The large data setAQA A-Level Maths: Revision notes

The AQA course expects you to be familiar with real data from the large data set, published by AQA in advance of the exam. Real data are rich enough to explore data presentation and interpretation: many records (rows, one per observation) and many variables (columns). Know, for each variable, its type, its units, how it was collected, and how missing readings are shown. Variables are categorical (labels, such as fuel type) or quantitative: discrete (counts) or continuous (measured on a scale, such as temperature). The type decides the diagram: bar chart or pie chart for categorical data, histogram, box plot or cumulative frequency graph for continuous data, and scatter diagram for two quantitative variables. Exam questions refer to the data set, so use the AQA files and their guidance notes to explore it yourself beforehand.

Use a spreadsheet or statistics package to explore the data: sort, filter and select a subset (for example one location, one year or one fuel type), then calculate summary statistics and draw diagrams. Useful features are sorting, filtering by a condition, and built-in functions for the mean, median, quartiles and standard deviation. You can also analyse a subset using a calculator with standard statistical functions: enter the data (or a frequency table) in statistics mode and read off $n$, $\sum x$, $\sum x^2$, the mean $\bar x$, the standard deviation $\sigma$, the median and the quartiles. Standard deviation is $\sigma=\sqrt{\frac{\sum x^2}{n}-\bar x^2}$.

Real data are untidy. Before analysing, look for missing values (blank cells or a code such as $-99$), impossible values, inconsistent units and duplicates. Missing codes must be removed, not treated as numbers: including a $-99$ would drag the mean down and inflate the standard deviation. Replacing blanks by 0 has the same effect, because the 0 adds nothing to the total but counts as an observation. A value that is unusual but possible is an outlier. Do not delete it automatically: check whether it is an error, then decide, and say what you did and why.

An outlier is usually defined by a rule that the question will give you:

• more than $1.5\times$ IQR above $Q_3$ or below $Q_1$; or
• more than $k$ standard deviations from the mean. Example. For a sample with $n=40$, $\sum x=612$ and $\sum x^2=9774$: $\bar x=15.3$ and $\sigma=\sqrt{244.35-234.09}=3.20$. Is 25.5 an outlier by the 2 standard deviation rule? The upper limit is $15.3+2(3.20)=21.7$, and $25.5>21.7$, so yes. Compare the median and IQR (resistant to outliers) with the mean and standard deviation (affected by them). If the mean is well above the median, the distribution is positively skewed.

Interpret real data in summary or graphical form and use it to investigate a question in context. Compare two samples using one measure of location (median or mean) and one of spread (IQR or standard deviation), in the context of the question, and note differences in skew. A sample may not represent the whole data set. State the limitations: how the sample was chosen, whether the data are only from some places or times, and whether the conclusion applies more widely. Correlation in a scatter diagram does not prove that one variable causes the other.