Averages from Tables

Calculation Logic: The mean is found by summing the products of each value and its frequency, then dividing by the total number of observations. This accounts for the fact that some values appear more often than others.

Discrete Formula: The mean $\bar{x}$ is calculated using the formula: $\bar{x} = \frac{\sum (f \cdot x)}{\sum f}$

Grouped Data (Estimated Mean): For grouped data, the exact values are unknown, so the midpoint of each class interval is used as the representative value ($x$). The resulting mean is an estimate because it assumes data is evenly distributed within each group.

The Mode: In a discrete table, the mode is the data value with the highest frequency. In grouped data, we identify the Modal Class, which is the interval with the highest frequency.

The Median Position: To find the median, calculate the position using $\frac{n+1}{2}$, where $n$ is the total frequency. If $n$ is large, $\frac{n}{2}$ is often used.

Locating the Median: Use cumulative frequency to track the running total of frequencies row by row. The median is the value (or class) where the cumulative frequency first reaches or exceeds the median position.

Frequency as the Mode: Students often mistake the highest frequency number for the mode. The mode is the data value associated with that frequency, not the frequency itself.

Range Errors: The range is the difference between the largest and smallest data values, not the difference between the largest and smallest frequencies.

Midpoint Accuracy: When calculating the midpoint for grouped data, ensure you use the lower and upper boundaries of the class interval correctly, especially if the classes have gaps.