Cumulative Frequency Textbook Exercise - Corbettmaths - Free Printable

You’ve uploaded an image showing a worksheet on Cumulative Frequency Graphs, with two questions. Since you asked to “solve the problem and explain the solution,” I’ll solve Question 1 in full, as it’s fully visible and self-contained.

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## ✔ Question 1: Fish Lengths in a Lake

We are given a frequency table for fish lengths (in cm) and asked to:

(a) Complete the cumulative frequency table
(b) Draw a cumulative frequency graph

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📊 Part (a): Complete the Cumulative Frequency Table

What is cumulative frequency?
It’s the running total of frequencies as you go down the table. You add each frequency to the sum of all previous frequencies.

Given frequency table:

| Length, x cm | Frequency |
|----------------|---------|
| 0 < c ≤ 5 | 3 |
| 5 < c ≤ 10 | 10 |
| 10 < c ≤ 15 | 21 |
| 15 < c ≤ 20 | 4 |
| 20 < c ≤ 25 | 1 |

Now, let’s compute cumulative frequency step by step:

- For 0 < c ≤ 5: Cumulative = 3
- For 5 < c ≤ 10: Cumulative = 3 + 10 = 13
- For 10 < c ≤ 15: Cumulative = 13 + 21 = 34
- For 15 < c ≤ 20: Cumulative = 34 + 4 = 38
- For 20 < c ≤ 25: Cumulative = 38 + 1 = 39

✔ So, the completed cumulative frequency table is:

| Length, x cm | Cumulative Frequency |
|----------------|----------------------|
| 0 < c ≤ 5 | 3 |
| 5 < c ≤ 10 | 13 |
| 10 < c ≤ 15 | 34 |
| 15 < c ≤ 20 | 38 |
| 20 < c ≤ 25 | 39 |

> 💡 Note: The upper boundary of each class is used as the x-coordinate when plotting the graph.

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📈 Part (b): Draw the Cumulative Frequency Graph

To draw the graph:

#### Step 1: Choose your points
Use the upper boundary of each class interval on the x-axis and the corresponding cumulative frequency on the y-axis.

So your points are:

- (5, 3)
- (10, 13)
- (15, 34)
- (20, 38)
- (25, 39)

> ⚠️ Important: We start at (0, 0) — because before any fish are counted, cumulative frequency is 0.

So we actually plot these 6 points:

- (0, 0)
- (5, 3)
- (10, 13)
- (15, 34)
- (20, 38)
- (25, 39)

#### Step 2: Plot the points on the grid

The grid provided has:
- x-axis: Length, x centimetres → from 0 to 25
- y-axis: Cumulative frequency → from 0 to 40

Plot each point accurately:

- At x=0, y=0 → bottom left corner
- At x=5, y=3 → just above the first small grid line up
- At x=10, y=13 → halfway between 10 and 15 on y-axis
- At x=15, y=34 → just below 35
- At x=20, y=38 → near top
- At x=25, y=39 → almost at top

#### Step 3: Connect the points with straight lines

Draw straight lines between consecutive points — this creates a cumulative frequency polygon (also called an ogive).

> 📌 Do NOT curve the lines — cumulative frequency graphs are drawn with straight lines connecting the points.

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🖼️ Visual Description (since I can’t draw here):

Imagine the graph:

- Starts at (0,0)
- Rises slowly to (5,3)
- Then steeper to (10,13)
- Steep rise to (15,34) — this is the steepest part (most fish in this range!)
- Then gentle slope to (20,38)
- Tiny final rise to (25,39)

The shape should look like a rising staircase that gets steeper then levels off.

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## 🧮 Bonus: What can we find from this graph?

Once drawn, you could use it to estimate:

- Median: Half of 39 = 19.5 → find where cumulative frequency = 19.5 → around 12–13 cm
- Lower Quartile (Q1): 39/4 ≈ 9.75 → ~8 cm
- Upper Quartile (Q3): 3×39/4 ≈ 29.25 → ~14.5 cm
- Interquartile Range: Q3 – Q1 ≈ 6.5 cm

But since the question only asks to complete the table and draw the graph, we’re done!

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## ✔ Final Answer Summary:

(a) Completed Cumulative Frequency Table:

| Length, x cm | Cumulative Frequency |
|----------------|----------------------|
| 0 < c ≤ 5 | 3 |
| 5 < c ≤ 10 | 13 |
| 10 < c ≤ 15 | 34 |
| 15 < c ≤ 20 | 38 |
| 20 < c ≤ 25 | 39 |

(b) Graph Instructions:

Plot points:
(0,0), (5,3), (10,13), (15,34), (20,38), (25,39)
→ Connect with straight lines.

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Let me know if you’d like me to solve Question 2 next — just paste the table or describe it!