Class 10 Maths · Chapter 13 · Topic 1 of 5
Read the classes before calculating
Understand class boundaries and frequencies before calculating.
Ideas you’ll use
- Class mark
- The midpoint of a class: xᵢ=(lower limit+upper limit)/2. It represents that whole class when estimating the mean.
Understand it, one idea at a time
Choose an idea. Follow its reason, calculation and clue in the picture.
A frequency histogram groups 20 observations
Frequency (count)
Class value (number)
- Equal-width class frequencies · Exact original data or calculated contributions.
Assumptions: Original fictional dataset. Continuous graph interpolation is a model where stated; actual observations are not recovered from it.
Read the plotted values
Equal-width class frequencies
| Class value (number) | Frequency (count) |
|---|---|
| 0 | 0 |
| 0 | 3 |
| 10 | 3 |
| 10 | 5 |
| 20 | 5 |
| 20 | 8 |
| 30 | 8 |
| 30 | 4 |
| 40 | 4 |
| 40 | 0 |
Intervals have width 10. Their heights 3, 5, 8, 4 show the class frequencies.
Read the boundary rule
For adjacent classes 0–10 and 10–20, include 10 in the second, not both.
Lower included, upper excluded in this modelIn the picture: The histogram bins share boundaries.
Read all the reasoning together
Read the boundary rule
For adjacent classes 0–10 and 10–20, include 10 in the second, not both.
Lower included, upper excluded in this modelAdd the frequencies
Our fictional classes have frequencies 3,5,8,4.
n=3+5+8+4=20Find representative midpoints
Average each class’s endpoints.
midpoints=5,15,25,35Remember what grouping hides
Raw values within each class are unknown, so grouped calculations estimate them.
Eight values in 20–30 need not all equal 25
Keep these values handy
| Class | f | Midpoint x |
|---|---|---|
| 0–10 | 3 | 5 |
| 10–20 | 5 | 15 |
| 20–30 | 8 | 25 |
| 30–40 | 4 | 35 |
Understand class boundaries and frequencies before calculating.
Worked example: Our original revision dataset
Classes 0–10, 10–20, 20–30, 30–40 have frequencies 3, 5, 8, 4. Midpoints are 5, 15, 25, 35; n=20All later methods in this chapter use this same clearly labelled fictional dataset. The eight observations in 20–30 are represented by 25, although their actual values need not all be 25.
Read the supporting explanation
Use a clear boundary rule. In 0–10, 10–20, 20–30, a value 10 belongs to 10–20, not both. Frequencies add to the total n. Grouping hides the exact individual values, so class-mark calculations give estimates of the original mean.
Go deeper: Make inclusive classes continuous
For measurements rounded to the nearest unit, classes 10–19, 20–29, 30–39 become 9.5–19.5, 19.5–29.5, 29.5–39.5. Half the 1-unit gap shifts both ends. Midpoints remain unchanged; median/mode use these continuous boundaries.
Your turn
Try the idea yourself. Show your reasoning, then compare.
Question
Why is the mean of the four midpoints alone usually wrong?
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