Statistics · Topic 2 of 13
Samples, Populations & Outliers
Theory
When gathering and analysing data, it is important to understand where the data comes from and whether it paints an accurate picture of what you are trying to study.
1. Populations vs Samples
Population: The entire, complete group of people or items that you want to draw conclusions about (e.g., Every eligible voter in the UK).
Sample: A smaller subset of the population. Gathering data from an entire population is usually too expensive or time-consuming, so researchers collect data from a sample to estimate the characteristics of the whole population.
2. Bias and Representative Samples
A sample must be representative, meaning it accurately reflects the diverse characteristics of the whole population. If a data collection method unfairly favours a certain group, it introduces bias.
Common Exam Trap:
You must be able to identify why a survey might be biased. For example, an online survey on social media is biased because it excludes people without internet access and relies on "self-selection" (only people with strong opinions usually bother to reply).
3. Outliers
Outliers are extreme values that sit far outside the general pattern of the rest of the data.
- You must carefully consider whether to keep or remove outliers. They should only be removed if they are impossible or clearly data entry errors (e.g., a person's age recorded as -5 or 250 years old), because including them would heavily distort calculations like the mean.
- If an outlier is unusual but physically possible, it should generally be kept.
Worked examples
Example 1
Example 1: The Streaming Service
A national streaming company wants to determine the favourite television genres of all adults living in Scotland. To gather data quickly, they survey 500 students at a university campus in Glasgow.
- (a) Identify the intended population.
- (b) Identify the sample.
- (c) Give two reasons why this sample is not representative of the intended population.
- (a) The population is all adults living in Scotland.
- (b) The sample is the 500 university students surveyed in Glasgow.
- (c) Reason 1: University students are generally younger, so their tastes will not represent the views of older adults.
- Reason 2: The survey was only conducted in one city (Glasgow), so it does not represent adults living in rural areas or other Scottish cities.
Example 2
Example 2: The Town Council Survey
A local town council wants to estimate the proportion of residents who support spending tax money to build a new cycle lane. They leave paper questionnaires on the counter of a local bicycle repair shop for customers to fill out. Explain why this data collection method will result in a biased sample.
The questionnaires are only available in a bicycle repair shop, meaning the survey will almost exclusively be answered by active cyclists. Cyclists are significantly more likely to support building a cycle lane than the general public, which makes the sample highly biased and unrepresentative of the whole town.
Example 3
Example 3: Medical Trial Outliers
A medical researcher is recording the resting heart rate (in beats per minute) of seven adult patients in a clinical trial. The recorded values are: 68, 72, 65, 71, 15, 74, 69. Identify the outlier in this dataset and explain how the researcher should handle it before calculating the mean.
The outlier is 15 bpm.
Since a resting heart rate of 15 bpm is medically impossible for a conscious adult, this is clearly a data entry or equipment error. The researcher should completely remove this value from the dataset before calculating the mean, as leaving it in would heavily distort the average and make it artificially low.