Histograms & Frequency Polygons

• The core principle of a histogram is the Area-Frequency Relationship. The frequency of a class is calculated as $Frequency = Frequency Density \times Class Width$.

• Frequency Density is the height of the bar on the vertical axis. It is used to ensure that the area remains proportional to the frequency even when class widths are unequal.

• For frequency polygons, the principle is Linear Interpolation. By connecting midpoints, we assume a linear change in frequency between the centers of adjacent groups, which helps in identifying the overall trend of the data.

• The Modal Class in both representations is the interval with the highest frequency density (the tallest bar or the highest peak of the polygon).

Constructing a Histogram

• Step 1: Calculate the Class Width for each interval by subtracting the lower boundary from the upper boundary ($Width = Upper - Lower$).
• Step 2: Calculate the Frequency Density using the formula: $Frequency Density = \frac{Frequency}{Class Width}$
• Step 3: Draw the horizontal axis for the continuous variable and the vertical axis for Frequency Density.
• Step 4: Draw rectangles for each class where the width matches the interval and the height matches the calculated frequency density.

Constructing a Frequency Polygon

• Step 1: Identify the Midpoint of each class interval using the formula: $Midpoint = \frac{Lower Boundary + Upper Boundary}{2}$
• Step 2: Plot points with coordinates $(Midpoint, Frequency)$. Note that for polygons, the vertical axis usually represents raw frequency if class widths are equal, or frequency density if they are not.
• Step 3: Connect the plotted points in sequence using straight line segments.

• Check Class Widths First: Always look at the table of data to see if the class widths are equal. If they are unequal, you must calculate frequency density; plotting raw frequency will result in an incorrect graph.

• Labeling the Axis: In a histogram, never label the vertical axis as 'Frequency' unless all class widths are exactly 1 unit wide. Use 'Frequency Density' to be mathematically accurate.

• Midpoint Accuracy: When drawing a frequency polygon, ensure your points are exactly in the middle of the interval. A common mistake is plotting at the start or end of the class range.

• Sanity Check: The total area of all bars in a histogram should equal the total frequency of the dataset. If the numbers don't align, re-check your frequency density calculations.

• The 'Gap' Error: Students often leave gaps between histogram bars as they do with bar charts. In continuous data, there is no 'space' between $10 \le x < 20$ and $20 \le x < 30$.

• Height Misconception: Assuming the tallest bar always has the highest frequency. In a histogram with unequal widths, a short, very wide bar can actually represent a much higher frequency than a tall, narrow one.

• Closing the Polygon: Some students try to 'close' the frequency polygon by connecting the last point back to the first or to the origin. Unless the frequency at those points is zero, the polygon should remain an open chain of segments.