Data Representation & Summary Stats — SL 4.1 Data representation and summary statistics (mean, median, quartiles, IQR, standard deviation)

Historical Context & Motivation

Humans have been collecting data for thousands of years — ancient civilizations tracked harvests, populations, and star positions. But raw numbers on their own can be overwhelming. Imagine scrolling through a spreadsheet of 10,000 exam scores: without some way to condense that information, you would have no idea whether the class performed well or poorly. The field of descriptive statistics was developed precisely to solve this problem — turning mountains of raw data into a handful of meaningful numbers and clear visuals that tell a story.

The central question that drives SL 4.1 is deceptively simple: how can we describe an entire data set using just a few numbers and a picture? The answer involves two complementary ideas — measures of central tendency (where is the 'middle' of the data?) and measures of spread (how spread out are the values?). Together with graphical representations, these tools let you compare data sets, spot outliers, and communicate findings clearly.

Core Principles & Definitions

Before diving into formulas, it helps to understand the big ideas that organize this topic. Every data set has a center — a single value that represents the 'typical' item — and a spread — a measure of how much the individual values differ from that center. Understanding both gives you a much richer picture than either one alone. A class that averages 75% on a test could have every student scoring between 70% and 80%, or it could have half scoring 50% and half scoring 100% — the average is the same, but the stories are completely different.

Visual Explanation — The Box-and-Whisker Plot

The box-and-whisker plot (or box plot) is one of the most compact and informative ways to display a data set. It shows the five-number summary — minimum, Q₁, median, Q₃, and maximum — in a single diagram. The 'box' spans from Q₁ to Q₃, so its width represents the IQR. A line inside the box marks the median. The 'whiskers' extend from the box to the smallest and largest values (or to a boundary defined by 1.5 × IQR from each quartile, with anything beyond marked as an outlier).

Looking at the diagram, you can see several things instantly. The left whisker (from 15 to 30) is shorter than the right whisker (from 60 to 67), suggesting the data might be slightly skewed to the left — there are values further below the median than above it. The box itself is relatively wide, indicating moderate variability in the central half of the data. If the median line were exactly centered in the box, the middle 50% of the data would be symmetric. In this case the median is closer to Q₃ than to Q₁, hinting that the lower half of the data is more spread out than the upper half.

Mathematical Framework

Now let's formalize the key formulas you will use in SL 4.1. Your GDC (graphing display calculator) can compute all of these, but understanding what each formula does is essential for interpreting results and answering exam questions that ask you to explain your reasoning.

Understanding Data Shape — Histograms and Distributions

Summary statistics tell you where the center is and how spread out the data is, but they don't tell you everything. A histogram groups data into intervals (called bins or classes) and uses bar heights to show frequency. The shape of the histogram — whether it is symmetric, skewed left, or skewed right — determines which measures of center and spread are most appropriate. For a symmetric distribution, the mean and median are approximately equal, and standard deviation is the natural measure of spread. For a skewed distribution, the median and IQR are preferred because they resist the pull of outliers.

The key insight from this diagram is practical: when you are told that a data set is skewed, you should report the median and IQR rather than the mean and standard deviation. The median sits near the bulk of the data, whereas the mean gets dragged toward the tail. In IB exams, you may be asked to justify your choice of summary statistic — always connect it to the shape of the distribution.

Worked Example — Full Analysis of a Data Set

A teacher records the scores (out of 50) of 12 students on a quiz: 18, 22, 25, 28, 30, 33, 35, 36, 38, 40, 42, 48. Find the mean, median, Q₁, Q₃, IQR, and standard deviation. Then determine whether the mean or median is a better measure of center.

Strengths & Limitations of Each Measure

Each summary statistic has strengths and weaknesses. Choosing the right one depends on the data's shape, the presence of outliers, and the question you are trying to answer. The table below compares the four main measures you need for SL 4.1.

Connection to Advanced Topics

The summary statistics you learn in SL 4.1 are the building blocks for nearly every other topic in IB Statistics and Probability. Understanding how the mean and standard deviation describe a data set prepares you for the normal distribution (SL 4.3), where these two numbers completely determine the bell-shaped curve. The concept of spread also connects to regression and correlation (SL 4.4), where you measure how two variables move together relative to their individual spreads.

If you continue to HL, you will encounter the chi-squared test and t-test, both of which rely heavily on means and standard deviations. Mastering the basics in SL 4.1 now means you will have the conceptual foundation to tackle these more powerful tools later.

Practice Problems

Summary