Key Points

A bar graph is a pictorial representation of data using rectangular bars of uniform width. It is used for discrete data, with equal spacing between bars, where the height of each bar represents the frequency or value of the variable.

A histogram is a graphical representation of a grouped frequency distribution with continuous class intervals. It consists of adjacent rectangular bars with no gaps in between, where the area of each bar is proportional to the corresponding frequency.

A bar graph represents discrete data and has gaps between bars, while a histogram represents continuous data and has no gaps. In a bar graph, only the height of the bar is meaningful, whereas in a histogram, the area (product of height and width) represents the frequency.

To draw a histogram for data with discontinuous intervals (e.g., 118-126, 127-135), you must make them continuous. Find the gap (e.g., $127 - 126 = 1$), halve it ($0.5$), then subtract this value from all lower limits and add it to all upper limits.

For a histogram where all class intervals have the same width, the height of each rectangular bar is taken to be equal to the frequency of that class. The widths of all bars are equal to the class size.

When class intervals have varying widths, the heights of the bars must be adjusted. The area of each rectangle, not its height, must be proportional to the frequency. A bar with a larger width will have its height reduced proportionally to maintain the correct area representation.

The height of a rectangle in a histogram with varying widths is adjusted using the formula: Adjusted Height = $\frac{\text{Frequency of the class}}{\text{Width of this class}} \times \text{Minimum class width}$.

A frequency polygon is a line graph obtained by joining the mid-points of the tops of the bars in a histogram. It provides a better visual representation of the shape of a distribution and is useful for comparing two or more datasets.

The class-mark is the mid-point of a class interval, used for plotting a frequency polygon. It is calculated as: Class-mark = $\frac{\text{Upper class limit} + \text{Lower class limit}}{2}$.

To draw a frequency polygon independently, first calculate the class-mark for each class interval. Then, plot the points (class-mark, frequency) on a graph and join them with straight line segments. The polygon is closed by connecting the ends to the horizontal axis at the mid-points of the imaginary classes preceding the first and succeeding the last class.