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The marks obtained by $80$ students of class $X$ in a mock test of Mathematics are given below in the table. Find median and the mode of the data :
| Marks | Number of Students |
|---|---|
| 0 and above | 80 |
| 10 and above | 77 |
| 20 and above | 72 |
| 30 and above | 65 |
| 40 and above | 55 |
| 50 and above | 43 |
| 60 and above | 28 |
| 70 and above | 16 |
| 80 and above | 10 |
| 90 and above | 8 |
| 100 and above | 0 |
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Correct Table
Marks | Number of Students | Class Interval | $f$ | $Cf$
0 and above | 80 | 0-10 | 3 | 3
10 and above | 77 | 10-20 | 5 | 8
20 and above | 72 | 20-30 | 7 | 15
30 and above | 65 | 30-40 | 10 | 25
40 and above | 55 | 40-50 | 12 | 37
50 and above | 43 | 50-60 | 15 | 52
60 and above | 28 | 60-70 | 12 | 64
70 and above | 16 | 70-80 | 6 | 70
80 and above | 10 | 80-90 | 2 | 72
90 and above | 8 | 90-100 | 8 | 80
Total | | | 80 | (I) (1 Mark)
$n = 80 \Rightarrow \frac{n}{2} = 40$
$50-60$ is the median class (II) (1/2 Mark)
Median $= 50 + (\frac{40-37}{15}) \times 10$ (III) (1 Mark)
$= 50 + 2 = 52$ (IV) (1/2 Mark)
$50-60$ is the modal class (V) (1/2 Mark)
Mode $= 50 + (\frac{15-12}{2\times15-12-12}) \times 10$ (VI) (1 Mark)
$= 50 + 5 = 55$ (VII) (1/2 Mark)
Marks | Number of Students | Class Interval | $f$ | $Cf$
0 and above | 80 | 0-10 | 3 | 3
10 and above | 77 | 10-20 | 5 | 8
20 and above | 72 | 20-30 | 7 | 15
30 and above | 65 | 30-40 | 10 | 25
40 and above | 55 | 40-50 | 12 | 37
50 and above | 43 | 50-60 | 15 | 52
60 and above | 28 | 60-70 | 12 | 64
70 and above | 16 | 70-80 | 6 | 70
80 and above | 10 | 80-90 | 2 | 72
90 and above | 8 | 90-100 | 8 | 80
Total | | | 80 | (I) (1 Mark)
$n = 80 \Rightarrow \frac{n}{2} = 40$
$50-60$ is the median class (II) (1/2 Mark)
Median $= 50 + (\frac{40-37}{15}) \times 10$ (III) (1 Mark)
$= 50 + 2 = 52$ (IV) (1/2 Mark)
$50-60$ is the modal class (V) (1/2 Mark)
Mode $= 50 + (\frac{15-12}{2\times15-12-12}) \times 10$ (VI) (1 Mark)
$= 50 + 5 = 55$ (VII) (1/2 Mark)