For the following data the value of `Q_(1) + Q_(3) - Q_(2),` is
`{:("Age in years" ,:,20,30,40,50,60,70,80),("No. of members" ,:,3,61,132,153,140,51,3):}`

Options

A. `Q_(1)`
B. `Q_(2)`
C. `Q_(3)`
D. `2Q_(2)`

Explanation

To solve the problem, we need to find the values of the first quartile (Q1), second quartile (Q2), and third quartile (Q3) for the given data, and then compute the expression \( Q1 + Q3 - Q2 \).

### Step 1: Organize the Data
We have the following data:

| Age (years) | No. of Members (frequency) |
|-------------|----------------------------|
| 20 | 3 |
| 30 | 61 |
| 40 | 132 |
| 50 | 153 |
| 60 | 140 |
| 70 | 51 |
| 80 | 3 |

### Step 2: Calculate the Cumulative Frequency
We will calculate the cumulative frequency for the data.

- For age 20: Cumulative Frequency = 3
- For age 30: Cumulative Frequency = 3 + 61 = 64
- For age 40: Cumulative Frequency = 64 + 132 = 196
- For age 50: Cumulative Frequency = 196 + 153 = 349
- For age 60: Cumulative Frequency = 349 + 140 = 489
- For age 70: Cumulative Frequency = 489 + 51 = 540
- For age 80: Cumulative Frequency = 540 + 3 = 543

The cumulative frequency table is:

| Age (years) | Cumulative Frequency |
|-------------|----------------------|
| 20 | 3 |
| 30 | 64 |
| 40 | 196 |
| 50 | 349 |
| 60 | 489 |
| 70 | 540 |
| 80 | 543 |

### Step 3: Calculate Total Frequency (n)
The total frequency \( n \) is the last cumulative frequency value, which is 543.

### Step 4: Calculate Q1
To find \( Q1 \):
- Use the formula: \( Q1 = \frac{n}{4} = \frac{543}{4} = 135.75 \)

Now, we need to find the cumulative frequency just greater than 135.75, which is 196 (corresponding to age 40). Thus, \( Q1 = 40 \).

### Step 5: Calculate Q2
To find \( Q2 \):
- Use the formula: \( Q2 = \frac{n}{2} = \frac{543}{2} = 271.5 \)

Now, we need to find the cumulative frequency just greater than 271.5, which is 349 (corresponding to age 50). Thus, \( Q2 = 50 \).

### Step 6: Calculate Q3
To find \( Q3 \):
- Use the formula: \( Q3 = \frac{3n}{4} = \frac{3 \times 543}{4} = 407.25 \)

Now, we need to find the cumulative frequency just greater than 407.25, which is 489 (corresponding to age 60). Thus, \( Q3 = 60 \).

### Step 7: Calculate \( Q1 + Q3 - Q2 \)
Now we can calculate the required expression:
\[ Q1 + Q3 - Q2 = 40 + 60 - 50 = 50 \]

### Final Answer
The value of \( Q1 + Q3 - Q2 \) is **50**.

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