Question
The following frequency distribution gives the monthly consumption of electricity of 68 consumers in a locality:
| Monthly Consumption (units) | Number of Consumers |
|---|---|
| 65–85 | 4 |
| 85–105 | 5 |
| 105–125 | 13 |
| 125–145 | 20 |
| 145–165 | 14 |
| 165–185 | 8 |
| 185–205 | 4 |
Draw a less-than ogive and a more-than ogive. Use them to find the median.
(CBSE 2024 Board Exam)
Solution — Step by Step
We add up frequencies from the top. Each row answers: “how many consumers use less than this upper boundary?”
| Less Than (units) | Cumulative Frequency |
|---|---|
| 85 | 4 |
| 105 | 9 |
| 125 | 22 |
| 145 | 42 |
| 165 | 56 |
| 185 | 64 |
| 205 | 68 |
Plot these points: (85, 4), (105, 9), (125, 22), (145, 42), (165, 56), (185, 64), (205, 68). Connect with a smooth curve. This is your less-than ogive.
Now we subtract from the total (68), going from bottom to top. Each row answers: “how many consumers use more than this lower boundary?”
| More Than (units) | Cumulative Frequency |
|---|---|
| 65 | 68 |
| 85 | 64 |
| 105 | 59 |
| 125 | 46 |
| 145 | 26 |
| 165 | 12 |
| 185 | 4 |
Plot: (65, 68), (85, 64), (105, 59), (125, 46), (145, 26), (165, 12), (185, 4). This is your more-than ogive.
Why do we look for where the two curves cross? Because at the median, exactly half the data lies below and half lies above — so the cumulative count from each direction must both equal .
Here, , so .
Draw a horizontal line at . It cuts both ogives at the same point — that point is the intersection. Drop a perpendicular from the intersection to the x-axis.
The perpendicular meets the x-axis at approximately 137.05 units.
This is the median monthly consumption.
Why This Works
The less-than ogive is an increasing S-curve (it can only stay flat or rise). The more-than ogive is a decreasing S-curve. They must cross exactly once, and that crossing point represents the value where cumulative frequency from below equals cumulative frequency from above — by definition, the median.
This is a clean graphical alternative to the formula method. Both give the same answer (), but the graphical method is often faster in board exams and requires no calculation beyond the cumulative tables.
Draw both ogives on the same graph paper with the same axes. The intersection is only visible if both are on the same plot. Students who draw them on separate sheets have to compare points manually — inefficient and error-prone.
Alternative Method — Using the Formula
We can verify using the median formula directly:
. The cumulative frequency just below 34 is 22 (for the class 105–125). So the median class is 125–145.
Where:
- (lower boundary of median class)
- (cumulative frequency before median class)
- (frequency of median class)
- (class width)
The graphical method gives ~137.05, the formula gives exactly 137. The tiny difference is due to reading error on the graph — both are acceptable in CBSE.
Common Mistake
Using class midpoints instead of boundaries for the ogive.
Less-than ogive points use upper class boundaries as x-coordinates (85, 105, 125…). More-than ogive points use lower class boundaries (65, 85, 105…). Many students plot midpoints (75, 95, 115…) out of habit from frequency polygons. This shifts both curves sideways and your intersection gives a wrong median. Always use boundaries for ogives.
The weightage for statistics in CBSE Class 10 is typically 11–12 marks, and ogive-based problems appear almost every year. Nail the table construction and your graph will follow naturally.