📊 Data Presentation & Diagrams
Raw engineering data — tolerances, test results, sensor readings — means little until it's organised. Frequency tables and diagrams turn a messy list of numbers into a shape you can actually interpret.
2Topics
8Worked Examples
4Eng. Cases
12Practice Qs
📋Frequency TablesTallying & grouping
📊DiagramsBar, histogram, polygon
⚙️ApplicationsReal engineering
🎮Quick-Fire GameTest your speed
📝PracticeGraded questions
🎯What you will learn
Organise raw data into frequency tables, choose sensible class widths for grouped data, and represent the results as bar charts, histograms and frequency polygons — the essential first step before any statistical analysis.
💡 A histogram is not just "a bar chart with no gaps" — for unequal class widths, the bar height must be frequency density, not frequency, or the shape lies to you.
🧭How to use this resource
1. Read the rules banner at the top of each tab
2. Experiment with the interactive tool
3. Check the key vocabulary panel
4. Study the visual diagram, then the worked examples
📋
Frequency Tables & Grouping
The first step with any raw dataset: count how often each value or range of values occurs.
🔧Class Width & Midpoint CalculatorEnter data range and number of classes
Enter the data range and desired number of classes, then press Calculate to get a sensible class width and boundaries…
📖 Key Vocabulary
Frequency — the number of times a value or class occurs in the data.
Class boundary — the true dividing line between adjacent classes, avoiding ambiguity at the edges.
Class width — the size of each grouped interval; usually kept constant across a table.
Tally chart — a quick counting method using tally marks, later converted to frequencies.
👁️ Visual aid: each raw value is sorted into exactly one class, and the tally becomes the frequency column.
✏️Worked Example 1 — Building a frequency tableType: Ungrouped
Question Data: 2,3,2,4,3,3,2,5,4,3. Build a frequency table.
Tally each value 2 appears 3×, 3 appears 4×, 4 appears 2×, 5 appears 1×
Check total
3+4+2+1 = 10 ✓ (matches the 10 data items)Answer Frequency table: 2→3, 3→4, 4→2, 5→1
📏Worked Example 2 — Choosing a class widthType: Class width
Question 40 shaft diameters range from 10.2 mm to 15.8 mm. Suggest a suitable class width for about 7 classes.
Range
15.8 − 10.2 = 5.6Divide by desired classes
5.6 ÷ 7 = 0.8Answer A class width of 0.8 mm gives exactly 7 classes: 10.2–11.0, 11.0–11.8, … up to 15.0–15.8.
🎯Worked Example 3 — Finding class midpointsType: Midpoints
Question Find the midpoint of the class 20–29 (class boundaries 19.5–29.5).
Apply formula
midpoint = (19.5+29.5)/2Answer midpoint = 24.5 — this single value represents every item in the class for later calculations.
⚙️Worked Example 4 — Engineering context (tolerance inspection)Type: Engineering
Question 50 machined bolts have diameters recorded to the nearest 0.01 mm, ranging from 11.94 mm to 12.08 mm. Group into 7 classes of equal width and state the boundaries of the first class.
Range
12.08 − 11.94 = 0.14Class width
0.14 ÷ 7 = 0.02Answer First class: 11.94–11.96 mm (boundaries 11.935–11.965 mm) — this framework lets QC engineers instantly see how many bolts cluster near the target diameter.
🧠 Common slip: letting classes overlap (e.g. 10–20 then 20–30) — always use boundaries that make it unambiguous which class a borderline value belongs to.
📊
Diagrams — Bar Charts, Histograms & Frequency Polygons
Three ways to turn a frequency table into a picture — each suited to a different type of data.
🔧Frequency Density CalculatorFor histograms with unequal class widths
Enter the frequency and class width, then press Calculate for the correct histogram bar height…
📖 Key Vocabulary
Bar chart — bars with gaps between them, one per category or discrete value.
Histogram — bars touching, used for continuous data; bar area (not height) represents frequency.
Frequency density — frequency ÷ class width; the correct y-axis for histograms with unequal widths.
Frequency polygon — midpoints of histogram bars joined by straight lines, useful for comparing distributions.
👁️ Visual aid: the histogram's bars touch because the underlying data is continuous; the polygon overlays the same shape as a simplified outline.
✏️Worked Example 1 — Bar chart vs histogramType: Choosing the diagram
Question State which diagram is appropriate for (a) number of defects per shift (5 categories: 0,1,2,3,4+), (b) cable lengths grouped 0–10m, 10–20m, etc.
(a) Discrete categories → bar chart, with gaps between bars
(b) Continuous grouped data → histogram, bars touching
Answer (a) bar chart, (b) histogram
📐Worked Example 2 — Equal class widthsType: Equal widths
Question A histogram has classes of equal width 5, with frequencies 4, 9, 14, 7. Find the frequency densities.
Apply formula
density = frequency ÷ widthCompute
4/5=0.8, 9/5=1.8, 14/5=2.8, 7/5=1.4Answer With equal widths, densities are proportional to frequencies — the histogram looks identical to a simple frequency bar chart.
⚖️Worked Example 3 — Unequal class widthsType: Unequal widths
Question A class 20–50 (width 30) has frequency 12; a class 50–60 (width 10) has frequency 8. Compare their frequency densities.
Class 1
12 ÷ 30 = 0.4Class 2
8 ÷ 10 = 0.8Answer Density 0.8 vs 0.4 — despite having a lower raw frequency, the narrower class is actually twice as "dense", which plotting raw frequency would hide.
⚙️Worked Example 4 — Engineering context (test result spread)Type: Engineering
Question Tensile strength test results (MPa) are grouped: 200–220 (f=5, width 20), 220–230 (f=8, width 10), 230–260 (f=6, width 30). Find the frequency densities to correctly draw the histogram.
Class 1
5 ÷ 20 = 0.25Class 2
8 ÷ 10 = 0.80Class 3
6 ÷ 30 = 0.20Answer Plotting these densities (not the raw frequencies 5, 8, 6) correctly reveals that the middle class is by far the most concentrated.
🧠 Common slip: plotting raw frequency as bar height when class widths differ — this always distorts the true shape of the distribution. Always convert to frequency density first.
⚙️
Engineering Applications
Four live scenarios — each with its own calculator. Enter your own numbers.
🔩 Quality Control — Class Width Planner
Before inspecting a batch, QC engineers plan sensible measurement classes so tolerance clustering is visible at a glance.
Class width ≈ (max − min) ÷ desired number of classes
📟 Try it — Class Width Planner
Enter min, max and desired classes, then press Calculate…
📊 Histogram Density — Unequal Class Correction
Any time class widths differ (common with real sensor data), the y-axis must be frequency density, not frequency.
Frequency density = frequency ÷ class width
📟 Try it — Density Corrector
Enter frequency and class width, then press Calculate…
📈 Defect Rate — Bar Chart Proportions
Comparing defect counts across production lines as percentages of total output makes disparate batch sizes comparable.
Percentage = (count ÷ total) × 100
📟 Try it — Defect Percentage
Enter defect count and total inspected, then press Calculate…
🌡️ Sensor Logging — Class Boundary Finder
Given a class's stated range (e.g. "20–29"), the true class boundaries matter for correctly assigning borderline sensor readings.
Boundaries = stated limits ± half the gap to the next class
📟 Try it — Boundary Finder
Enter the stated limits and the gap, then press Calculate…
🎮
Quick-Fire Data Presentation Game
12 questions on tables and diagrams. How many can you get?
⚡ Data Presentation Challenge
Score: 0Question: 1/12Streak: 0🔥
Press Start to begin!
📝
Practice Questions
Graded from warm-up to challenge. Attempt fully before revealing the working.
Question 1 🥉2 marks
Data: 5,6,5,7,6,6,5,8. Find the frequency of each value.
5→3, 6→3, 7→1, 8→1 (total 8 ✓)
Question 2 🥉2 marks
Find the midpoint of the class 30–39 (boundaries 29.5–39.5).
(29.5+39.5)/2 = 34.5
Question 3 🥉2 marks
State which diagram suits: number of employees per department (5 departments).
Discrete categories → bar chart
Question 4 🥉2 marks
A class has frequency 10 and width 5. Find its frequency density.
10 ÷ 5 = 2
Question 5 🥈3 marks
Data ranges from 5 to 68. Suggest a class width for approximately 9 classes.
Range = 68−5 = 63
63 ÷ 9 = 7
63 ÷ 9 = 7
Question 6 🥈4 marks
Two classes: 10–20 (width 10, f=6) and 20–25 (width 5, f=6). Which has the higher frequency density?
Class 1: 6/10=0.6
Class 2: 6/5=1.2
Class 2 (20–25) has the higher density despite equal raw frequency
Class 2: 6/5=1.2
Class 2 (20–25) has the higher density despite equal raw frequency
Question 7 🥈3 marks
A class is stated as "40–49" with the next class starting at 50. Find the true upper class boundary.
Gap = 50−49 = 1, half-gap = 0.5
Upper boundary = 49.5
Upper boundary = 49.5
Question 8 🥈4 marks
A histogram class 0–15 (width 15) has frequency density 0.4. Find its frequency.
frequency = density × width = 0.4 × 15 = 6
Question 9 🥇4 marks
Machined shaft diameters (mm) range 19.85–20.20. Choose a class width for 7 classes and state the first class's boundaries.
Range = 0.35, width = 0.35/7 = 0.05
First class: 19.85–19.90 mm
First class: 19.85–19.90 mm
Question 10 🥇5 marks
Three classes: 0–10(w=10,f=8), 10–15(w=5,f=9), 15–30(w=15,f=6). Rank them by frequency density (highest first).
Densities: 8/10=0.8, 9/5=1.8, 6/15=0.4
Order: 10–15 (1.8) > 0–10 (0.8) > 15–30 (0.4)
Order: 10–15 (1.8) > 0–10 (0.8) > 15–30 (0.4)
Question 11 🥇5 marks
A batch of 420 bolts is inspected; 18 are rejected. Find the rejection rate as a percentage and explain how it would compare across batches of different sizes.
(18/420)×100 = 4.29% (3 s.f.)
Using percentages (not raw counts) makes batches of different sizes directly comparable on a bar chart.
Using percentages (not raw counts) makes batches of different sizes directly comparable on a bar chart.
Question 12 🥇5 marks
A frequency table has classes 0–20(f=10), 20–30(f=15), 30–70(f=12). Explain why plotting raw frequencies would misrepresent this data, and calculate the correct heights.
Widths differ (20,10,40) so raw frequency ignores how "spread out" each class is.
Densities: 10/20=0.5, 15/10=1.5, 12/40=0.3
Correct heights: 0.5, 1.5, 0.3 — plotting raw f (10,15,12) would wrongly make the last class look nearly as tall as the first.
Densities: 10/20=0.5, 15/10=1.5, 12/40=0.3
Correct heights: 0.5, 1.5, 0.3 — plotting raw f (10,15,12) would wrongly make the last class look nearly as tall as the first.