Simulator by Aarthy Balasubramanian
Class 10 · Statistics
Median of grouped data, from the ogive
Build the running totals, plot the less-than ogive, read the median off the graph, then use the same graph to derive the formula.
Marks of 50 students in a unit test (out of 60)
| Marks | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 |
|---|---|---|---|---|---|---|
| Students | 3 | 6 | 12 | 16 | 8 | 5 |
How many scored less than a given mark?
Pick an upper limit. The shaded bars are the classes counted.
Ask the class
How many students scored less than 40? Can you get it from "less than 30" without adding everything again?
Show answer
Yes: 21 + 16 = 37. Add the next class to the running total. A running total like this is called the cumulative frequency.
Build the less-than table
Each row counts every student who scored less than the upper limit of a class.
Ask the class
Why does the table start with "less than 0"? What should the last cumulative frequency equal?
Show answer
Nobody scored less than 0, so the running total starts at 0. The last value must equal N = 50, the total number of students.
Plot the less-than ogive
Each row of the table is one point: (upper limit, cumulative frequency).
Ask the class
Why do we plot the upper limits and not the middle of each class? Why can the ogive never go down?
Show answer
"Less than 30" counts everyone up to the end of the class 20–30, so the point sits at 30. A running total can only stay the same or grow, so the curve only rises.
Read the median from the graph
The median is the middle score. Among 50 students, 25 score below it and 25 above, so go to N/2 = 25 on the vertical axis, across to the curve, then down to the marks.
Ask the class
The graph gives about 32.5, but a hand-drawn graph is never exact. How could we get the exact value?
Show answer
Zoom into the median class 30–40, where the ogive is a straight line, and use similar triangles. That is the next stage.
Derive the formula from the graph
Median of grouped data
Median = l + N/2 − cff × h
l lower limit of the median class · N total frequency · cf cumulative frequency of the class before it · f frequency of the median class · h class width
Joining two ogive points with a straight line assumes the students in a class are spread evenly across it, so the median is an estimate.