LEARNING OBJECTIVES
What you will be able to do
- classify and tabulate data
- compare distributions responsibly
- calculate averages and measures of spread
- draw and interpret statistical diagrams
- analyse scatter diagrams
- use cumulative frequency curves
- draw and interpret histograms
AT A GLANCE
INTRODUCTION · THE BIG IDEA
Organise, represent and interpret data while recognising what the evidence can—and cannot—show.
Statistics turns data into information. A calculation or graph summarises a data set, but no single summary captures everything.
Good interpretation compares both centre and spread, considers how the data were collected, and avoids claiming more than the evidence supports.
SECTION 01
Data, tables and responsible comparison
Data may be discrete, such as number of messages, or continuous, such as mass. Tally and two-way tables organise counts and reveal categories.
Compare distributions using an average and a measure of spread. A lower median with a smaller interquartile range, for example, means lower typical values with greater consistency.
Association does not prove causation. A biased sample, small sample or hidden variable can limit a conclusion.
DETAILED EXPLANATION
- Primary data is collected for the investigation; secondary data already exists.
Comparing two delivery services
- Service A: median 31 min, IQR 8 min.
- Service B: median 28 min, IQR 15 min.
- Compare median for typical speed and IQR for consistency.
Answer: B is typically faster, but A is more consistent. Neither measure alone supports both conclusions.
SECTION 02
Averages and measures of spread
The mean uses every value and is affected by extremes. The median is the central ordered value and is resistant to extremes. The mode is the most frequent value.
Range uses only the extremes. Interquartile range, Q₃ − Q₁, describes the middle half of the data.
For grouped data, use class midpoints to estimate the mean because exact values are unknown. The modal class has the greatest frequency.
Estimated mean from grouped data
- Classes 0–10, 10–20, 20–30 have frequencies 4, 7, 5.
- Midpoints are 5, 15 and 25.
- Σfx = 4(5) + 7(15) + 5(25) = 250.
- Σf = 16.
Answer: Estimated mean = 250/16 = 15.625.
SECTION 03
Charts and statistical diagrams
Bar charts have separated bars for categories. Histograms have touching bars because they represent continuous intervals. A simple frequency diagram displays how often each value or class occurs.
Pie-chart sector angles are proportional to frequency. Pictograms use a key to show how many items each symbol represents, including fractions of a symbol. Stem-and-leaf diagrams retain individual values and require ordered leaves plus a key.
Composite and dual bar charts compare related groups; choose a consistent scale and include a legend.
Pie chart sector
- In a survey of 80 students, 26 choose swimming.
- Sector angle = 26/80 × 360°.
- = 117°.
Answer: The swimming sector has angle 117°.
SECTION 04
Scatter diagrams and lines of best fit
Positive correlation rises left to right, negative correlation falls, and zero correlation shows no clear linear trend. Correlation strength describes how closely points follow a pattern.
A line of best fit is a single ruled line through the middle of the data with a roughly even balance of points on both sides.
Interpolation estimates within the observed range and is usually safer than extrapolation beyond it. An outlier does not follow the main pattern.
Using a line of best fit
- A line passes approximately through (2, 18) and (8, 42).
- Gradient = (42 − 18)/(8 − 2) = 4.
- Using y = 4x + c and (2, 18), c = 10.
- For x = 6, y = 4(6) + 10.
Answer: The estimated value is 34.
SECTION 05
Cumulative frequency
Cumulative frequency is a running total. Plot each total at the upper class boundary and join points with a smooth increasing curve.
The median is read at N/2, the lower quartile at N/4 and the upper quartile at 3N/4. Percentiles use the corresponding percentage of N.
Reading quartile positions
- A curve represents 120 observations.
- Q₁ is read at cumulative frequency 30.
- Median is read at 60.
- Q₃ is read at 90.
Answer: Read the corresponding horizontal-axis values and subtract Q₁ from Q₃ for the IQR.
SECTION 06
Histograms and frequency density
A histogram represents frequency by area, not simply bar height. Unequal class widths therefore require frequency density on the vertical axis.
Class width is the difference between boundaries. Frequency equals bar area, so it is density multiplied by class width.
Finding a histogram bar height
- The class 20 ≤ x < 35 has frequency 45.
- Class width = 35 − 20 = 15.
- Frequency density = 45/15.
Answer: The bar height is 3 on the frequency-density axis.
QUICK CHAPTER SUMMARY
The ideas to carry forward
- Classify data before choosing a display.
- Compare both centre and spread.
- Grouped means are estimates based on midpoints.
- Charts need accurate scales, labels and keys.
- Correlation is association, not proof of cause.
- Cumulative frequency gives medians, quartiles and percentiles.
- Histogram area represents frequency.
QUICK REVISION CHECKLIST
Can you do each of these without your notes?
- classify and tabulate data
- compare distributions responsibly
- calculate averages and measures of spread
- draw and interpret statistical diagrams
- analyse scatter diagrams
- use cumulative frequency curves
- draw and interpret histograms