Statistics Quest

Track 1 · Statistical Thinking · Lesson 1 of 10

What statistics is for

Explain how statistics turns variable data into evidence for decisions, and tell descriptive from inferential questions.

20 minFoundation7 graded exercisesFree

A regional manager sees that this month's sales are 4% below last month's and wants to change the team's targets. Before anyone acts, someone has to ask: is a 4% dip unusual, or is it the normal month-to-month wobble? Answering that question honestly is what statistics is for.

Statistics is the discipline of learning from data when the data vary. It gives us ways to collect data so that they can answer a question, to summarise what the data show, and to say how far the conclusion can be trusted.

If every employee had the same commute, every patient responded identically to a drug and every week had the same sales, we would need one measurement and no statistics. Statistics exists because of variability: individuals differ, measurements contain error, and samples differ from one another.

Histogram of 240 commute times ranging from 4 to 148 minutes, with most values between about 15 and 45 minutes and a long tail to the right.Commute time (minutes)Count
Typical one-way commute of 240 employees at a fictional company (synthetic data).
Show the data as a table
FromToCount
411.216
11.218.425
18.425.643
25.632.840
32.84029
4047.236
47.254.412
54.461.613
61.668.83
68.8766
7683.24
83.290.45
90.497.61
97.6104.80
104.81122
112119.20
119.2126.42
126.4133.62
133.6140.80
140.81481

The chart shows variability in action. In this (synthetic) dataset commutes range from 4 minutes to 148 minutes. The mean is about 37.3 minutes and the median 31.5 minutes. No single employee is the average: the summary describes the group, and the spread around it is just as much part of the story as the centre.

The statistical investigation cycle

  1. Pose a question that the data can answer, and say who or what it is about.
  2. Plan and collect data: decide what to measure, on whom, and how (a survey, records, an experiment).
  3. Analyse: organise, summarise and visualise the data; quantify variability.
  4. Conclude and communicate: answer the question, state how certain the answer is, and say what the data cannot tell you.

Statistical work has two broad aims.

  • Descriptive statistics summarise the data you actually have: “the median commute of these 240 employees is 31.5 minutes.”
  • Inferential statistics use a sample to draw conclusions about something larger, with a statement of uncertainty: “based on 400 surveyed customers, between 58% and 68% of all our customers would renew.”

The difference is not the arithmetic but the claim. A description is about the data in hand; an inference reaches beyond them and must say how far it could be wrong.

Worked example

Describing a whole company versus inferring from a sample

The salaries dataset contains every one of the 600 employees of a fictional company. (a) An HR analyst reports the mean monthly base salary. Is that descriptive or inferential? (b) A consultant surveys 60 of the company's employees and uses their mean salary to estimate the company-wide mean. Which kind is that?

  1. (a) The 600 records are the whole group of interest. The mean, 16,397 SAR per month in this synthetic dataset (median 13,145 SAR), simply describes that group. There is no sampling uncertainty because nobody was left out.

  2. (b) The consultant sees only 60 of 600 people and wants to say something about all 600. A different 60 would give a different mean, so the estimate must come with a statement of uncertainty. That is inference.

  3. The lesson. Ask what is the claim about? If it reaches beyond the data actually observed, it is inferential and needs a measure of uncertainty.

(a) Descriptive; (b) inferential.

Quick check

Which of these is a statistical question?

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

Descriptive or inferential?

Not started

Classify each statement as descriptive (about the data in hand) or inferential (a conclusion about a larger group drawn from a sample).

Of the 48 matches our team played last season, we won 29.
A survey of 1,200 adults suggests that between 31% and 37% of adults in the country cycle at least once a month.
The median processing time of last month's 3,412 permit applications was 9 days.
Testing 50 batteries from a shipment of 20,000, the supplier estimates the shipment's average life at 410 hours.
In the call centre's records for Monday, 14% of calls were abandoned.

Exercise 2 · Practice, difficulty 1 of 3

Which questions need statistics?

Not started

Select all the questions that are statistical questions — ones whose answer depends on data that vary.

Select all that apply

Exercise 3 · Practice, difficulty 1 of 3

The investigation cycle

Not started

A municipality wants to know whether residents support extending library opening hours. Put the stages of the investigation in a sensible order.

Put the steps in order (use the arrows)

  1. 1Agree the question: what share of adult residents support longer opening hours?
  2. 2Decide whom to ask and how: draw a random sample of households from the address register
  3. 3Report the answer, its uncertainty and the survey's limitations to the council
  4. 4Collect the responses and check them for errors and missing answers
  5. 5Summarise the responses and estimate the share of supporters with a margin of error

Exercise 4 · Guided, difficulty 1 of 3

How much do waiting times vary?

Not started

An emergency department recorded the waiting time (minutes) of eight patients who arrived on Tuesday evening:

34, 41, 58, 16, 28, 42, 45, 51

The simplest measure of variability is the range = largest value − smallest value. What is the range of these waiting times?

minutes

Exercise 5 · Application, difficulty 2 of 3

Long commutes

Not started

The HR team is considering a flexible-start policy for employees whose commute is longer than 45 minutes. Using the commute dataset (240 employees), what percentage of employees have commute_min greater than 45? Give your answer to 1 decimal place.

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

Exercise 6 · Interpretation, difficulty 2 of 3

A 4% dip in sales

Not started

A retail chain's regional manager notices that this month's sales are 4% lower than last month's and proposes replacing the store team. What is the most useful statistical response?

Choose one

Exercise 7 · Interpretation, difficulty 2 of 3

“My commute proves the average is wrong”

Not started

The commute dataset gives a mean commute of about 37 minutes for the 240 employees. A team leader objects: “My commute is 20 minutes, and so is my deputy's, so the 37-minute figure must be wrong.” In two or three sentences, explain why the team leader's argument does not hold.

0 words

What you can now do

  • Explain why variability is the reason statistics exists
  • Recognise a statistical question and the stages of an investigation
  • Tell descriptive summaries from inferential claims that need a statement of uncertainty
  • Judge a change against normal variation before acting on it