Cumulative Frequency & Quartile Calculator

Generate precise Ogive Curves, estimate Median, $Q_1$, $Q_3$, and Interquartile Range (IQR) for Cambridge IGCSE Math (0580).

1. Input Grouped Data
Class Boundary Upper Bound Frequency ($f$)
0 < x ≤ 10
10 < x ≤ 20
20 < x ≤ 30
30 < x ≤ 40
40 < x ≤ 50
Calculated Summary
Total Frequency ($N$): 60
Median ($Q_2$): --
Lower Quartile ($Q_1$): --
Upper Quartile ($Q_3$): --
IQR ($Q_3 - Q_1$): --
2. Interactive Cumulative Frequency Diagram (Ogive Curve)
Step-by-Step Solution Steps:
  • Click "Plot & Calculate" to generate working steps.

Key Exam Notes: Cumulative Frequency for IGCSE

1. Plotting Points Correctly

Always plot Cumulative Frequency ($c.f.$) on the vertical axis (Y-axis) against the Upper Class Boundary on the horizontal axis (X-axis). Never use midpoints for Cumulative Frequency diagrams!

2. Reading Quartiles & IQR

  • Median ($Q_2$): Read value at $\frac{1}{2}N$ on the Y-axis.
  • Lower Quartile ($Q_1$): Read value at $\frac{1}{4}N$ on the Y-axis.
  • Upper Quartile ($Q_3$): Read value at $\frac{3}{4}N$ on the Y-axis.
  • Interquartile Range (IQR): $IQR = Q_3 - Q_1$.

3. Common Mark Scheme Pitfalls

Ensure your curve starts at $c.f. = 0$ from the lowest boundary (e.g., origin). Draw a smooth S-shaped curve (Ogive) through all plotted points—do not use straight lines unless requested!

IGCSE Exam Masterclass: Worked Example

Past Paper Question Analysis (Cambridge IGCSE 0580)

Question: The table below shows the examination marks of 80 students.

Marks ($m$) Frequency ($f$) Cumulative Frequency ($c.f.$)
0 < m ≤ 2088
20 < m ≤ 401624 ($8+16$)
40 < m ≤ 603054 ($24+30$)
60 < m ≤ 801872 ($54+18$)
80 < m ≤ 100880 ($72+8$)
Exam Solution & Mark Scheme Walkthrough:
  • (a) Total Students ($N$): $N = 80$.
  • (b) Find the Median position: $\frac{1}{2}N = \frac{80}{2} = 40$. Read from $c.f. = 40$ on the Y-axis across to the curve $\rightarrow$ Estimated Median $\approx 51$ marks.
  • (c) Find the Interquartile Range (IQR):
    • Lower Quartile ($Q_1$ pos): $\frac{1}{4}N = 20 \rightarrow$ Read $X$-axis $\approx 36$
    • Upper Quartile ($Q_3$ pos): $\frac{3}{4}N = 60 \rightarrow$ Read $X$-axis $\approx 64$
    • $\mathbf{IQR = Q_3 - Q_1} = 64 - 36 = \mathbf{28}$ marks.
  • (d) Typical Percentile Question: "Find the number of students who scored more than 75 marks."
    → Locate $75$ on the X-axis, read up to curve $\rightarrow c.f. \approx 68$.
    → Number of students scoring more than 75 = $80 - 68 = \mathbf{12\text{ students}}$.

Top 4 Mistakes to Avoid in IGCSE Examinations

❌ Plotting Midpoints

Mistake: Plotting cumulative frequency against class midpoints.

Correct: Always plot points at the Upper Class Boundary. (e.g., for $20 < m \le 40$, plot at $X = 40$).

❌ Incorrect Graph Origin

Mistake: Starting the curve from the first frequency point without connecting to $Y=0$.

Correct: Curve must start at $(X_{\text{min}}, Y=0)$ — the lower boundary of the very first class.

❌ Straight Lines vs Curves

Mistake: Joining points with strict straight lines (like a polygon).

Correct: Draw a smooth S-shaped curve (Ogive) passing through all cumulative frequency points.

❌ Misreading "More Than"

Mistake: Giving the raw cumulative frequency value as the final answer for "students scoring above $X$".

Correct: Always subtract from total frequency: $\text{Students Above} = \text{Total } N - c.f.\text{ value}$.

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