Exploring Data: Seeing the Story Numbers Tell

The 1854 Map That Stopped a Plague

In the summer of 1854, cholera was killing hundreds of people in the Soho district of London. The prevailing theory was that cholera spread through "bad air" — invisible miasma rising from the sewers. A physician named John Snow did not buy it. He thought the disease was waterborne, but he had no way to prove it with the medical technology of his time.

So Snow did something radical: he collected data. He went door to door, recording every cholera death and plotting each one on a map. When he was finished, the pattern was unmistakable. The deaths clustered tightly around a single water pump on Broad Street. Households that used a different water source — even ones just a few blocks away — were barely affected.

Snow convinced the local authorities to remove the handle from the Broad Street pump. The epidemic ended.

This is what statistics does. It takes a confusing, frightening mess of individual events and reveals the structure hidden inside it. But before you can see any structure, you have to understand what kind of data you have — and how to display it so that the patterns become visible.

Data: The Raw Material

Data are values collected about individuals — people, animals, objects, or events. Each individual has one or more characteristics we measure, called variables.

Variables come in two fundamental types, and confusing them is one of the most common errors in statistics:

Categorical variables place individuals into groups or categories. Examples: eye color, political party, yes/no responses, letter grades, zip codes. You can count how many fall into each category, but you cannot meaningfully average them. The "average zip code" in your class is nonsense.

Quantitative variables measure or count something numerical. Examples: height, temperature, test scores, income, number of siblings. You can perform arithmetic on these — means, medians, and standard deviations all make sense.

Here is the critical test: can you meaningfully add or average the values? If yes, it is quantitative. If no, it is categorical.

Frequency Tables: Counting What You See

The simplest way to organize data is to count. A frequency table lists each value (or category) and how often it occurs.

Suppose you survey 30 students about their favorite subject and get: Math (8), English (6), Science (10), History (4), Art (2). That is a frequency table. The relative frequency expresses each count as a proportion or percentage of the total:

Relative frequencies let you compare groups of different sizes. If School A has 200 students and 50 prefer science (25%), while School B has 800 students and 160 prefer science (20%), the comparison is only meaningful in relative terms.

For quantitative data, you group values into classes (also called bins or intervals). If you measured the heights of 50 students, you might group them into intervals: 60-62 inches, 63-65 inches, 66-68 inches, and so on. The choice of intervals matters — too few and you lose detail; too many and the pattern drowns in noise.

Displaying Categorical Data: Bar Charts

A bar chart displays categorical data using bars whose heights (or lengths) represent frequencies or relative frequencies. The bars do not touch — the gaps between them signal that the categories are distinct, not continuous.

Bar charts can be vertical or horizontal. They can display counts or percentages. They can compare groups side by side. What they must never do is mislead — and misleading bar charts are everywhere.

The most common trick: truncating the vertical axis. If a bar chart starts at 90 instead of 0, a difference of 3% can look like a difference of 300%. Every time you see a graph in the news, check where the y-axis starts.

Displaying Quantitative Data: Histograms, Dotplots, and Stemplots

Quantitative data needs different displays because the values have a natural order and often form continuous distributions.

A histogram looks like a bar chart but has no gaps between bars — the touching bars indicate continuous data. Each bar covers an interval (bin), and its height shows how many values fall in that range. Histograms reveal the shape of a distribution: is it symmetric, skewed left, skewed right, or bimodal?

A dotplot places a dot above a number line for each data value. Simple and effective for small data sets — you can see every individual value and spot clusters and gaps immediately.

A stemplot (stem-and-leaf plot) splits each value into a "stem" (leading digits) and a "leaf" (final digit). For the data set {23, 25, 27, 31, 34, 34, 38, 42, 45}:

The stemplot preserves the actual data values while showing the distribution shape — a histogram loses the individual values once they are binned.

Describing Distributions: Shape, Center, Spread

When you look at a histogram or dotplot, three features should immediately register:

Shape: Is the distribution symmetric (mirror image around the center), skewed right (tail extends right — common for income, home prices), or skewed left (tail extends left — less common but seen in exam scores where most students do well)? Is there one peak (unimodal), two peaks (bimodal), or more?

Center: Where is the "typical" value? We will formalize this with mean and median in Unit 2, but for now, estimate where the middle of the data lies.

Spread: How much variability exists? Are values tightly clustered or widely dispersed? A class where everyone scores between 85 and 95 has very different spread from one where scores range from 40 to 100 — even if the averages are identical.

Outliers: Are there values that fall far from the rest of the data? An outlier demands investigation: is it a data entry error, an unusual but legitimate observation, or a signal of something important?

Context Matters: The Same Numbers Tell Different Stories

A histogram of household income in the United States is dramatically right-skewed — most households earn moderate incomes, while a long tail stretches toward very high earners. This shape is not accidental; it reflects the economic structure of the country.

A histogram of human body temperatures is nearly symmetric and bell-shaped, clustering tightly around 98.6 degrees Fahrenheit. This shape reflects biology — homeostasis keeps temperature within a narrow range.

A histogram of the number of goals scored per soccer game is left-skewed, with a peak at 0-1 goals and a tail stretching toward rare high-scoring games.

The graph tells you what the data look like. The context tells you what the data mean. Statistics without context is numerology — patterns without meaning.

Why This Matters

Every statistical analysis begins here — with understanding what kind of data you have and displaying it honestly. The cholera map worked because John Snow chose the right display for his data. A frequency table would have told him cholera was deadly; only the spatial plot showed him why.

In the units ahead, you will learn to calculate precise measures of center and spread, design studies that produce trustworthy data, and build probability models that let you make predictions. But none of that machinery works if the data going in is misclassified, poorly displayed, or misunderstood.

The first rule of statistics: look at your data. The second rule: look again.