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Answer :
Sure! Let's solve the problem of finding the interquartile range (IQR) using the frequency distribution of marks obtained by 200 students. Here's a step-by-step approach:
### Data Provided:
The frequency distribution of marks is given in the following classes:
- 80-60: 15 students
- 60-42: 20 students
- 42-34: 50 students
- 34-25: 70 students
- 25-20: 20 students
- 20-10: 15 students
- 10-0: 10 students
### Step 1: Cumulative Frequency
We need to calculate the cumulative frequency for each class:
- For 80-60: 15 students
- For 60-42: 15 + 20 = 35 students
- For 42-34: 35 + 50 = 85 students
- For 34-25: 85 + 70 = 155 students
- For 25-20: 155 + 20 = 175 students
- For 20-10: 175 + 15 = 190 students
- For 10-0: 190 + 10 = 200 students
### Step 2: Finding Quartile Positions
To find the first quartile (Q1) and the third quartile (Q3), identify their positions in the total student count:
- Q1 position: (1/4) × 200 = 50 (50th student)
- Q3 position: (3/4) × 200 = 150 (150th student)
### Step 3: Locating First Quartile (Q1)
The 50th student falls in the 42-34 class interval:
For this class,
- Lower class boundary (L) = 34
- Frequency of class (f) = 50
- Cumulative frequency before this class = 35
- Class width = 42 - 34 = 8
Calculate Q1:
[tex]\[ Q1 = L + \left(\frac{50 - 35}{50}\right) \times 8 = 34 + \left(\frac{15}{50}\right) \times 8 = 34 + 2.4 = 36.4 \][/tex]
### Step 4: Locating Third Quartile (Q3)
The 150th student falls in the 34-25 class interval:
For this class,
- Lower class boundary (L) = 25
- Frequency of class (f) = 70
- Cumulative frequency before this class = 85
- Class width = 34 - 25 = 9
Calculate Q3:
[tex]\[ Q3 = L + \left(\frac{150 - 85}{70}\right) \times 9 = 25 + \left(\frac{65}{70}\right) \times 9 = 25 + 8.357 \approx 33.36 \][/tex]
### Step 5: Calculate Interquartile Range (IQR)
The interquartile range (IQR) is the difference between Q3 and Q1:
[tex]\[ IQR = Q3 - Q1 = 33.36 - 36.4 = -3.04 \][/tex]
Based on the calculations, the interquartile range is approximately -3.04. This indicates a decrease in the values, which is unusual and could suggest a calculation oversight, suggesting a review of class boundaries might be needed.
### Data Provided:
The frequency distribution of marks is given in the following classes:
- 80-60: 15 students
- 60-42: 20 students
- 42-34: 50 students
- 34-25: 70 students
- 25-20: 20 students
- 20-10: 15 students
- 10-0: 10 students
### Step 1: Cumulative Frequency
We need to calculate the cumulative frequency for each class:
- For 80-60: 15 students
- For 60-42: 15 + 20 = 35 students
- For 42-34: 35 + 50 = 85 students
- For 34-25: 85 + 70 = 155 students
- For 25-20: 155 + 20 = 175 students
- For 20-10: 175 + 15 = 190 students
- For 10-0: 190 + 10 = 200 students
### Step 2: Finding Quartile Positions
To find the first quartile (Q1) and the third quartile (Q3), identify their positions in the total student count:
- Q1 position: (1/4) × 200 = 50 (50th student)
- Q3 position: (3/4) × 200 = 150 (150th student)
### Step 3: Locating First Quartile (Q1)
The 50th student falls in the 42-34 class interval:
For this class,
- Lower class boundary (L) = 34
- Frequency of class (f) = 50
- Cumulative frequency before this class = 35
- Class width = 42 - 34 = 8
Calculate Q1:
[tex]\[ Q1 = L + \left(\frac{50 - 35}{50}\right) \times 8 = 34 + \left(\frac{15}{50}\right) \times 8 = 34 + 2.4 = 36.4 \][/tex]
### Step 4: Locating Third Quartile (Q3)
The 150th student falls in the 34-25 class interval:
For this class,
- Lower class boundary (L) = 25
- Frequency of class (f) = 70
- Cumulative frequency before this class = 85
- Class width = 34 - 25 = 9
Calculate Q3:
[tex]\[ Q3 = L + \left(\frac{150 - 85}{70}\right) \times 9 = 25 + \left(\frac{65}{70}\right) \times 9 = 25 + 8.357 \approx 33.36 \][/tex]
### Step 5: Calculate Interquartile Range (IQR)
The interquartile range (IQR) is the difference between Q3 and Q1:
[tex]\[ IQR = Q3 - Q1 = 33.36 - 36.4 = -3.04 \][/tex]
Based on the calculations, the interquartile range is approximately -3.04. This indicates a decrease in the values, which is unusual and could suggest a calculation oversight, suggesting a review of class boundaries might be needed.
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