Question

What is the lower limit of the median class ?

Answer 1

Step 1: Data Comprehension.
The provided data is a frequency distribution table:

Number of students per Teacher	Number of Schools
20 - 25	5
25 - 30	15
30 - 35	25
35 - 40	30
40 - 45	15
45 - 50	10

Step 2: Cumulative Frequency Calculation.
The cumulative frequency is calculated to find the median class. It is the running total of frequencies.
- Class 20-25: Cumulative frequency = 5
- Class 25-30: Cumulative frequency = 5 + 15 = 20
- Class 30-35: Cumulative frequency = 20 + 25 = 45
- Class 35-40: Cumulative frequency = 45 + 30 = 75
- Class 40-45: Cumulative frequency = 75 + 15 = 90
- Class 45-50: Cumulative frequency = 90 + 10 = 100
Step 3: Median Class Determination.
The total number of schools is 100. The median position is \( \frac{100}{2} = 50 \).
The cumulative frequency for class 30-35 is 45, and for class 35-40 is 75. The median class is the one where the cumulative frequency first exceeds 50, which is the 35-40 class.
Step 4: Lower Limit Identification.
The lower limit of the median class (35-40) is 35.
Final Answer:
The lower limit of the median class is \( \boxed{35} \).
 

Answer 2

Step 1: Analyze the provided frequency distribution.
The given data represents the number of schools corresponding to a specific range of students per teacher:

Number of students per Teacher	Number of Schools
20 - 25	5
25 - 30	15
30 - 35	25
35 - 40	30
40 - 45	15
45 - 50	10

Step 2: Determine the modal class.
The modal class is identified as the interval with the highest frequency. In this dataset, the interval 35-40 has the highest frequency of 30 schools.
Therefore, the modal class is 35-40.
Step 3: Identify the upper limit of the modal class.
For the modal class 35-40, the upper limit is 40.
Final Answer:
The upper limit of the modal class is \( \boxed{40} \).
 

Answer 3

Step 1: Calculate the Median.
First, determine the median class by calculating the cumulative frequency.

Cumulative frequency table:
- 20-25: 5
- 25-30: 5 + 15 = 20
- 30-35: 20 + 25 = 45
- 35-40: 45 + 30 = 75
- 40-45: 75 + 15 = 90
- 45-50: 90 + 10 = 100

The total number of schools is 100, so the median is the \( \frac{100}{2} = 50 \)-th school.
The cumulative frequency for the 30-35 class is 45, and for the 35-40 class is 75. Thus, the median class is 35-40.

The median formula is:
$$ \text{Median} = L + \left( \frac{\frac{N}{2} - F}{f} \right) \times h $$
Where:
- \(L\) = 35 (lower limit of the median class)
- \(N\) = 100 (total frequency)
- \(F\) = 45 (cumulative frequency of the class before the median class)
- \(f\) = 30 (frequency of the median class)
- \(h\) = 5 (class width)

Substituting the values:
$$ \text{Median} = 35 + \left( \frac{50 - 45}{30} \right) \times 5 $$
$$ \text{Median} = 35 + \left( \frac{5}{30} \right) \times 5 $$
$$ \text{Median} = 35 + \frac{25}{30} $$
$$ \text{Median} = 35 + 0.8333 $$
$$ \text{Median} = 35.83 $$

Step 2: Calculate the Mode.
The modal class is the class with the highest frequency, which is 35-40 with a frequency of 30.

The mode formula is:
$$ \text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h $$
Where:
- \(L\) = 35 (lower limit of the modal class)
- \(f_1\) = 30 (frequency of the modal class)
- \(f_0\) = 25 (frequency of the class before the modal class)
- \(f_2\) = 15 (frequency of the class after the modal class)
- \(h\) = 5 (class width)

Substituting the values:
$$ \text{Mode} = 35 + \left( \frac{30 - 25}{2(30) - 25 - 15} \right) \times 5 $$
$$ \text{Mode} = 35 + \left( \frac{5}{60 - 25 - 15} \right) \times 5 $$
$$ \text{Mode} = 35 + \left( \frac{5}{20} \right) \times 5 $$
$$ \text{Mode} = 35 + \left( 0.25 \right) \times 5 $$
$$ \text{Mode} = 35 + 1.25 $$
$$ \text{Mode} = 36.25 $$

Final Answer:
Median: \( \boxed{35.83} \).
Mode: \( \boxed{36.25} \).