The mode from given grouped data would be
`{:("Class Interval",10-20,20-30,30-40,40-50,50-60),("Frequency",5,7,6,8,4):}`

Options

A. 42
B. 43
C. 44
D. 45

Answer

To find the mode of the given grouped data, we will follow these steps:

### Step 1: Identify the Class Intervals and Frequencies
We have the following class intervals and their corresponding frequencies:
- Class Intervals: 10-20, 20-30, 30-40, 40-50, 50-60
- Frequencies: 5, 7, 6, 8, 4

### Step 2: Determine the Modal Class
The modal class is the class interval with the highest frequency.
- Here, the highest frequency is 8, which corresponds to the class interval 40-50.

### Step 3: Assign Values for the Mode Formula
We need to assign values for the mode formula:
- Let \( L \) = lower limit of the modal class = 40
- \( F_1 \) = frequency of the modal class = 8
- \( F_0 \) = frequency of the class preceding the modal class = 6 (frequency of class 30-40)
- \( F_2 \) = frequency of the class succeeding the modal class = 4 (frequency of class 50-60)
- Height (h) = width of the class interval = 50 - 40 = 10

### Step 4: Use the Mode Formula
The formula for mode in grouped data is:
\[ \text{Mode} = L + \left(\frac{F_1 - F_0}{2F_1 - F_0 - F_2}\right) \times h \]

Substituting the values we have:
\[ \text{Mode} = 40 + \left(\frac{8 - 6}{2 \times 8 - 6 - 4}\right) \times 10 \]

### Step 5: Simplify the Expression
Calculating the numerator:
\[ F_1 - F_0 = 8 - 6 = 2 \]

Calculating the denominator:
\[ 2F_1 - F_0 - F_2 = 2 \times 8 - 6 - 4 = 16 - 6 - 4 = 6 \]

Now substituting back:
\[ \text{Mode} = 40 + \left(\frac{2}{6}\right) \times 10 \]
\[ \text{Mode} = 40 + \left(\frac{1}{3}\right) \times 10 \]
\[ \text{Mode} = 40 + \frac{10}{3} \]
\[ \text{Mode} = 40 + 3.33 \approx 43.33 \]

### Step 6: Final Answer
Rounding to the nearest whole number, the mode is approximately 43.

### Conclusion
Thus, the mode of the given grouped data is **43**.

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