Analysis of Data

Data is either primary (you collect it yourself) or secondary (collected by someone else), and either discrete (counted, e.g. number of goals) or continuous (measured, e.g. time, height).

A good sample should be representative of the population it's drawn from. Common sampling methods:

• Random sampling — every member of the population has an equal chance of being chosen.
• Stratified sampling — the population is split into groups (strata), and a sample is taken from each group in proportion to its size.
• Systematic sampling — you select every $n$th item from a list.
• Quota sampling — you sample until you've filled fixed quotas for different groups (not necessarily randomly within each group).

The three averages: mean (sum ÷ count), median (middle value when ordered), mode (most frequent value). Each has strengths — the median isn't distorted by extreme outliers the way the mean is, which matters when a dataset is skewed.

Measures of spread: the range (max − min) is simple but sensitive to outliers; the interquartile range (IQR) = $Q_3 - Q_1$ ignores the top and bottom quarter of the data, so it's more robust; standard deviation measures the typical distance of values from the mean and uses every data point.

For grouped data, you don't know the exact values, so you use the midpoint of each class interval to estimate the mean, and identify the modal class (the class with the highest frequency) rather than a single mode.

Different diagrams suit different questions:

• Stem-and-leaf diagrams — show every data value while grouping by size; easy to read off the median and range.
• Box plots (box-and-whisker) — show the minimum, $Q_1$, median, $Q_3$ and maximum in one picture; ideal for comparing the spread and skew of two or more datasets side by side.
• Cumulative frequency graphs — plot running totals against the upper class boundary; used to read off the median and quartiles for grouped data, and to estimate how many values lie above or below a given value.
• Histograms — like bar charts but for continuous, grouped data with unequal class widths; the area of each bar (not the height) represents frequency, so the vertical axis is frequency density = frequency ÷ class width.

Exam questions often ask you to compare two datasets. Always compare both an average (which is typically higher/lower) and a measure of spread (which is more consistent/variable) — and write your comparison in context, not just as bare numbers.

An outlier is a value that's unusually far from the rest of the data. A common rule: a value is an outlier if it's more than $1.5 \times \text{IQR}$ beyond $Q_1$ or $Q_3$. Outliers can be genuine (worth investigating) or errors (worth removing) — the context decides which.