$250$ apples of a box were weighed and the distribution of masses of the apples is given in the following table:

| Mass (in grams) | $80 - 100$ | $100 - 120$ | $120 - 140$ | $140 - 160$ | $160 - 180$

Question

$250$ apples of a box were weighed and the distribution of masses of the apples is given in the following table:

| Mass (in grams) | $80 - 100$ | $100 - 120$ | $120 - 140$ | $140 - 160$ | $160 - 180$ |
|---|---|---|---|---|---|
| Number of apples | $20$ | $60$ | $70$ | $x$ | $60$ |

(i) Find the value of $x$ and the mean mass of the apples.
(ii) Find the modal mass of the apples.

Accepted Answer

Step 1: (i) $20 + 60 + 70 + x + 60 = 250 \Rightarrow x = 250 - 210 = 40$.
Step 2: Using direct method with $x_i$ as class marks:

| Mass | $f_i$ | $x_i$ | $f_i x_i$ |
|---|---|---|---|
| $80 - 100$ | $20$ | $90$ | $1800$ |
| $100 - 120$ | $60$ | $110$ | $6600$ |
| $120 - 140$ | $70$ | $130$ | $9100$ |
| $140 - 160$ | $40$ | $150$ | $6000$ |
| $160 - 180$ | $60$ | $170$ | $10200$ |
| **Total** | $250$ | | $33700$ |
Step 3: Mean mass $= \frac{33700}{250} = 134.8$ g.
Step 4: (ii) Modal class is $120 - 140$ (highest frequency $= 70$). Mode $= 120 + \frac{(70 - 60)}{(140 - 60 - 40)} 	imes 20$.
Step 5: $= 120 + \frac{10}{40} 	imes 20 = 120 + 5 = 125$ gm.

Question

Solution