Learning Objectives
- Classify and tabulate categorical, discrete and continuous data.
- Construct and interpret frequency tables, pictograms, bar graphs and pie charts.
- Choose a suitable representation for a given dataset.
- Identify misleading statistical diagrams and unsupported conclusions.
- Plan the main stages of a fair statistical investigation.
Key Terms
- Frequency
- The number of times a value or category occurs.
- Population
- The full group about which information is required.
- Sample
- A subset of the population that is observed.
- Bias
- A systematic influence that makes results unrepresentative.
- Primary data
- Data collected first-hand for the investigation.
- Secondary data
- Data previously collected by another person or organisation.
8th Edition Chapter Map
- Frequency tables
- Pictograms
- Bar graphs
- Pie charts
- Evaluation of statistical representations
- Statistical investigations
Statistics And Types Of Data
Statistics involves collecting, organising, displaying and interpreting data. Categorical data records labels or groups, such as transport method. Numerical data records numbers. Discrete data takes separate countable values, while continuous data can take any value in an interval and is usually measured.
The population is the entire group of interest; a sample is a smaller group studied to learn about the population. A sample should be sufficiently large and representative. A biased sampling method systematically favours some members, leading to unreliable conclusions.
Frequency Tables
Frequency is the number of times a value or category occurs. Tally marks help record raw data, and the total frequency must match the number of observations. Categories should be mutually exclusive so one response cannot belong to two categories, and collectively exhaustive so every response can be placed.
| Books read | Tally | Frequency |
|---|---|---|
| 0 | ||| | 3 |
| 1 | ||||/ || | 7 |
| 2 | ||||/ |||| | 9 |
| 3 or more | ||||/ | | 6 |
Pictograms
A pictogram uses symbols with a stated key. Partial symbols represent corresponding fractions of the key. Pictograms are visually accessible for small simple datasets but become inefficient for large frequencies or awkward fractional values.
Worked Example: Pictogram Key
If one symbol represents 8 students, 2\frac12 symbols represent 2.5\times8=20 students. Always include the key when drawing a pictogram.
Bar Graphs
Bar graphs show categorical or discrete data. Bars have equal widths and separated gaps; frequency is represented by height. The vertical scale should be uniform and both axes labelled. A truncated vertical axis may exaggerate differences, so read the starting value before interpreting.
Pie Charts
A pie chart represents a whole as 360^\circ. Sector angle is proportional to frequency.
Worked Example: Constructing A Pie Chart
In a survey of 120 students, 42 choose football, 30 cricket, 18 hockey and the rest athletics.
- Football angle =42/120\times360=126^\circ.
- Cricket =90^\circ.
- Hockey =54^\circ.
- Athletics frequency =30, angle =90^\circ.
The angles total 360^\circ, providing a useful check.
Choosing And Evaluating A Representation
| Display | Useful for | Limitation |
|---|---|---|
| Frequency table | Exact organised values | Patterns may be less visually obvious |
| Pictogram | Simple small datasets and young audiences | Partial symbols can be awkward |
| Bar graph | Comparing category sizes | Scale choices can mislead |
| Pie chart | Showing proportions of one whole | Difficult to compare similar sectors accurately |
Statistical diagrams can mislead through unequal scales, three-dimensional effects, omitted categories, unequal bar widths or pictorial symbols whose area grows faster than the represented value. A correct interpretation distinguishes what the data shows from conclusions the data cannot support.
Planning A Statistical Investigation
- Question: define a clear measurable question.
- Collection: choose suitable primary or secondary data and a fair sampling method.
- Organisation and display: use a frequency table and an appropriate diagram.
- Interpretation: identify patterns, compare groups and acknowledge limitations.
Questionnaires should avoid leading wording, ambiguous terms and overlapping response options. A convenience sample may be easy to collect but may not represent the population. Random or systematic sampling is usually more defensible when properly applied.
Examination Guidance
- Include titles, labels, units, uniform scales and a key where required.
- For a pie chart, check sector angles total 360^\circ.
- Read the graph scale before extracting values.
- When evaluating data, mention sample size, sampling method and possible bias.
- Do not calculate mean, median or mode in this D1 chapter; those are treated in the 8th-edition Book 3 chapter on averages.
Common Mistakes
- Drawing touching bars for categorical data.
- Using an incomplete key in a pictogram.
- Finding a pie-chart angle as frequency divided by 360.
- Making a causal claim from a simple association.
- Ignoring an axis that does not start at zero.
Chapter Practice
1. A category has frequency 35 in a survey of 140. Find its pie-chart angle.
35/140\times360=90^\circ.
2. A pie-chart sector measures 72^\circ and represents 18 people. Find the total number surveyed.
72/360=1/5, so 18 is one fifth and the total is 90.
3. Explain one problem with surveying only students in the school library about weekly reading time.
The sample is likely biased toward students who read or use the library more than the general school population.
Further Statistical Practice
4. In a sample of 200 people, 54 prefer option A. Find the pie-chart angle.
54/200\times360=97.2^\circ.
5. Give one advantage and one disadvantage of a pie chart compared with a bar graph.
A pie chart shows each category as part of the whole clearly. However, it is difficult to compare sectors of similar size or read exact frequencies without labels.
6. A survey question asks, “Don’t you agree that school meals should be healthier?” Explain the problem and improve it.
The wording is leading and encourages agreement. A neutral alternative is, “How satisfied are you with the healthiness of school meals?” followed by balanced response options.
7. A bar chart uses pictures of cars whose height and width both double for a value that doubles. Explain why this is misleading.
Doubling both dimensions makes the picture’s area four times as large, so the visual impression exaggerates a twofold increase.
