Mean, Median & Mode

Calculating the Mean

• Formula: The mean $\bar{x}$ is found using $\bar{x} = \frac{\sum x}{n}$, where $\sum x$ is the sum of all values and $n$ is the number of values.
• Frequency Tables: For data in tables, use $\bar{x} = \frac{\sum (xf)}{\sum f}$, where $x$ is the data value and $f$ is its frequency.

Finding the Median

• Step 1: Arrange all data points in ascending or descending order.
• Step 2: Determine the position using $\frac{n+1}{2}$. If $n$ is odd, the median is the value at that exact position. If $n$ is even, the median is the mean of the two middle values.

Estimating from Grouped Data

• Midpoints: Because exact values are unknown in grouped data, the midpoint of each class interval is used as a representative value for the mean calculation.
• Modal Class: Identify the interval with the highest frequency rather than a single value.

• The Sanity Check: Always verify that your calculated average falls between the minimum and maximum values of the data set; if it doesn't, a calculation error occurred.

• Units and Rounding: Ensure the mean includes the same units as the data. In exams, do not round the mean to a whole number unless specifically instructed, as averages often result in decimals (e.g., 2.4 children).

• Grouped Data Language: Look for the word 'estimate'. If a question asks for an 'estimate of the mean', it is a signal that you must use midpoints for grouped frequency data.

• Ordering for Median: A frequent mistake is picking the middle number from an unordered list. The data MUST be sorted first.

• Frequency vs. Value: In frequency tables, students often mistake the highest frequency for the mode. The mode is the data value associated with that frequency, not the frequency itself.

• Even Data Sets: Forgetting that an even number of data points requires averaging the two middle values to find the median.