What statistics is, really.
Two kinds of statistics. Descriptive describes the data we have; inferential makes claims about a population from a sample. Naming the difference is the move that makes the rest of the chapter make sense.
A poll headline scrolls past: "45% of Americans favor the new bill, ±3%." Two sentences worth of math are hiding inside it. The 45% is what the pollsters measured in a specific group of people they actually called. The ±3% is what they claim the answer would be if they had called every American.
Those two pieces are different kinds of statistics. The 45% is descriptive — it describes a data set the pollsters can hold in their hand. The ±3% is inferential — it stretches a claim about that data set into a claim about a much larger group nobody actually interviewed.
This whole topic is about telling those two apart. Lessons 2 and 3 sharpen the descriptive side (mean, median, mode, range, standard deviation). Lessons 4 and 5 introduce the bell curve that powers most inferential claims. Lesson 6 closes the loop with margin of error, the math behind that ±3%.
Descriptive describes; inferential infers.
Inferential statistics use a small sample to make claims about a much larger population we haven't measured. The pollster calls 1,000 voters and claims to know how 100 million voters will vote. That leap is the whole point of the inferential side, and the math that makes the leap legitimate (standard deviation, margin of error, the normal distribution) shows up in Lessons 5 and 6.
Same numbers can power either side. The difference is what we're claiming with them. The descriptive side stops at "here is what this data set looks like." The inferential side takes one more step into "here is what we think the larger population looks like."
Descriptive statistics summarize a data set we have measured. Inferential statistics use that data to make claims about a larger population from which the data was a sample.
Five words you'll see all week
- Data set The collection of numbers we actually have in hand. 10 weights, 12 class sizes, 7 dog weights, 100 test scores — each is a data set.
- Population The full group we'd ideally like to know about. "All American adult voters," "all households in Phoenix," "all 11th graders." Usually too big to measure directly.
- Sample A subset of the population that we can measure. A poll of 1,000 voters is a sample from the population of all voters. The size of the sample drives how confident we can be in inferential claims.
- Descriptive claim A claim about the data set itself. "The median weight is 162 pounds." Exact for this data, no margin of error.
- Inferential claim A claim that extrapolates from the sample to the population. "The median weight of all 11th-grade boys in the US is about 162 pounds, ±5." Always has a margin of error.
Same number, two claims, two kinds of statistics.
"A high-school basketball coach measures the height of every player on the team. The 15 players average 6 feet 1 inch tall. The coach posts the average on the team bulletin board."
Identify the data set, sample, and population.
Data set: the 15 height measurements. Sample: these same 15 players (if we're thinking of them as a sample of something larger). Population: depends on what claim we want to make.
Read the descriptive claim.
"The 15 players on this team average 6 feet 1 inch." That is exact. We measured them. The claim is about this team, no leap.
→ descriptiveListen for the inferential leap.
Now the coach tells a reporter: "The average high-school basketball player in our state is about 6 feet 1." Same number, different claim. The coach is using the team as a sample to infer something about the population of all high-school basketball players in the state.
→ inferentialSpot the difference.
The descriptive claim cannot be wrong (we measured every player on the team). The inferential claim can be wrong, because the team might not be representative of the state. The honest version of the inferential claim would include a margin of error: "about 6 feet 1, give or take two inches."
→ inferential claims always carry a ±Three claims. Descriptive or inferential?
A teacher reports: "The 28 students in this class averaged 84% on the midterm." Is this descriptive (D) or inferential (I)?
A poll reports: "52% of likely U.S. voters favor the candidate, ±3%." Pollsters interviewed 1,200 people. Is this descriptive (D) or inferential (I)?
A drug company tests its new medicine on 200 volunteers and reports: "Patients in the trial experienced an average 15% reduction in symptoms." Is this descriptive (D) or inferential (I)?
Three fast questions before you move on.
Q1. Which of the following is a descriptive claim?
Q2. Why does the inferential side always come with a margin of error?
Q3. A nutrition study weighs every student in a specific high school and reports the mean weight as 152 pounds. The principal then tells the school board: "The average high-school student in our city weighs about 152 pounds." The principal's claim is...
Naming the kinds first, then doing the math.
Every claim in this topic is one of these two kinds. The descriptive moves come in Lessons 2 and 3 (mean, median, mode, range, standard deviation). The inferential moves come in Lessons 4 through 6 (the bell curve, the empirical rule, and the margin of error that puts a ± on every poll headline you will ever read).
Next: Lesson 2 introduces the three classic measures of center — mean, median, mode — and a powerful visual intuition for the mean as the value that balances the data.
Continue to Lesson 02Different angle? Need another rep? These are optional — tap any that look helpful.