Question

What is the lower limit of the median class ?

Answer 1

Step 1: Understanding the problem:
We are asked to find the lower limit of the median class based on the given frequency distribution table. The median class is the class interval that contains the median value of the data.

Step 2: Finding the cumulative frequency:
The first step in finding the median class is to calculate the cumulative frequency for each class. The cumulative frequency is the sum of the frequencies of all classes up to and including the current class.
The frequency distribution is as follows:
\[ \begin{array}{|c|c|} \hline \text{Number of students per Teacher} & \text{Number of Schools} \\ \hline 20 - 25 & 5 \\ 25 - 30 & 15 \\ 30 - 35 & 25 \\ 35 - 40 & 30 \\ 40 - 45 & 15 \\ 45 - 50 & 10 \\ \hline \end{array} \] Now, calculate the cumulative frequency:
For the class \(20 - 25\), the cumulative frequency is \(5\).
For the class \(25 - 30\), the cumulative frequency is \(5 + 15 = 20\).
For the class \(30 - 35\), the cumulative frequency is \(20 + 25 = 45\).
For the class \(35 - 40\), the cumulative frequency is \(45 + 30 = 75\).
For the class \(40 - 45\), the cumulative frequency is \(75 + 15 = 90\).
For the class \(45 - 50\), the cumulative frequency is \(90 + 10 = 100\).

Step 3: Finding the median class:
The total number of schools is 100, and the median corresponds to the \( \frac{100}{2} = 50 \)-th data point.
The median class is the class interval where the cumulative frequency first exceeds 50.
From the cumulative frequency table, we can see that the cumulative frequency for the class \(30 - 35\) is 45, and the cumulative frequency for the class \(35 - 40\) is 75. Therefore, the median class is \(35 - 40\).

Step 4: Conclusion:
The lower limit of the median class is 35.

Answer 2

Step 1: Understanding the problem:
We are asked to find the upper limit of the modal class based on the given frequency distribution table. The modal class is the class interval with the highest frequency.

Step 2: Identifying the modal class:
The first step in finding the modal class is to identify which class has the highest frequency. The frequency distribution is as follows:
\[ \begin{array}{|c|c|} \hline \text{Number of students per Teacher} & \text{Number of Schools} \\ \hline 20 - 25 & 5 \\ 25 - 30 & 15 \\ 30 - 35 & 25 \\ 35 - 40 & 30 \\ 40 - 45 & 15 \\ 45 - 50 & 10 \\ \hline \end{array} \] From the table, we can see that the class \(35 - 40\) has the highest frequency of 30 schools.
Thus, the modal class is \(35 - 40\).

Step 3: Finding the upper limit of the modal class:
The upper limit of the modal class is the upper value of the class interval \(35 - 40\), which is 40.

Step 4: Conclusion:
The upper limit of the modal class is 40.

Answer 3

Step 1: Understanding the problem:
We are asked to find both the median and the modal values of the data based on the frequency distribution table. Let’s break it down into two parts: first finding the median, then finding the mode.

Step 2: Finding the median of the data:
To find the median, we first need to know the median class, which is the class where the cumulative frequency first exceeds the middle value of the total number of observations. The median corresponds to the \( \frac{N}{2} \)-th value, where \( N \) is the total number of observations.
From the previous solution, we know that the total number of schools is 100, so the median corresponds to the \( \frac{100}{2} = 50 \)-th data point.
We already calculated the cumulative frequency as follows:
\[ \begin{array}{|c|c|c|} \hline \text{Number of students per Teacher} & \text{Number of Schools} & \text{Cumulative Frequency} \\ \hline 20 - 25 & 5 & 5 \\ 25 - 30 & 15 & 20 \\ 30 - 35 & 25 & 45 \\ 35 - 40 & 30 & 75 \\ 40 - 45 & 15 & 90 \\ 45 - 50 & 10 & 100 \\ \hline \end{array} \] The median class is the one where the cumulative frequency first exceeds 50, which is the class \( 35 - 40 \), because the cumulative frequency reaches 75 for this class.
The formula for the median is:
\[ \text{Median} = L + \left( \frac{\frac{N}{2} - F}{f} \right) \times h \] where:
- \( L \) is the lower limit of the median class,
- \( N \) is the total frequency (100 in this case),
- \( F \) is the cumulative frequency of the class before the median class,
- \( f \) is the frequency of the median class,
- \( h \) is the class width.
Substituting the values:
- \( L = 35 \) (lower limit of the median class),
- \( N = 100 \),
- \( F = 45 \) (cumulative frequency of the class before),
- \( f = 30 \) (frequency of the median class),
- \( h = 5 \) (class width).
Now, calculating the median:
\[ \text{Median} = 35 + \left( \frac{\frac{100}{2} - 45}{30} \right) \times 5 = 35 + \left( \frac{50 - 45}{30} \right) \times 5 \] \[ \text{Median} = 35 + \left( \frac{5}{30} \right) \times 5 = 35 + \frac{25}{30} = 35 + \frac{5}{6} = 35.8333 \] Thus, the median is approximately \( 35.83 \).

Step 3: Finding the mode of the data:
The mode corresponds to the class with the highest frequency. From the frequency distribution table, we can see that the class \( 35 - 40 \) has the highest frequency of 30 schools.
The formula for the mode is:
\[ \text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h \] where:
- \( L \) is the lower limit of the modal class,
- \( f_1 \) is the frequency of the modal class,
- \( f_0 \) is the frequency of the class before the modal class,
- \( f_2 \) is the frequency of the class after the modal class,
- \( h \) is the class width.
Substituting the values:
- \( L = 35 \) (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).
Now, calculating the mode:
\[ \text{Mode} = 35 + \left( \frac{30 - 25}{2 \times 30 - 25 - 15} \right) \times 5 = 35 + \left( \frac{5}{60 - 25 - 15} \right) \times 5 \] \[ \text{Mode} = 35 + \left( \frac{5}{20} \right) \times 5 = 35 + \frac{25}{20} = 35 + 1.25 = 36.25 \] Thus, the mode is \( 36.25 \).

Step 4: Conclusion:
1. The median of the data is approximately \( 35.83 \).
2. The mode of the data is \( 36.25 \).