Chapter Overview
🧭Your visual roadmap through the chapter
Data, Types & Collection
Classifying data and how it's gathered
Data Organization & Summary
Arrays, frequency distributions, tally, tabulation
Graphical Representation
Bar/pie charts, histograms, ogives, stem-leaf, time series
Data is the source of information. This chapter covers the different types of data, how it is collected, how raw data is organized and summarized, and how it is represented graphically.
Recognising data types and choosing the right organizing/graphing method is the foundation for all later statistical analysis and interpretation topics.
Classify data correctly; build frequency distributions and tables; construct/read histograms, ogives and stem-leaf plots; identify fractiles from a graph.
Key Definitions
📘Source-based terminology, exam-ready
Data vs Information
Data refers to facts, figures or values. Information is derived when facts are processed, structured and analysed for decision making.
Key word: raw vs processed
Numerical / Categorical
Numerical (quantitative) data allows arithmetic operations; categorical (qualitative) data does not.
Key word: can you calculate it?
Discrete / Continuous
Discrete variables take only certain, separate values (a count). Continuous variables can take any value in a range.
Key word: count vs measure
Ordinal / Nominal
Ordinal data has a natural order. Nominal data has no natural order — reordering doesn't change its meaning.
Key word: order matters or not
Cross-sectional / Time Series
Cross-sectional: data at one distinct point in time. Time series: observations of one phenomenon tracked over time.
Key word: one moment vs over time
Primary / Secondary Data
Primary data is collected first-hand for the investigation. Secondary data was previously collected and may be relevant.
Key word: first-hand vs reused
Population / Sample
Population data comes from investigating every member. Sample data is a representative selection within the population.
Key word: everyone vs a selection
Structured / Unstructured
Structured data has a pre-defined model or organization. Unstructured data has no organized form.
Key word: organized or not
Core Concept
🎯Explained in exam-oriented language
CONCEPT → Organizing Raw Data
MEANING
Raw data is rearranged into arrays, frequency distributions and tables so it becomes easier to understand.
PURPOSE
"The human mind cannot assimilate data" in its raw form — structure makes it usable.
APPLICATION
Summarising large datasets (e.g. 100 salesmen's distances) before analysis.
CONCEPT → Graphical Representation
MEANING
Organized data is displayed visually — some methods simple (bar, pie), others more sophisticated (histogram, ogive).
PURPOSE
Show relationships, proportions and distributions at a glance.
APPLICATION
Choosing the right chart for categorical vs grouped numerical data.
Key Rules & Principles
📏Every source rule, exam-ready
RULE 1 Constructing grouped classes
Identify the minimum and maximum term to create classes
Keep the number of classes relatively small (6–7, or 15–20 for large datasets)
Classes should generally be of equal width for easier comparison
Class limits must be unambiguous (e.g. "0 to less than 10")
The upper limit of one class is the lower limit of the next — no gaps
RULE 3 Principles of data tabulation
Simplicity: classify and keep details to a minimum
Title: a clear title explaining the table's purpose
Source, Units and Headings must be clearly labelled
Totals and subtotals shown where meaningful
RULE 4 Unequal class widths in a histogram
Either calculate frequency density (frequency ÷ class width), or
Adjust the frequency to what it would be for the standard width, then use that as the bar height
RULE 2 Data quality
Useful data should be: complete, relevant & significant, timely, appropriately detailed, clear, properly communicated, and reliable/consistent
Formula Zone
🧮Every formula supported by the source text
RELATIVE FREQUENCY
The sum of all relative frequencies in a distribution always equals 1.
FREQUENCY DENSITY
Used to build a histogram bar height when class widths are unequal.
CLASS WIDTH, BOUNDARY & MID-POINT
Class boundary: midpoint between one class's upper limit and the next class's lower limit (±0.5 for discrete data). Mid-point = (upper limit + lower limit) ÷ 2.
FRACTILES (PERCENTILE / DECILE / QUARTILE)
Decile = 1/10 of the way through; Quartile = 1/4 of the way through. Q2 = 50th percentile = the median.
Calculation Roadmap
🗺️General path for any data-organization question
1· Read Question
Identify the raw data and what's required
2· Identify Method
Array, frequency table, chart, or fractile?
3· Organize
Sort, tally, or group into classes
4· Calculate/Plot
Apply the formula, or plot the chart
5· Interpret
Read off / state the final answer
Step-by-step Methods
🪜Kept separate — do not mix methods
METHOD A · Building a Grouped Frequency Distribution
- 1Find the minimum and maximum values (the range)
- 2Decide the number of classes; divide range by classes for width
- 3Set unambiguous class limits (e.g. "480 to under 500")
- 4Tally each data value into its class, then count for frequency
METHOD C · Histogram with Unequal Class Widths
- 1Note each class's frequency and width
- 2Calculate frequency density = frequency ÷ class width for each class
- 3Plot frequency density as the bar height (not raw frequency)
- 4Alternative: adjust frequency to a common width, then use as height
METHOD B · Cumulative Frequency (Less/More Than)
- 1List classes in order with their individual frequencies
- 2"Less than" series: add each class's frequency to the running total of all previous classes
- 3"More than" series: start at the total and subtract each class's frequency going forward
- 4Use for plotting an ogive
METHOD D · Reading a Fractile from an Ogive
- 1Find total number of items (n) in the distribution
- 2Calculate the position: e.g. Q1 is the 25% × n th item
- 3Find that position on the vertical (cumulative frequency) axis
- 4Read across to the curve, then down to the horizontal axis for the value
METHOD E · Constructing a Stem-and-Leaf Plot
- 1Sort the data into ascending order
- 2Split each value into a stem (leading digits) and a leaf (last digit)
- 3List stems once down the left; write each value's leaf beside its stem
- 4Define the units for stem and leaf clearly (e.g. stem=10s, leaf=1s)
Different Cases / Scenarios
🔀Recognise the data, pick the right chart or method
IF data is categorical (e.g. proportions of a whole)
→ USE a bar chart or pie chart
IF data is a grouped frequency distribution
→ USE a histogram (density if class widths differ)
IF you need a running/cumulative total plotted
→ USE an ogive
IF you need to keep the actual raw data values visible
→ USE a stem-and-leaf plot
IF you need to show a variable's trend over time
→ USE a time series graph
IF class widths in a histogram are NOT equal
→ USE frequency density, not raw frequency, as bar height
IF asked for a percentile/decile/quartile value
→ Find its position, then read it off an ogive
IF data was collected specifically for this investigation
→ It is primary data; if reused from elsewhere, it's secondary
Comparison Tables
⚖️Don't confuse these!
| Numerical (Quantitative) | Categorical (Qualitative) | |
|---|---|---|
| Arithmetic possible? | Yes | No |
| Examples | Age, salary, temperature | Eye colour, location, gender |
| Discrete | Continuous | |
|---|---|---|
| Values | Certain, separate values (a count) | Any value within a range |
| Examples | Number of contracts, children | Height, weight |
| Ordinal | Nominal | |
|---|---|---|
| Order | Has a natural order | No natural order |
| Examples | A/B/C grade, small-medium-large | Gender, pass-fail |
| "Less Than" Cumulative Series | "More Than" Cumulative Series | |
|---|---|---|
| Direction | Adds frequencies going up through classes | Subtracts frequencies going up through classes |
| Starts at | 0 | Total frequency |
Graph & Data Interpretation
📈How to read the source's charts and tables
Reading an ogive
Each point shows how many items fall below the corresponding value on the horizontal axis (a "less than" ogive). To find a fractile, locate its position on the vertical axis, trace across to the curve, then down to read the value.
Reading a histogram
Bar height = frequency when class widths are equal, or frequency density when they are not. There are no gaps between bars, since the data is continuous/grouped.
Reading a stem-and-leaf plot
The stem (left of the line) holds the leading digit(s); each leaf (right of the line) is one data point's last digit. Reading a stem with all its leaves reconstructs every original value in that class.
Reading pie & component bar charts
A pie chart shows each category's share of the whole (slice size = proportion/percentage). A percentage component bar chart shows every bar at equal height, split into segments by percentage share instead.
Worked Calculation Logic
✍️Method students must reproduce in the exam
GIVEN → FORMULA → SUBSTITUTION → CALCULATION → ANSWER
An ogive shows a distribution of 40 items. What is the 1st quartile of the distribution?
GIVEN: n = 40 items; Q1 = the item 25% of the way through
FORMULA: Position of Q1 = 25% × n
SUBSTITUTION: Position = 0.25 × 40
CALCULATION: Position = the 10th item — locate 10 on the vertical (cumulative frequency) axis, trace to the curve, then down to the horizontal axis
FINAL ANSWER: Reading off the ogive at the 10th item gives a 1st quartile of approximately 110
Exam Question Patterns
📝Classification — identify data as numerical/categorical, discrete/continuous, etc.
Frequency distribution — build a grouped table from raw data
Chart selection — choose the appropriate chart for given data
Fractile reading — find a percentile/decile/quartile from an ogive
Density calculation — histogram with unequal class widths
Interpretation — reading stem-leaf, pie, or component bar charts
Exam Traps
⚠️⚠ WATCH OUT
- ✗ Discrete vs continuous confusion — a count (discrete) is not the same as a measurement (continuous).
- ✗ Using raw frequency as histogram height when class widths are unequal — use frequency density instead.
- ✗ Ambiguous class limits — always state classes as "X to under Y" to avoid double-counting a boundary value.
- ✗ Less than vs more than series mix-up — one accumulates upward, the other counts down from the total.
- ✗ Misreading fractile position — e.g. using n instead of the correct fraction of n (25% for Q1).
Quick Decision Tree
🌳Chapter 7 in 60 Seconds
⚡Data types: numeric/categorical, discrete/continuous, ordinal/nominal Primary = first-hand; secondary = reused Relative freq = class f ÷ total f Frequency density = f ÷ class width
Charts: bar/pie (categorical), histogram/ogive (grouped) Ogive = cumulative frequency curve Q2 = median = 50th percentile
Trap: unequal widths need frequency density
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