Estimating the Mean from Grouped Data
When data are grouped into classes or intervals, the exact values are not listed one by one. To estimate the mean, we treat each class as if all its values were concentrated at the class midpoint.
Method
- Find the midpoint of each class
- Midpoint = ((\text{lower bound} + \text{upper bound}) / 2)
- Multiply each midpoint by its frequency
- This gives the estimated total for that class.
- Add the frequency products
- Compute (\sum (\text{midpoint} \times \text{frequency})).
- Add all frequencies
- This gives the total number of data values.
- Divide
- Estimated mean = (\dfrac{\sum (\text{midpoint} \times \text{frequency})}{\sum \text{frequency}})
Important idea
The result is an estimate, not the exact mean, because the original data values inside each group are unknown. Use the midpoint as a representative value for each class.
Check your work
- Make sure every class has a midpoint.
- Make sure frequencies are matched to the correct class.
- Check that the final answer is reasonable and simplified.
If the answer is a decimal, round only if the exercise asks you to.