Find the median of the following: x 10-20 20-30 30-40 40-50 50-60 60-70 70-80 frequency 12 30 34 65 45 5 18\

To find the median of a frequency distribution, we first need to follow these steps:

### Steps to Calculate the Median:

1. **Identify the cumulative frequency** to locate the median class.
2. **Determine the total number of observations**.
3. **Find the median class**.
4. **Use the formula for the median** for grouped data.

### Step 1: Create a cumulative frequency table.

| Class Interval | Frequency (f) | Cumulative Frequency (CF) |
|----------------|---------------|---------------------------|
| 10-20          | 12            | 12                        |
| 20-30          | 30            | 12 + 30 = 42              |
| 30-40          | 34            | 42 + 34 = 76              |
| 40-50          | 65            | 76 + 65 = 141             |
| 50-60          | 45            | 141 + 45 = 186            |
| 60-70          | 5             | 186 + 5 = 191             |
| 70-80          | 18            | 191 + 18 = 209            |

### Step 2: Calculate the total number of observations.
The total number of observations, \( N = 209 \).

### Step 3: Find the median class.
The median is located at the \(\frac{N}{2}\)-th observation. So, \(\frac{209}{2} = 104.5\). The cumulative frequency just greater than 104.5 is 141, which corresponds to the class interval **40-50**. This is the **median class**.

### Step 4: Apply the median formula for grouped data.
The formula for the median is:

\[
\text{Median} = L + \left(\frac{\frac{N}{2} - CF}{f}\right) \times h
\]

Where:
- \( L = 40 \) (lower boundary of the median class),
- \( \frac{N}{2} = 104.5 \),
- \( CF = 76 \) (cumulative frequency of the class preceding the median class),
- \( f = 65 \) (frequency of the median class),
- \( h = 10 \) (class width).

Now, substitute the values into the formula:

\[
\text{Median} = 40 + \left(\frac{104.5 - 76}{65}\right) \times 10
\]
\[
\text{Median} = 40 + \left(\frac{28.5}{65}\right) \times 10
\]
\[
\text{Median} = 40 + (0.4385) \times 10
\]
\[
\text{Median} = 40 + 4.385
\]
\[
\text{Median} \approx 44.39
\]

### Final Answer:
The median of the data is approximately **44.39**.

Would you like more details on this process or have any questions?

Here are 5 questions related to the concept of finding the median:
1. How is the cumulative frequency table constructed?
2. What is the difference between grouped and ungrouped data when finding the median?
3. Can the median class change if the frequency distribution is adjusted slightly?
4. Why do we use the lower boundary of the median class in the formula?
5. How would the median be affected if we increased the class width?

**Tip**: Always ensure you identify the median class correctly by checking the cumulative frequencies carefully.

Find the median of the following: x 10-20 20-30 30-40 40-50 50-60 60-70 70-80 frequency 12 30 34 65 45 5 18

Learn how to find the median of a grouped data frequency distribution with intervals 10-80 and frequencies 12, 30, 34, 65, 45, 5, 18. This step-by-step solution covers cumulative frequency tables and the median formula, suitable for students in Grades 9-12.

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