Question

What is the lower limit of the median class ?

Answer 1

Step 1: Understand the given data. 
We are given the following 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: Calculate the cumulative frequency.
We need to calculate the cumulative frequency to determine the median class. The cumulative frequency is the sum of the frequencies from the first class up to the current class.
- For 20-25: Cumulative frequency = 5
- For 25-30: Cumulative frequency = 5 + 15 = 20
- For 30-35: Cumulative frequency = 20 + 25 = 45
- For 35-40: Cumulative frequency = 45 + 30 = 75
- For 40-45: Cumulative frequency = 75 + 15 = 90
- For 45-50: Cumulative frequency = 90 + 10 = 100
Step 3: Find the median class.
The total number of schools is 100, so the median corresponds to the \( \frac{100}{2} = 50 \)-th school.
From the cumulative frequency table, we see that the cumulative frequency reaches 45 for the class 30-35 and reaches 75 for the class 35-40.
Therefore, the median class is the class where the cumulative frequency first exceeds 50, which is the class 35-40.
Step 4: Identify the lower limit of the median class.
The lower limit of the median class is the lower boundary of the class 35-40, which is **35**.
Final Answer:
The lower limit of the median class is \( \boxed{35} \).
 

Answer 2

Step 1: Understand the given data. 
We are given the following 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: Identify the modal class.
The modal class is the class interval with the highest frequency. From the table, we can see that the class with the highest frequency is 35-40 with a frequency of 30 schools.
Thus, the modal class is 35-40.
Step 3: Find the upper limit of the modal class.
The upper limit of the modal class 35-40 is 40.
Final Answer:
The upper limit of the modal class is \( \boxed{40} \).
 

Answer 3

Step 1: Find the Median of the Data.
To find the median, we need to first calculate the cumulative frequency and determine the median class.
The cumulative frequency table is as follows:
- For 20-25: Cumulative frequency = 5
- For 25-30: Cumulative frequency = 5 + 15 = 20
- For 30-35: Cumulative frequency = 20 + 25 = 45
- For 35-40: Cumulative frequency = 45 + 30 = 75
- For 40-45: Cumulative frequency = 75 + 15 = 90
- For 45-50: Cumulative frequency = 90 + 10 = 100
The total number of schools is 100, so the median corresponds to the \( \frac{100}{2} = 50 \)-th school.
From the cumulative frequency table, we see that the cumulative frequency reaches 45 for the class 30-35 and reaches 75 for the class 35-40.
Therefore, the median class is the class 35-40.
To find the median, we use the formula: $$ \text{Median} = L + \left( \frac{\frac{N}{2} - F}{f} \right) \times h $$
Where: - \(L\) is the lower limit of the median class (35). - \(N\) is the total frequency (100). - \(F\) is the cumulative frequency of the class before the median class (45). - \(f\) is the frequency of the median class (30). - \(h\) is the class width (5). Substituting the values into the formula: $$ \text{Median} = 35 + \left( \frac{50 - 45}{30} \right) \times 5 $$
$$ \text{Median} = 35 + \left( \frac{5}{30} \right) \times 5 $$
$$ \text{Median} = 35 + \frac{25}{30} $$
$$ \text{Median} = 35 + 0.8333 $$
$$ \text{Median} = 35.83 $$
Step 2: Find the Modal of the Data.
The modal class is the class with the highest frequency. From the table, we can see that the class with the highest frequency is 35-40 with a frequency of 30 schools.
To find the mode, we use the formula: $$ \text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h $$
Where: - \(L\) is the lower limit of the modal class (35). - \(f_1\) is the frequency of the modal class (30). - \(f_0\) is the frequency of the class before the modal class (25). - \(f_2\) is the frequency of the class after the modal class (15). - \(h\) is the class width (5). Substituting the values into the formula: $$ \text{Mode} = 35 + \left( \frac{30 - 25}{2(30) - 25 - 15} \right) \times 5 $$
$$ \text{Mode} = 35 + \left( \frac{5}{60 - 25 - 15} \right) \times 5 $$
$$ \text{Mode} = 35 + \left( \frac{5}{20} \right) \times 5 $$
$$ \text{Mode} = 35 + \left( 0.25 \right) \times 5 $$
$$ \text{Mode} = 35 + 1.25 $$
$$ \text{Mode} = 36.25 $$
Final Answer:
The median of the data is \( \boxed{35.83} \).
The mode of the data is \( \boxed{36.25} \).