Using Cumulative Frequency Diagrams

 

What Can I Learn from a Cumulative Frequency Diagram?

Cumulative frequency diagrams aren't just for plotting data—they help us estimate some really useful facts like:

• The median
• The lower quartile (Q1)
• The upper quartile (Q3)
• The interquartile range (IQR)
• Percentiles (like the 10th or 90th percentile)

These are all estimates since grouped data doesn’t show exact individual values.

 

Understanding the Basics

• The y-axis shows cumulative frequency (how many values are less than or equal to a point).
• The x-axis shows the variable (like time, height, or score).
• The curve should rise steadily and never drop back down.
• All values are estimated from the curve—not exact

 

How to Estimate the Median

Step-by-step:

1. Find half the total number of values: $\frac{n}{2}$
2. From that number on the y-axis, draw a horizontal line to the curve.
3. From that point on the curve, draw a vertical line down to the x-axis.
4. The value you hit on the x-axis is your median estimate.

 

Lower and Upper Quartiles

Just like the median, but for the 25% and 75% points.

• Lower Quartile (Q1): $\frac{n}{4}$
• Upper Quartile (Q3): $\frac{3n}{4}$

Follow the same steps:

1. Start on the y-axis at $\frac{n}{4}$​ or $\frac{3n}{4}$
2. Go across to the curve
3. Drop down to find the estimate on the x-axis

 

Interquartile Range (IQR)

Once you know Q1 and Q3:

$$\text{IQR} = \text{Q3} - \text{Q1}$$

This tells you how spread out the middle 50% of your data is.

 

 

 

 

Estimating Percentiles

To find, for example, the 10th percentile:

1. Calculate $\frac{10}{100} \times n$ = the 10% position.
2. Follow the same method as for median/quartiles.

You can find the 90th percentile, 60th percentile, or any other in just the same way.

 

Example

A bird sanctuary tracks the flight time (in seconds) of 80 birds using gliding techniques.

They plot the data in a cumulative frequency diagram.

 

 

(a) Estimate the median, Q1 and Q3.

 

 

There are 80 birds:

• Median $\to \frac{80}{2} ​= 40$
• Q1 $\to \frac{80}{4} = 20$
• Q3 $\to \frac{3 \times 80}{4} = 60$

Go to the y-axis at 20, 40 and 60.

• Draw horizontal lines to meet the curve
• Then vertical lines down to find estimates on the x-axis

Let's say the diagram gives us:

• Q1 = 9.5 s
• Median = 11.2 s
• Q3 = 13.3 s

$$\text{IQR} = 13.3 - 9.5 = 3.8 \text{ seconds}$$

(b) Estimate how many birds flew longer than 14 seconds

 

 

• From 14 seconds on the x-axis, draw a vertical line up to the curve.
• Then draw a horizontal line across to the y-axis.

Let’s say you land on 72.

That means 72 birds flew up to 14 seconds.

So the number of birds who flew longer than 14 seconds:

$$80 - 72 = 8 \text{ birds}$$