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A histogram is a graphical representation that illustrates the frequency of data within specific intervals or "bins". It is particularly helpful for visualizing the distribution of numerical data, such as lead times in days.

In a histogram, adjacent bars represent different intervals (e.g., 0-5 days, 5-10 days), with the height of each bar illustrating the frequency of values within that interval. Unlike a bar graph, a histogram does not have gaps between bars as it deals with continuous data.

When creating a histogram, follow these steps:

• Choose class intervals that cover the entire range of data without overlapping.
• Count how many data points fall within each interval to determine the frequency.
• Draw bars for each interval with heights corresponding to the frequencies.

In the given Internet retailer example, the histogram would show bars at five intervals: 0 to 5 days, 5 to 10 days, 10 to 15 days, 15 to 20 days, and 20 to 25 days, with heights of 6, 7, 12, 8, and 7 respectively.

A frequency polygon is another way to show the distribution of data. It is particularly useful for comparing multiple data sets.

By connecting the midpoints of each class interval with straight lines, we get a polygonal shape that presents an overview of how the frequencies are distributed. This can be more visually intuitive than a histogram in some cases.

To draw a frequency polygon, consider these steps:

• Find the midpoint for each class interval (e.g., for 0-5 days, the midpoint is 2.5 days).
• Plot these midpoint values along the x-axis against their corresponding frequencies on the y-axis.
• Connect these points using straight lines.
• Ensure to start and end the polygon on the baseline by adding a point at the start of the first interval and the end of the last interval at zero frequency.

In our case, the frequency polygon would connect points like (2.5, 6), (7.5, 7), (12.5, 12), (17.5, 8), and (22.5, 7), illustrating the data's symmetrical distribution.

Calculating midpoints is crucial in constructing frequency polygons and interpreting data intervals. The midpoint of a class interval provides a central value to represent the entire class.

The midpoint is found using the formula: \[ \text{Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2} \] This formula calculates the value lying in the middle of an interval. Midpoints simplify comparisons and provide a visual anchor for frequency polygons.

Let's apply this with an example from the table:

• For the first interval, "0 up to 5 days", the midpoint calculation is \( \frac{0 + 5}{2} = 2.5 \text{ days} \).
• Carry out similar calculations for the other intervals like \( \frac{5 + 10}{2} = 7.5\), \( \frac{10 + 15}{2} = 12.5 \), and so forth.

These midpoints (2.5, 7.5, 12.5, 17.5, and 22.5) are essential for plotting frequency polygons and understanding the central trend of data.