Statistical Reasoning
What this domain tests
Statistical Reasoning covers data collection and organization, one-variable distributions, measures of central tendency and dispersion, and graphical representation. The exam asks you to read and produce graphs, compute summary measures, and interpret what they mean in context.
Core rules and concepts
- Collecting data: distinguish a census from a sample, and identify bias in a sampling method. A representative sample reflects the population's composition.
- Organizing data: frequency tables, stem-and-leaf plots, and histograms for grouped data. Choose class intervals of equal width and count each value once.
- Measures of central tendency: mean (sum divided by count), median (middle value of ordered data), and mode (most frequent value). The median is resistant to outliers; the mean is not.
- Measures of dispersion: range (maximum minus minimum), and interquartile range (Q3 minus Q1). The IQR describes the spread of the middle half of the data.
- Graphical representation: bar charts for categorical data, histograms for continuous grouped data, and box-and-whisker plots to display median, quartiles, and extremes.
Worked example
A class records the number of books read: 2, 3, 3, 4, 5, 5, 5, 6, 8, 12. Find the mean, median, mode, range, and IQR.
Sum: $2+3+3+4+5+5+5+6+8+12=53$ over 10 values, so the mean is 5.3.
The data are already ordered; the median is the average of the 5th and 6th values: $\frac{5+5}{2}=5$. The mode is 5 (three occurrences). The range is $12-2=10$.
For quartiles, split into two halves of five. Q1 is the median of the lower half (2, 3, 3, 4, 5), which is 3; Q3 is the median of the upper half (5, 5, 6, 8, 12), which is 6. The IQR is $6-3=3$. The value 12 is an outlier candidate, yet the median stays at 5, illustrating resistance.
Traps that cost marks
- Computing the median without first ordering the data.
- Confusing the range with the IQR, or reporting Q3 as the maximum.
- Using a bar chart for continuous grouped data where a histogram is required.
- Describing an association from a graph without checking the scale of the axes.
- Reporting a mean from grouped data as if it were exact when class midpoints were used.