Consider the following statements in respect of histogram.
I. The algebraic sum of deviations of a set of n values from its atithmetic mean is zero.
II In the case of frequency distribution, mode is the value of variable which corresponds to maximam frequency Which of the above statement (s) is/are correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor 2

Answer

To solve the question regarding the correctness of the statements about histograms, we will analyze each statement step by step.

### Step 1: Analyze the first statement
**Statement I:** "The algebraic sum of deviations of a set of n values from its arithmetic mean is zero."

1. **Understanding the mean:** The arithmetic mean (denoted as \( \bar{x} \)) of a set of \( n \) observations \( x_1, x_2, ..., x_n \) is calculated as:
\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \]

2. **Calculating the deviations:** The deviation of each observation from the mean is given by \( x_i - \bar{x} \).

3. **Sum of deviations:** The algebraic sum of these deviations is:
\[ \sum_{i=1}^{n} (x_i - \bar{x}) = \sum_{i=1}^{n} x_i - \sum_{i=1}^{n} \bar{x} \]
Since \( \bar{x} \) is a constant, we can write:
\[ \sum_{i=1}^{n} \bar{x} = n \cdot \bar{x} \]
Therefore, the sum of deviations becomes:
\[ \sum_{i=1}^{n} x_i - n \cdot \bar{x} \]

4. **Substituting the mean:** From the definition of the mean, we know that \( \sum_{i=1}^{n} x_i = n \cdot \bar{x} \). Thus:
\[ \sum_{i=1}^{n} (x_i - \bar{x}) = n \cdot \bar{x} - n \cdot \bar{x} = 0 \]

**Conclusion for Statement I:** The first statement is correct.

### Step 2: Analyze the second statement
**Statement II:** "In the case of frequency distribution, mode is the value of variable which corresponds to maximum frequency."

1. **Understanding mode:** The mode is defined as the value that appears most frequently in a data set. In a frequency distribution, it is the value (or class interval) that has the highest frequency.

2. **Definition confirmation:** The statement correctly describes the mode in the context of frequency distributions.

**Conclusion for Statement II:** The second statement is also correct.

### Final Conclusion
Both statements I and II are correct. Therefore, the correct answer is that both statements are true.

### Summary of the Solution
- Statement I is correct because the algebraic sum of deviations from the mean is zero.
- Statement II is correct because the mode is defined as the value corresponding to the maximum frequency in a frequency distribution.

### Final Answer
Both statements are correct.

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