Chapter 11: Statistical sampling and data presentation
The chapter problem: is the bus company's claim believable?
A bus company says its 07:30 service from the estate to the town centre takes 25 minutes on average. A passenger group does not trust the claim. Over one week, the group records the time and the distance of 30 journeys on that service.
Here are the 30 journey times, in minutes:
The group also records how far each journey went. The distances rise from about km to about km as the journey times rise.
The group cannot time every journey for a year. It has a sample of 30, and it must say what that sample suggests about all journeys on the service. The company says the mean is 25 minutes. By the end of the chapter you will decide whether the sample supports that claim. You will also see what one unusual value does to the picture.
First contact — no working needed yet. Read the 30 times without adding them up. Predict whether the mean of the sample is above 25 minutes, below 25 minutes, or very close to 25 minutes. Write your prediction down.
Then count how many of the 30 journeys took longer than 30 minutes. Predict that number before you count. Keep both predictions for the Review.
If you are not sure where to start
Look at the middle of the list, not at the ends. About half the values sit below 23 and half sit above 23, so the centre is near 23. The value 58 is much larger than the rest, and a value like that pulls the mean upwards. A quick estimate is useful here: it does not have to be exact.
For the count, look at the last eight numbers. Only a few of them are larger than 30.
By the end of this chapter you will be able to
- explain the difference between a population and a sample, and why a sample is often the only practical choice;
- define a sampling frame and use simple random, systematic, stratified and opportunity sampling;
- choose or criticise a sampling technique for a real problem, and explain why two samples can lead to different conclusions;
- read and draw diagrams for single-variable data, and explain why the area of a histogram bar represents frequency;
- work out a frequency density and use it to find a frequency or a probability;
- read a scatter diagram, describe the correlation and interpret a regression line, including why correlation does not imply causation;
- calculate the mean, median, mode, range and interquartile range;
- calculate the standard deviation, including from summary statistics, and compare distributions using it;
- recognise an outlier, decide whether to keep, correct or remove it, and clean a data set;
- criticise a statistical diagram or a published claim;
- work with a large data set, using a spreadsheet or a calculator.
Where it is used
News reports quote averages, surveys and pass rates every day, and each number rests on a sample. A hospital measures waiting times to see whether a target is met. A factory tests part of a batch, because testing all of it would destroy it. A market researcher asks a few hundred people what a whole town thinks. Sport, weather forecasting, medicine and government all turn a sample into a claim about a much larger group. Knowing how the sample was chosen, and how much it can vary, is what separates a careful claim from a careless one.
Your turn to connect it
Think of a claim you have met that used a number from a survey or a sample. It could be a waiting time, an average price, a pass rate, or a health fact. Write down the claim, and write down what you would want to know about the sample before you believed it. Keep your answer for the Review.
Roadmap
- 11.1 Population, sample and informal inference
- 11.2 Sampling techniques and their critique
- 11.3 Diagrams for single-variable data and the histogram
- 11.4 Scatter diagrams, regression lines and correlation
- 11.5 Measures of central tendency and variation, including standard deviation
- 11.6 Outliers, data cleaning and critiquing presentation
- 11.7 Working with the large data set
- Review