If the mean of the following data is 26, then find the missing frequency x.
`{:("Class Interval",f),(" "0-10,4),(" "10-20,X),(" "20-30,9),(" "30-40,5),(" "40-50,6):}`

Answer

To find the missing frequency \( x \) in the given data where the mean is 26, we can follow these steps:

### Step 1: Identify the Class Intervals and Frequencies
We have the following class intervals and frequencies:
- Class Interval: 0-10, Frequency: 4
- Class Interval: 10-20, Frequency: \( x \)
- Class Interval: 20-30, Frequency: 9
- Class Interval: 30-40, Frequency: 5
- Class Interval: 40-50, Frequency: 6

### Step 2: Calculate the Class Marks
The class mark \( X_i \) for each class interval can be calculated using the formula:
\[ X_i = \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \]
Calculating the class marks:
- For 0-10: \( X_1 = \frac{0 + 10}{2} = 5 \)
- For 10-20: \( X_2 = \frac{10 + 20}{2} = 15 \)
- For 20-30: \( X_3 = \frac{20 + 30}{2} = 25 \)
- For 30-40: \( X_4 = \frac{30 + 40}{2} = 35 \)
- For 40-50: \( X_5 = \frac{40 + 50}{2} = 45 \)

### Step 3: Set Up the Frequency Table
Now we can set up the frequency table with class marks and frequencies:
\[ \begin{array}{|c|c|c|} \hline \text{Class Interval} & f_i & X_i \\ \hline 0-10 & 4 & 5 \\ 10-20 & x & 15 \\ 20-30 & 9 & 25 \\ 30-40 & 5 & 35 \\ 40-50 & 6 & 45 \\ \hline \end{array} \]

### Step 4: Calculate \( \Sigma f_i \) and \( \Sigma f_i X_i \)
- \( \Sigma f_i = 4 + x + 9 + 5 + 6 = 24 + x \)
- Now calculate \( \Sigma f_i X_i \):
- For 0-10: \( 4 \times 5 = 20 \)
- For 10-20: \( x \times 15 = 15x \)
- For 20-30: \( 9 \times 25 = 225 \)
- For 30-40: \( 5 \times 35 = 175 \)
- For 40-50: \( 6 \times 45 = 270 \)

Thus,
\[ \Sigma f_i X_i = 20 + 15x + 225 + 175 + 270 = 690 + 15x \]

### Step 5: Use the Mean Formula
The mean is given by:
\[ \text{Mean} = \frac{\Sigma f_i X_i}{\Sigma f_i} \]
Substituting the values we have:
\[ 26 = \frac{690 + 15x}{24 + x} \]

### Step 6: Cross Multiply and Solve for \( x \)
Cross multiplying gives:
\[ 26(24 + x) = 690 + 15x \]
Expanding both sides:
\[ 624 + 26x = 690 + 15x \]
Rearranging gives:
\[ 26x - 15x = 690 - 624 \]
\[ 11x = 66 \]
Dividing by 11:
\[ x = 6 \]

### Conclusion
The missing frequency \( x \) is 6.

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