept 1 — The Mean
The mean is the most commonly used measure of central tendency. It uses every data value in its calculation, which makes it powerful but also sensitive to extreme values (outliers). When a data set is roughly symmetric and contains no outliers, the mean is the preferred measure.
📖 Key Definition — Mean The mean \(\bar{x}\) of a data set is the sum of all values divided by the number of values: \[ \bar{x} = \frac{\sum x}{n} \] For a frequency distribution: \(\bar{x} = \frac{\sum f \cdot x}{\sum f}\), where \(f\) is the frequency of each value \(x\). For grouped data, use the midpoint of each class interval as \(x\).
The mean is the "balance point" of the