What Is Statistics?

In this lesson you’ll learn the difference between a population and a sample, what parameters and statistics are, how descriptive and inferential statistics differ, and why a well-chosen sample of 1,000 can tell you about a country of 300 million.

The Greek-versus-Roman split is a convention worth learning now: Greek letters describe populations, Roman letters describe samples. It tells you at a glance whether a number is a truth about everyone or an estimate from a handful of people.

A population is every individual you want to know about. Every voter in a country. Every light bulb a factory produces. Every patient with a particular condition.

A sample is the subset you actually measure.

The population is almost never measurable. It’s too big, too expensive, or partly in the future (the factory hasn’t made next month’s bulbs yet). Sometimes measuring destroys the item, as with testing how long a bulb lasts.

So statistics does something that sounds impossible and works anyway: measure a few, conclude about the many.

A parameter is a number describing a population. It’s a fixed, usually unknown truth.

A statistic is a number computed from a sample. It’s known, but it varies depending on which sample you happened to draw.

	Population	Sample
Size	$N$	$n$
Mean	$\mu$	$\bar{x}$
Standard deviation	$\sigma$	$s$
Proportion	$p$	$\hat{p}$

The whole project of inference is using the right column to estimate the left one.

Descriptive statistics summarizes data you have. “The 40 students in this class averaged 78 on the exam” is a complete, certain fact about those 40 students. Nothing is being guessed.

Inferential statistics reaches beyond the data. “Based on these 40 students, the average for all students at this university is probably between 74 and 82” is a claim about people you never tested, and it comes with a stated level of confidence.

Descriptive statistics can’t be wrong, only misleading. Inferential statistics can genuinely be wrong, and a good analysis tells you how often it would be.

The surprising fact underneath all of this: sample quality matters far more than sample size relative to the population.

A random sample of 1,000 people describes a country of 300 million about as well as it describes a town of 30,000. That feels wrong, but it’s true, and you’ll see why when you reach the Central Limit Theorem. The precision of an estimate depends on $n$, not on the fraction of the population you captured.

What does wreck a sample is bias, meaning some members of the population were systematically more likely to be included.

• Selection bias - the sampling method skips a group. A phone survey on landlines misses young people.
• Voluntary response bias - people opt in, and those with strong opinions opt in more. Online polls are notorious for this.
• Nonresponse bias - the people who decline to answer differ from the ones who do.
• Convenience sampling - asking whoever is easy to reach, like your own friends.

A simple random sample gives every individual an equal chance of selection. It’s the gold standard because it has no systematic tilt.

The famous cautionary tale is the 1936 Literary Digest poll, which mailed 10 million ballots and got 2.4 million back, predicting a landslide for Alfred Landon. Franklin Roosevelt won in one of the largest landslides in US history. The magazine’s list came from car registrations, telephone directories, and its own subscribers, all of which skewed wealthy during the Depression. A sample of 2.4 million lost to George Gallup’s sample of about 50,000, because Gallup’s was representative and the Digest’s was not.

A big biased sample is worse than a small fair one, because the size lends false confidence.

Example 1: Identify the population and sample.

A city wants to know the average commute time of its 400,000 workers. Researchers survey 800 workers at randomly chosen bus stops.

Solution.

• Population: all 400,000 workers in the city.
• Sample: the 800 workers surveyed.
• Parameter of interest: $\mu$, the true mean commute time for all workers.
• Statistic: $\bar{x}$, the mean commute time of the 800.

But there’s a problem. Sampling at bus stops only reaches people who take the bus, and their commutes likely differ from those who drive or walk. That’s selection bias, and the estimate will be off no matter how large the sample gets.

Example 2: Parameter or statistic?

1. The US Census reports the median household income of all US households.
2. A poll of 1,500 adults finds 62% support a policy.
3. A factory tests 50 of the 10,000 bulbs made today and finds 3 defective.

Solution.

1. Parameter. A census attempts to measure everyone, so this describes the population.
2. Statistic. It’s from a sample of 1,500. Written $\hat{p} = 0.62$.
3. Statistic. From a sample of 50, so $\hat{p} = 3/50 = 0.06$.

Example 3: Descriptive or inferential?

1. “The average of these 12 test scores is 84.”
2. “We estimate the drug reduces recovery time by 2 days, plus or minus 0.6 days.”
3. “This histogram shows the distribution of our 500 survey responses.”

Solution.

1. Descriptive. A summary of the data in hand.
2. Inferential. A claim about the drug in general, with uncertainty attached.
3. Descriptive. A picture of the collected data, nothing beyond it.

Example 4: Spot the bias.

A restaurant emails a satisfaction survey to everyone who joined its loyalty program and reports that 94% of customers are satisfied.

Solution. Two problems stack up here.

First, selection bias: loyalty program members are people who liked the restaurant enough to sign up. Dissatisfied customers mostly never joined.

Second, voluntary response bias: among members, the ones who bother to reply skew toward strong feelings, and satisfied regulars are more likely to engage with a brand they like.

The 94% describes “loyalty members who chose to respond,” not “customers.” The number may be perfectly accurate and still not answer the question asked.

Example 5: Does the population size matter?

A pollster wants a margin of error of about 3 points. Does she need a bigger sample for a country of 300 million than for a city of 300,000?

Solution. No, essentially not. For any population much larger than the sample, the margin of error depends on $n$ alone. Roughly 1,100 respondents gives about a 3-point margin either way.

This is why national polls of 1,000 to 1,500 people are standard. It also explains a common complaint, “they never asked me,” which is true and irrelevant: the math never required asking a large fraction of anyone.

Election polling is the most visible use, and the most visibly criticized. When polls miss, the cause is almost always bias rather than sample size: some group was harder to reach, or less willing to state a preference. Pollsters spend most of their effort on representativeness, not volume.

Medical trials sample patients to make claims about future patients, which is why the eligibility criteria matter so much. A drug tested only on adults aged 25 to 55 has a population that doesn’t include children or the elderly, and extending the conclusion to them is a leap beyond the data.

Quality control samples a production line because testing everything is impossible when the test destroys the product. A crash-test rating comes from wrecking a few cars, not all of them.

In software, A/B tests split users into a sample that sees version A and a sample that sees version B. The whole design rests on the split being random, which is why assigning by something like signup date quietly ruins the experiment.

Census data is the rare case of attempting a true population measurement, and even that ends up using statistical adjustment for people it misses.