TheIBAxolotl
Frequency tables
A frequency table lists each value (or class interval) with how often it occurs. For continuous data the classes are written like \(10 \le t < 20\) so every value belongs to exactly one class. Adding a running total column gives the cumulative frequency.
Histograms
A histogram shows grouped continuous data with bars that touch. In this course the classes have equal widths, so the bar height is the frequency. The tallest bar shows the modal class. Look at the shape: symmetric, skewed to the right (long tail of large values), or skewed to the left.
Cumulative frequency graphs
Plot cumulative frequency against the upper boundary of each class and join the points with a smooth curve (or straight segments). To read statistics:
- Median: go across at \(\frac{n}{2}\) and read down.
- \(Q_1\) and \(Q_3\): go across at \(\frac{n}{4}\) and \(\frac{3n}{4}\).
- Percentiles: the \(p\)th percentile is at \(\frac{p}{100} \times n\).
- "How many are less than \(x\)?": go up from \(x\) and read across.
Quartiles from a list
For raw data, sort first. The median splits the list in half; \(Q_1\) is the median of the lower half and \(Q_3\) the median of the upper half (excluding the overall median when \(n\) is odd, which matches the GDC). The interquartile range \(IQR = Q_3 - Q_1\) measures the spread of the middle half.
Worked example
Marks of 11 students: 3, 5, 6, 8, 9, 10, 12, 13, 15, 18, 30.
- Median: 6th value \(= 10\).
- Lower half 3, 5, 6, 8, 9 gives \(Q_1 = 6\); upper half 12, 13, 15, 18, 30 gives \(Q_3 = 15\).
- \(IQR = 9\). Fences: \(6 - 13.5 = -7.5\) and \(15 + 13.5 = 28.5\), so 30 is an outlier.
Box-and-whisker plots
A box plot shows the five-number summary: minimum, \(Q_1\), median, \(Q_3\), maximum. Outliers are drawn as crosses and the whisker then stops at the largest value that is not an outlier. In the example, the whisker would end at 18 with a cross at 30.
Comparing data sets
When comparing two box plots, always make two comments:
- A centre comparison: "Class A has a higher median (64 vs 58), so it typically scored higher."
- A spread comparison: "Class B has a larger IQR (22 vs 12), so its marks are more varied."
Use the numbers and put them in context.
GDC tips
- One-variable statistics gives \(Q_1\), median and \(Q_3\) for raw lists or for a value list with a frequency list.
- Most GDCs draw box plots (including a version that marks outliers) from the same lists.
Common mistakes
- Plotting cumulative frequency at class midpoints instead of upper boundaries.
- Leaving gaps between histogram bars for continuous data.
- Comparing ranges instead of IQRs when there are outliers.
- Giving comparisons without context or numbers.
Exam tips
- Draw reading lines on the cumulative frequency graph to show method.
- Answers read from graphs are accepted within a tolerance, so draw carefully with a ruler.
- "Compare" needs a centre and a spread statement.