Averages from Grouped Data

What Is Grouped Data?

Grouped data is when values are presented in class intervals (groups) rather than as individual numbers.

• Useful for large sets of data.
• Each group or class interval represents a range of values.
• Frequencies show how many data points fall in each group.

Data can be:

• Discrete (e.g. shoe sizes)
• Continuous (e.g. height, time)

 

Why Do We Estimate the Mean from Grouped Data?

We estimate the mean because:

• The exact data values aren’t given.
• We assume each value in a class interval is equal to its midpoint.

Use the formula:

$$\text{Estimated Mean} = \frac{\sum (\text{midpoint} \times \text{frequency})}{\sum \text{frequency}}$$

 

How to Estimate the Mean

Step-by-Step:

1. Find the midpoint of each class interval.
2. Multiply midpoint by frequency (fx).
3. Add up all the fx values.
4. Add up the frequencies.
5. Divide total fx by total frequency.

 

Example

The table shows the time 30 students spent on homework each week:

 

Time (hours)	Frequency
$0 \leq t < 2$	3
$2 \leq t < 4$	5
$4 \leq t < 6$	10
$6 \leq t < 8$	7
$8 \leq t < 10$	5

Find tthe mean time spent on homework.

 

Step 1: Add midpoints and calculate fx

 

Time (hours)	Frequency	Midpoint	fx
$0 \leq t < 2$	3	1	3
$2 \leq t < 4$	5	3	15
$4 \leq t < 6$	10	5	50
$6 \leq t < 8$	7	7	49
$8 \leq t < 10$	5	9	45
Total	20		162

 

Step 2: Estimate the mean

$$\text{Estimated Mean} = \frac{162}{30} = 5.4 \text{ hours}$$

 

Modal Class

• The class with the highest frequency.
• In this case, the modal class is:

$$4 \leq t < 6$$

 

Finding the Median Class

1. Find total frequency, $n = 30$
2. Find the $\frac{n+1}{2} = \frac{31}{2} = 15.5$th value
3. Use a running total:
   • Up to class 1: 3
   • Up to class 2: 3 + 5 = 8
   • Up to class 3: 8 + 10 = 18

The 15.5th value lies in the 3rd class:

$$4 \leq t < 6$$