Averages from Grouped Data

• The Midpoint Assumption is the logical foundation for these calculations. We assume that the data points within a class are uniformly distributed, making the midpoint the most statistically sound representative for the entire group.

• The Midpoint ($x$) is calculated by finding the average of the lower and upper bounds of the class: $x = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}$.

• The total frequency ($\sum f$) represents the total number of data points ($n$) in the entire set, which is essential for finding the average.

Calculating the Estimated Mean

• Step 1: Identify the midpoint ($x$) for every class interval in the table.
• Step 2: Multiply each midpoint by its corresponding frequency ($f$) to find the $fx$ value for that row.
• Step 3: Sum all the $fx$ values to find the total estimated sum of all data points ($\sum fx$).
• Step 4: Divide the total sum by the total frequency ($\sum f$).

Formula: $\text{Estimated Mean} = \frac{\sum (f \cdot x)}{\sum f}$

Identifying the Modal Class

• Look for the highest value in the frequency column. The class interval associated with this frequency is the modal class.

Finding the Median Interval

• Calculate the position of the median using $\frac{n+1}{2}$, where $n$ is the total frequency.
• Use a cumulative frequency (running total) to find which class interval contains this specific position.

• Discrete vs. Continuous Intervals: For discrete data (e.g., ages 10-19, 20-29), ensure the midpoint accounts for the full range. For continuous data (e.g., $10 \le x < 20$), the midpoint is simply the average of the boundaries.

• Dividing by the wrong total: A frequent error is dividing $\sum fx$ by the number of rows (classes) in the table rather than the total frequency (the sum of the $f$ column).

• Incorrect Midpoints: Students often use the upper bound or lower bound instead of the center. Always verify the midpoint calculation: $(L + U) / 2$.

• Confusing Median Position with Value: Finding that the median is the 15th value is only the first step; you must then identify which interval that 15th value falls into.