In a histogram, the area of each rectangle (or bar) represents the frequency of that interval or class. If we say that the area of the rectangle is proportional to its frequency, it means that if the area increases or decreases, the frequency does the same.
However, to determine if the lengths of the rectangles are also proportional to the frequencies, we need to consider the width of the rectangles. In a histogram, the area of a rectangle is calculated as:
Area = Width × Height
Here, the height of the rectangle corresponds to the frequency of that interval, and the width is the size of the interval. If the widths of all rectangles are equal, then yes, we can say that the lengths (or heights) of the rectangles will also be proportional to their frequencies.
On the other hand, if the widths of the rectangles vary, the relationship becomes more complex. In that case, the height of a rectangle divided by its width will give you a frequency density, and the relationship between lengths and frequencies won’t hold since you cannot directly compare the heights of rectangles with different widths.
In summary, if the histogram has equal widths, then the lengths of the rectangles are proportional to their frequencies. If the widths differ, then we cannot say that the lengths are directly proportional to the frequencies.