Planning and Data Collection: Problems with Data and Outliers
Welcome to your study notes on Problems with Data and Outliers! When carrying out the Statistical Enquiry Cycle for CCEA GCSE Statistics (2260), we start by planning and collecting our data. However, real-world data is rarely perfect. From missing questionnaire answers to numbers that look completely wild, data often has flaws that we need to understand, spot, and clean before doing any serious calculations.
Don't worry if this sounds a bit tricky at first. We will break down every single step, formula, and visual rule so that you can tackle these questions with total confidence in your exam!
1. Common Problems in Raw Data
When you collect raw data—whether from a survey, an experiment, or secondary records—several issues can crop up:
• Non-Response and Missing Data
This happens when participants choose not to answer a question, leave pages blank, refuse to take part, or drop out of a study.
Why is this a major problem? Non-response causes non-response bias. If the people who do not reply share specific traits (for example, people working night shifts who are never home to answer the door, or people with very low incomes who prefer not to state their earnings), your remaining sample is no longer representative of the whole population.
• Recording and Measurement Errors
These are mistakes made during the measurement or data entry stages. Examples include:
- Typing mistakes (e.g. typing a person's height as \(1800\text{ cm}\) instead of \(180\text{ cm}\)).
- Faulty or uncalibrated equipment (e.g. weighing scales that add \(2\text{ kg}\) to every reading).
- Misreading gauges or misinterpreting questions.
• Ambiguous or Unusable Responses
These occur when respondents give answers that cannot be processed. Examples include selecting two options when asked to "tick one box only", writing illegible handwriting, or giving vague, inappropriate comments on an open-ended survey.
Key Takeaway: Non-response is not just about having fewer results; its biggest danger is causing bias by making your sample unrepresentative.
2. Understanding Outliers
What is an Outlier?
An outlier is an extreme value that lies an abnormal distance away from the other values in a dataset.
Everyday Analogy: Imagine five teenagers in a room have \(\text{£}5\), \(\text{£}10\), \(\text{£}12\), \(\text{£}15\), and \(\text{£}18\) in their pockets. Suddenly, a billionaire walks in with \(\text{£}1{,}000{,}000\). That billionaire's cash is an outlier!
How Outliers Affect Summary Statistics
Outliers do not affect all statistical measures in the same way:
1. Mean and Range (Sensitive / Distorted)
The mean and the range are heavily influenced by extreme values. In our pocket money example, the billionaire pulls the mean up to hundreds of thousands of pounds, which gives a misleading picture of what a typical person in the room has. The range also explodes from \(\text{£}13\) to almost \(\text{£}1{,}000{,}000\).
2. Median and Interquartile Range (Resistant / Robust)
The median (the middle number) and the interquartile range (IQR) are resistant (or robust) to outliers. Because they focus only on the middle of the ordered data, an extreme value at either end does not distort them.
Key Takeaway: When a dataset contains outliers, the median and IQR are much better measures of average and spread than the mean and range.
3. Identifying Outliers: The Official Rules & Formulae
For Higher Tier CCEA GCSE Statistics, you need to know how to calculate exact boundary values (sometimes called "outlier fences") to decide mathematically whether a value is an outlier.
Method A: Using Quartiles and the Interquartile Range (\(\text{IQR}\))
First, find the lower quartile (\(Q_1\)), upper quartile (\(Q_3\)), and calculate the interquartile range:
\(\text{IQR} = Q_3 - Q_1\)
A value is classified as an outlier if it falls beyond either of these two thresholds:
Lower Boundary: \(\text{Outlier} < Q_1 - 1.5 \times \text{IQR}\)
Upper Boundary: \(\text{Outlier} > Q_3 + 1.5 \times \text{IQR}\)
Step-by-Step Example (Quartile Method):
A class takes a test. The results show \(Q_1 = 30\) and \(Q_3 = 50\). A student scored \(85\). Is this score an outlier?
Step 1: Calculate the \(\text{IQR}\).
\(\text{IQR} = 50 - 30 = 20\)
Step 2: Multiply the \(\text{IQR}\) by \(1.5\).
\(1.5 \times 20 = 30\)
Step 3: Work out the upper boundary.
\(\text{Upper Boundary} = Q_3 + 1.5 \times \text{IQR} = 50 + 30 = 80\)
Step 4: Compare the score to the boundary.
Because \(85 > 80\), the score of \(85\) is officially an outlier!
Method B: Using the Mean (\(\bar{x}\)) and Standard Deviation (\(\sigma\))
When data is roughly symmetrical or normally distributed, a common convention is to identify outliers as values that lie more than \(2\) standard deviations (or in some contexts, \(3\) standard deviations) away from the arithmetic mean:
Lower Boundary: \(\text{Outlier} < \bar{x} - 2\sigma\)
Upper Boundary: \(\text{Outlier} > \bar{x} + 2\sigma\)
4. Visualising Outliers on Diagrams
Box Plots (Box-and-Whisker Diagrams)
When you draw a box plot for a dataset that contains an outlier:
1. Do NOT extend the whisker to the outlier! The whisker must stop at the lowest or highest value within the non-outlier range.
2. Mark the outlier as an isolated point, an asterisk (\(*\)), or a small cross (\(\times\)) at its exact coordinate along the axis.
Scatter Diagrams
On a scatter plot showing bivariate data, an outlier is a plotted point that sits noticeably far away from the main cluster of points and the line of best fit.
5. Data Cleaning and Dealing with Outliers
What is Data Cleaning?
Data cleaning (also called data scrubbing) is the systematic process of finding, fixing, or removing corrupt, inaccurate, incomplete, or duplicate records from a dataset before running an analysis.
What should you do when you find an outlier?
Follow this three-step decision rule:
Step 1: Investigate the value
Check whether the outlier is a genuine real-world observation (e.g. an Olympic sprinter running 100 metres in under 10 seconds) or an obvious mistake (e.g. an adult's age recorded as \(300\) years).
Step 2: If it is an identifiable error
- If possible, check the original paper/data collection sheet and correct the value.
- If it cannot be corrected and makes no sense, discard / remove the unusable entry.
Step 3: If it is genuine, valid data
- DO NOT delete it! Real-world extremes are part of natural variation.
- Use resistant summary statistics (like the median and IQR) rather than the mean and range.
- Alternatively, perform your statistical calculations twice (once including the outlier, and once without it) and compare the results in your final report.
6. Common Exam Pitfalls to Avoid
Trap 1: Writing "just delete the outlier"
Examiners frequently penalise students who say outliers should automatically be thrown away. Outliers should only be removed if they are proven errors or outside the target population.
Trap 2: Formula mix-ups for outlier boundaries
Remember: subtract \(1.5 \times \text{IQR}\) from \(Q_1\), and add \(1.5 \times \text{IQR}\) to \(Q_3\). Never add or subtract from the median, and never multiply \(Q_3\) directly by \(1.5\) without using the \(\text{IQR}\)!
Trap 3: Drawing box plot whiskers across outliers
Always stop the whisker at the last regular data value and plot the outlier as a separate cross (\(\times\)).
Trap 4: Forgetting why non-response matters
If an exam question asks about the problem with missing questionnaire replies, mention bias and lack of representativeness, not just that the sample size became smaller.
Quick Revision Checklist
• Can you explain what non-response bias is? (\(\checkmark\))
• Do you know which averages are distorted by outliers (Mean, Range) and which are resistant (Median, IQR)? (\(\checkmark\))
• Can you use \(Q_1 - 1.5 \times \text{IQR}\) and \(Q_3 + 1.5 \times \text{IQR}\) to find outlier boundaries? (\(\checkmark\))
• Can you state the rule for plotting an outlier on a box plot? (\(\checkmark\))
• Do you know when to keep an outlier and when to clean/remove it? (\(\checkmark\))