Track 1 · Statistical Thinking · Lesson 2 of 10

Populations, samples, parameters and statistics

Identify the population, sample, parameter and statistic in a study, use the standard notation, and compute a sample proportion.

25 minFoundation7 graded exercisesFree

A telecom operator cannot ask all 2 million of its customers whether they will renew, so it asks 800 of them. Every decision based on that survey depends on understanding the gap between the 800 people who answered and the 2 million the company actually cares about.

The population is the entire group you want to draw conclusions about: all customers, all patients with a condition, every part a machine produces this month. A sample is the subset you actually observe.

A number that describes the population is a parameter. It is usually unknown, because measuring everyone is too costly or impossible. A number computed from the sample is a statistic. We use statistics to estimate parameters.

Quantity	Population parameter	Sample statistic
Number of units	$N$	$n$
Mean	$\mu$ (“mu”)	$\bar{x}$ (“x-bar”)
Standard deviation	$\sigma$ (“sigma”)	$s$
Proportion	$p$	$\hat{p}$ (“p-hat”)

Standard notation: Greek letters (or plain letters) for population parameters, decorated letters for sample statistics.

\hat{p} = \frac{x}{n}

The sample proportion

$\hat{p}$: the sample proportion, our estimate of the population proportion $p$
$x$: the number of sampled units with the characteristic of interest (“successes”)
$n$: the sample size

Worked example

Renewal intentions

The telecom operator randomly selects 800 of its 2,000,000 customers; 548 say they will renew their contract. Identify the population, sample, parameter and statistic, and estimate the number of customers who would renew.

Population: all 2,000,000 customers ( $N = 2{,}000{,}000$ ). Sample: the 800 customers surveyed ( $n = 800$ ).
Parameter: $p$ , the proportion of all customers who intend to renew — unknown.
Statistic: $\hat{p} = 548/800 = 0.685$ , so 68.5% of the sample intend to renew.
Estimate for the population: $0.685 \times 2{,}000{,}000 = 1{,}370{,}000$ customers. Another random sample of 800 would give a slightly different $\hat{p}$ , so this estimate comes with uncertainty (Uncertainty and significance shows how much).

$\hat{p} = 0.685$ estimates $p$ ; about 1.37 million customers would renew.

Before sampling you need a sampling frame: the list from which the sample is actually drawn (a customer database, an address register, a staff directory). If the frame misses part of the population — customers on prepaid plans who are not in the billing database, employees on contract who are not in the directory — no amount of careful sampling from the frame can represent them.

Dataset

Engagement survey population and response rates

6 rows · one row = one department · synthetic data

About this dataset Open in the lab

Department	Headcount (employees)	Responses (responses)	Response rate (%)
Operations	420	189	45
Customer Service	330	165	50
Sales	260	143	55
Technology	210	151	71.9
Finance	120	82	68.3
Human Resources	60	48	80

A whole population and a sample: the fictional Palmstone Group invited all employees in six departments to an engagement survey. headcount is the population in each department; responses is how many answered.

In this (synthetic) company the population is 1,400 employees and 778 responded. Any average computed from the 778 responses is a statistic; the corresponding average over all 1,400 employees is the parameter HR really wants. Whether the statistic is a good estimate depends on how the sample came about — a theme of Bias and representative samples.

Quick check

A national statistics office reports that the average household size in the country, from its full census, is 4.8 people. Is 4.8 a parameter or a statistic?

Exercises

The lesson is complete when every exercise is answered correctly. Hints are free; viewing a worked solution before solving means the exercise won't count towards mastery.

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Exercise 1 · Practice, difficulty 1 of 3

Parameter or statistic?

Not started

Classify each bold number as a parameter (describes the whole population) or a statistic (computed from a sample).

Exercise 2 · Practice, difficulty 1 of 3

Notation

Not started

Match each symbol to its meaning.

\mu

\bar{x}

p

\hat{p}

\sigma

s

Exercise 3 · Guided, difficulty 1 of 3

From sample proportion to population estimate

Not started

A water utility serves 36,000 households. It surveys a random sample of 500 households, and 216 of them say they would sign up for a smart meter.

(a) Compute the sample proportion $\hat{p}$ (3 decimal places). (b) Estimate how many of the utility's households would sign up (nearest whole number).

Exercise 4 · Application, difficulty 2 of 3

Overall response rate

Not started

The engagement_population table lists, for each department, the number of employees invited (headcount) and the number who responded (responses). What percentage of all invited employees responded to the survey? Give your answer to 1 decimal place.

Data: engagement_population · Open in the lab to explore and calculate.

Exercise 5 · Practice, difficulty 2 of 3

Identify the population

Not started

A city's transport department wants to know how many minutes residents spend travelling to work. It mails a questionnaire to 2,000 addresses chosen at random from the electricity billing list; 1,140 are returned. What is the population of interest?

Exercise 6 · Interpretation, difficulty 2 of 3

The sample mean missed

Not started

An airline's booking system holds the weight of every checked bag on a route last year — the whole population. Their mean is $\mu = 18.6$ kg, and individual bags typically differ from that mean by about 4 kg (some weigh 10 kg, some 27 kg). For a training exercise, a trainee draws a random sample of 40 of last year's bag records and finds $\bar{x} = 19.3$ kg. What is the best conclusion?

Exercise 7 · Interpretation, difficulty 2 of 3

Describe the study

Not started

A sports federation wants to know the average weekly training hours of registered youth footballers in the country. It randomly selects 25 clubs and records the training hours of every registered youth player in them (612 players); the mean is 6.4 hours per week. In two to four sentences, identify the population, the sample, the parameter and the statistic.

What you can now do

Identify the population, sampling frame, sample and respondents in a study
Distinguish parameters (μ, σ, p) from statistics (x̄, s, p̂)
Compute a sample proportion and scale it up to a population estimate
Expect statistics to differ from parameters because of sampling variability