The correct option for the question about the characteristics of a histogram is **A. I, II, and III**. Let's break down why this is the correct answer and analyze each statement in detail. ### Explanation of the Correct Answer 1. **Statement I: Rectangular bars of equal width** - **Explanation**: In a histogram, each class interval (or bin) is represented by a rectangular bar. The width of these bars is uniform across the histogram. This uniformity is crucial because it allows for a clear visual representation of the frequency distribution of the data. If the widths were not equal, it would misrepresent the data, making it difficult to interpret the frequency of each class interval accurately. 2. **Statement II: The height of each rectangular bar is proportional to the frequency of the corresponding class interval** - **Explanation**: The height of each bar in a histogram represents the frequency (or count) of data points that fall within that specific class interval. This means that if a class interval has a higher frequency, its corresponding bar will be taller. This proportionality is essential for understanding the distribution of the data visually. It allows viewers to quickly assess which intervals have more or fewer occurrences. 3. **Statement III: Rectangular bars have common sides with no gaps in between** - **Explanation**: In a histogram, the bars are adjacent to each other, meaning they share common sides and there are no gaps between them. This characteristic distinguishes histograms from bar charts, where gaps may exist between bars. The absence of gaps in histograms emphasizes the continuous nature of the data being represented, which is particularly important for interval or ratio data. ### Why Other Options Are Incorrect - **Option B: I and II** - This option omits Statement III, which is crucial for defining a histogram. Without the requirement for bars to be adjacent, the definition would not fully capture the essence of a histogram, as it would allow for gaps that are not characteristic of histograms. - **Option C: II and III** - This option excludes Statement I, which is also essential. If the bars are not of equal width, the histogram would not accurately represent the frequency distribution. The uniform width is a fundamental property of histograms. - **Option D: I and III** - This option leaves out Statement II, which is critical for understanding how the height of the bars relates to the data. Without this information, one cannot interpret the histogram correctly, as the height is what conveys the frequency of each class interval. ### Summary of Key Points - A histogram consists of rectangular bars that are all of equal width. - The height of each bar corresponds to the frequency of the data within that class interval, making it a visual representation of data distribution. - The bars in a histogram are adjacent, with no gaps, emphasizing the continuous nature of the data. - Understanding these three characteristics is essential for correctly interpreting and constructing histograms. By keeping these points in mind, you can confidently approach questions related to histograms and their properties in your exams.