Data Analysis: Displaying, Summarising and Comparing Data
Core concepts
You must be fluent with the vocabulary of one- and two-variable data: categorical (nominal or ordinal) versus numerical (discrete or continuous) variables; frequency and percentage frequency tables; and the standard displays - dot plots, stem-and-leaf plots, histograms, bar charts, segmented bar charts, parallel boxplots and scatterplots. Know how to describe a distribution by shape (symmetric, positively skewed, negatively skewed), centre (mean, median, mode), spread (range, interquartile range, standard deviation) and outliers. Be able to compute the five-number summary, identify outliers using the 1.5 × IQR fences, and compare two or more groups in context. For bivariate data, interpret the correlation coefficient r (direction and strength only, never causation), describe form and strength, and read a least-squares regression line for prediction and interpolation.
Traps to watch
- Confusing median with mean when the data are skewed; the mean is pulled toward the tail.
- Forgetting that IQR uses Q3 − Q1, not the range between the lowest and highest.
- Reporting r without a context sentence, or implying that a strong correlation proves cause.
- Extrapolating far beyond the observed data range.
- Mixing up percentage frequency (count ÷ total × 100) with relative frequency.
Worked example
Data: 3, 5, 5, 6, 8, 9, 20. The median is 6; Q1 = 5 and Q3 = 9, so IQR = 4. Fences are 5 − 6 = −1 and 9 + 6 = 15, so 20 is an outlier. Because of that outlier, report the median and IQR rather than the mean and standard deviation when describing the centre and spread.
Rule of thumb
When outliers or strong skew appear, describe with median and IQR; when data are roughly symmetric with no outliers, mean and standard deviation are appropriate.