Descriptive Statistics: What It Is and How It Works

Population, sample and the words you need

The open-access textbook Introductory Statistics 2e, published by OpenStax, describes statistics as the practice of collecting, analyzing, interpreting and presenting data.

Within that practice, descriptive statistics has a narrow job: putting collected data in order and compressing it. The starting point is the population, every person, thing or object an investigation covers; observing all of it is rare, so the work shifts to a sample, a portion selected from that larger population. Treccani defines a sample as a group of elementary units forming a subset of the population. Out of this come two words that keep the accounting straight. A parameter is a numerical feature of the entire population. A statistic, in the technical sense of the term, is a number describing a property of the sample, and it serves to estimate the parameter that nobody manages to calculate directly.

Practical example: the population is the 900 students of a school, and the investigation measures how many hours they sleep. The average hours worked out on the 60 students interviewed are a statistic; the true average across all 900 is the parameter, and it stays unknown.

A variable is the characteristic or measure recorded for each unit of the population: hours of sleep, eye color, a math grade. Survey data ends up in a table with one row per unit and one column per variable, the shape used by relational databases as well.

Descriptive and inferential statistics

The border between the two halves of statistics runs through the question being answered. The descriptive half works on the data already in hand and summarizes it. The inferential half, Treccani writes, starts from samples of data chosen scientifically to reach conclusions that hold for the entire population, wider than the sample observed; and to get there, OpenStax adds, it uses formal methods grounded in probability that establish how much those conclusions can be trusted.

	Descriptive statistics	Inferential statistics
Question	What does this data look like?	What can we say about the population?
Object	The data collected	The sample, to estimate the parameter
Tools	Classification, summary measures, charts	Formal probability-based methods

The two halves complete each other. One set of data, Treccani notes, may be treated as a population or as a sample depending on the context: the grades of a single class are the population when the question concerns that class, and a sample when the question concerns the school.

Types of variables

Before any calculation comes the matter of what kind of data is on the table. OpenStax separates qualitative data, which arises from classifying or describing attributes of the population, from quantitative data, which is always numbers obtained by counting or by measuring an attribute.

Inside the quantitative family the split continues. Data obtained by counting, such as a number of objects, is discrete: a discrete variable takes only certain specific values and skips the ones in between. Data that admits fractions, decimals or even irrational values, typically the product of a measurement, is continuous: a continuous variable can land on any value inside an interval. The distinction echoes the one drawn between number sets in math lessons.

Kind of data	How it arises	Examples
Qualitative	By classifying or describing an attribute	Eye color, category of membership
Quantitative discrete	By counting	Number of siblings, number of books read
Quantitative continuous	By measuring	Height, reaction time, temperature

Qualitative variables then carry a level of measurement. Data Science for Psychologists, the university course notes by Corrado Caudek, calls the simplest level a nominal scale, the one that files observations into categories carrying no ranking at all. An ordinal scale files them the same way, except its categories also run in a sequence, as happens with ordinal numbers.

Practical example: the color of a T-shirt sits on a nominal scale, because red and green do not rank one above the other. A questionnaire rating of «not at all», «a little», «very much» sits on an ordinal scale: the categories stay categories, yet they run in a direction.

How a sample is drawn

Who ends up in the sample gets settled before the data is collected, and the method chosen has to be declared alongside the results. Reading a piece of research, the method and the name of the publisher are both worth checking: the same care it takes to weigh up the sources of a school report.

Method	How it works
Simple random	n units are drawn, and any set of n members is as likely to come up as any other set of n
Stratified	The population divides into strata that are homogeneous inside, and each stratum yields a proportionate number of units
Cluster	The population divides into clusters, by geographical closeness for instance, and a few entire clusters are picked at random with all their units
Systematic	You start from a point chosen at random in a list of the population and take one unit every n, at regular intervals

Stratified and cluster sampling look alike in shape and differ in outcome: the first takes a slice of every subgroup, the second takes a few subgroups whole. In cluster sampling each cluster counts as a single sampling unit.

Summarizing data with a few numbers

A list of 300 values cannot be read. Measures of central tendency summarize it, using three definitions that do not coincide: the mean comes from adding up all the values and dividing by their number; the median is the value that, once the data is ordered, splits it into two halves of equal size; the mode is the value occurring most frequently. On one set of data the three measures can land at points far apart, and a summary reporting only one of them hides the shape of the distribution.

The charts of descriptive statistics

OpenStax lists the charts used to summarize and organize data: the dot plot, the bar chart, the stem-and-leaf plot, the histogram, the pie chart, the frequency polygon and the box plot. Three of them cover most of what school work asks for.

The bar chart mostly represents qualitative data: one bar per category, as long as the frequency of that category.

The histogram is a Cartesian chart used when the range of a variable’s values is divided into classes, of equal or different width. It is made of adjacent rectangles, resting against one another, arranged along a horizontal axis and a vertical one.

The horizontal axis of a histogram carries the measured quantity, the vertical one the frequency, which may be absolute, relative or percentage. To stay readable the chart generally uses between 5 and 15 bars, meaning between 5 and 15 classes.

The box plot, or box-and-whisker diagram, shows visually how the data concentrates. Five values draw it: the minimum and the maximum at the two ends, the first and third quartile, and the median in between. The box covers the central part of the distribution and the whiskers reach the ends, so the eye catches which zone the values crowd into.

Practical example: for the favorite colors of a class you use a bar chart, because the data is qualitative. For the heights of the same class, split into bands of 5 centimeters, you use a histogram. If what matters is where the heights concentrate, a box plot says it with five numbers.