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Find the value of Q1 and Q3 given data below

To find the values of the first quartile (Q1) and third quartile (Q3) for this grouped frequency distribution, we need to use the formula for quartiles in a grouped frequency distribution.

### Steps:

1. **Total number of students (N):**
   \[
   N = 2 + 3 + 6 + 12 + 13 + 11 + 7 = 54
   \]

2. **Cumulative frequency (cf):**
   We create a cumulative frequency table to assist in locating Q1 and Q3.

| Marks obtained | No. of students | Cumulative frequency (cf) |
|----------------|-----------------|---------------------------|
| 10-20          | 2               | 2                         |
| 20-30          | 3               | 5                         |
| 30-40          | 6               | 11                        |
| 40-50          | 12              | 23                        |
| 50-60          | 13              | 36                        |
| 60-70          | 11              | 47                        |
| 70-80          | 7               | 54                        |

### Quartile formulas:
- **First quartile (Q1)** corresponds to \( N/4 \).
- **Third quartile (Q3)** corresponds to \( 3N/4 \).

3. **Q1 Position:**
   \[
   N/4 = 54/4 = 13.5
   \]
   The cumulative frequency just greater than 13.5 is 23, which corresponds to the class interval **40-50**.

4. **Q3 Position:**
   \[
   3N/4 = 3 \times 54 / 4 = 40.5
   \]
   The cumulative frequency just greater than 40.5 is 47, which corresponds to the class interval **60-70**.

### Quartile Formula for Grouped Data:
The formula for the quartiles is:
\[
Q_k = L + \left( \frac{kN}{4} - \text{cf before class}}{f_k} \right) \times h
\]
where:
- \( L \) = lower boundary of the class containing the quartile,
- \( N \) = total number of data points,
- \( \text{cf before class} \) = cumulative frequency before the class containing the quartile,
- \( f_k \) = frequency of the class containing the quartile,
- \( h \) = class width.

#### For \( Q_1 \):
- \( L = 40 \), \( f_k = 12 \), \( \text{cf before class} = 11 \), \( h = 10 \)
\[
Q_1 = 40 + \left( \frac{13.5 - 11}{12} \right) \times 10
\]
\[
Q_1 = 40 + \left( \frac{2.5}{12} \right) \times 10 = 40 + 2.08 = 42.08
\]

#### For \( Q_3 \):
- \( L = 60 \), \( f_k = 11 \), \( \text{cf before class} = 36 \), \( h = 10 \)
\[
Q_3 = 60 + \left( \frac{40.5 - 36}{11} \right) \times 10
\]
\[
Q_3 = 60 + \left( \frac{4.5}{11} \right) \times 10 = 60 + 4.09 = 64.09
\]

### Conclusion:
- \( Q_1 \approx 42.08 \)
- \( Q_3 \approx 64.09 \)

Would you like further details on this solution or explanations of any part of the process?

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Here are some follow-up questions:
1. What is the difference between the lower and upper quartiles?
2. How can you use quartiles to determine the interquartile range?
3. Why is the cumulative frequency important in grouped data analysis?
4. How does the class width affect the calculation of quartiles?
5. How can we use quartiles to detect outliers in data?

**Tip:** Quartiles divide data into four equal parts, and their calculation is crucial for understanding the spread and distribution of data.

This page explains how to find the first and third quartiles (Q1 and Q3) using a grouped frequency distribution of student marks. It covers key statistical concepts like cumulative frequency, class intervals, and the quartile formula. Learn step-by-step calculations for students scoring between 10 and 80 marks, with Q1 ≈ 42.08 and Q3 ≈ 64.09.

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