Mode & Mean from Grouped Data

Calculating the Estimated Mean

• Step 1: Find Midpoints: For every class interval, calculate the midpoint $x = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}$.
• Step 2: Calculate $f \cdot x$: Multiply the frequency of each class by its corresponding midpoint to find the total value contributed by that group.
• Step 3: Sum the Totals: Calculate the sum of all $f \cdot x$ products ($\\sum f \cdot x$) and the sum of all frequencies ($\\sum f$).
• Step 4: Divide: Apply the formula to find the mean estimate: $\bar{x} = \frac{\sum (f \cdot x)}{\sum f}$

Identifying the Modal Class

• Scan the frequency column of the data table to find the highest numerical value.
• The class interval associated with this highest frequency is the Modal Class; no further calculation is required unless an interpolated mode is requested.

• Grouped vs. Ungrouped: In ungrouped data, the mean is exact ($\frac{\sum x}{n}$). In grouped data, the mean is an estimate because the specific values of $x$ are approximated by class midpoints.

• The Range Check: Always verify that your calculated mean falls within the boundaries of the overall dataset; if your mean is smaller than the lowest class or larger than the highest, a calculation error has occurred.

• Frequency Summation: Ensure you divide by the total frequency (the sum of the $f$ column), NOT the number of rows in the table.

• Midpoint Accuracy: Double-check midpoints for intervals with unusual widths (e.g., $0-10$ has a midpoint of $5$, but $11-20$ might have a midpoint of $15.5$ depending on whether the data is discrete or continuous).

• Reasonableness Test: Look at the histogram or frequency table; the mean should generally be pulled toward the side with the highest frequencies.

• Confusing $n$ with Row Count: Students often divide by the number of classes (rows) instead of the total number of observations (sum of frequencies).

• Using Class Limits: A common error is using only the upper or lower limit of a class for calculation instead of the midpoint.

• Misidentifying the Mode: Students sometimes list the highest frequency value itself as the mode, rather than the class interval that contains it.