Exact or only close?
Grouping the marks moves every student, so the grouped mean is approximate.
Two tables, one class
Table 13.1 lists the exact marks of the 30 students. Table 13.3 puts the same students in six classes. In the first table every student stands at his or her own mark. In the second, every student stands at a class mark.
We found 62 from the grouped table. Now we use the exact marks.
Will the beam of the exact marks balance at the same place as the beam of the grouped marks?
What this lesson covers
The idea
The mean computed from grouped data is approximate and may differ from the exact mean of the ungrouped data, because of the mid-point assumption.
Two tables, one class
Table 13.1 lists the exact marks of the 30 students. Table 13.3 puts the same students in six classes. In the first table every student stands at his or her own mark. In the second, every student stands at a class mark.
We found 62 from the grouped table. Now we use the exact marks.
Will the beam of the exact marks balance at the same place as the beam of the grouped marks?
- Yes, exactly the same place
- No, but very close
- No, far apart
Balance both beams
Two tabs: the 30 exact marks, and the same 30 students moved to their class marks. Balance each beam. The −0.1 and +0.1 buttons move the pivot by a tenth.
The grouped mean is approximate
The mean found from grouped data is approximate. It can differ from the exact mean of the ungrouped data, because we assume every student stands at the class mark.
Exact marks: x̄ = 1779 / 30 = 59.3. Class marks: x̄ = 1860 / 30 = 62. The exact mean is 59.3. The grouped mean, 62, is an approximation to it.
Notes
The mean of grouped data is approximate. It may differ from the exact mean of the ungrouped data, because of the mid-point assumption.
Check yourself
Table 13.1 gives Σfx = 1779 for the 30 exact marks. Find their exact mean.
Answer: 59.3
1779/30 = 59.3.
Why did the grouped table give 62 while the exact marks give 59.3?
In the class 25–40, 3 students scored exactly 36 each. In the grouped table they stand at the class mark 32.5. By how many marks is the class mark below their real mark?
Answer: 3.5
36 − 32.5 = 3.5. Moves like this, some up and some down, make the grouped mean differ a little from the exact mean.
A teacher has only the grouped table (Table 13.3). Which mean can she find?
- Each student was moved from his or her own mark to the class mark. — correct. Yes! The mid-point assumption moves students, so the mean can move too.
- The grouped table has a different formula.. It is the same formula, Σfx / Σf. Only the x values changed: class marks instead of exact marks.
- Somebody made a mistake in adding.. Both sums are right. The two answers differ because of the mid-point assumption.
- The approximate mean, 62, using class marks — correct. Yes! She does not know the exact marks, so she uses class marks and gets an approximate mean.
- The exact mean, 59.3. To get 59.3 she needs every exact mark, which the grouped table does not show.
- No mean at all. She can find a mean by assuming each class stands at its class mark. It is approximate.