Mean, Median & Mode

Calculating from a List

• Mean: Sum all values and divide by $n$. Formula: $\bar{x} = \frac{\sum x}{n}$.
• Median: Sort the data. Find the position using $\frac{n+1}{2}$. If the result is a decimal (e.g., $4.5$), average the values at positions $4$ and $5$.
• Mode: Count the frequency of each value; the one with the highest count is the mode.

Calculating from Frequency Tables

• Mean: Multiply each value ($x$) by its frequency ($f$), sum these products ($\sum fx$), and divide by the total frequency ($\sum f$).
• Median: Use cumulative frequency to find which row contains the $\frac{n+1}{2}$ position.
• Mode: Identify the value ($x$) corresponding to the highest frequency ($f$).

Estimating from Grouped Data

• Estimated Mean: Since exact values are unknown, use the midpoint of each class interval as a representative value ($x$) and apply the $\frac{\sum fx}{\sum f}$ formula.
• Modal Class: The interval with the highest frequency density (or highest frequency if intervals are equal width).

• When to use Median: In real-world scenarios like house prices or salaries, a few multi-millionaires can drastically inflate the mean. The median provides a more 'typical' value that represents the 50th percentile of the population.

• When to use Mode: If a shoe store needs to know which size to stock most, the mean size (e.g., 8.42) is useless. The mode (e.g., size 9) tells them exactly which specific item is in highest demand.

• The 'Frequency' Trap: A common mistake is giving the frequency itself as the mode or median. Always remember that the answer must be a data value (the $x$ column), not the count (the $f$ column).

• Ordering for Median: Never calculate the median without first checking if the data is ordered. In exams, data is often presented in a random list to test this specific procedural step.

• Sanity Checks: After calculating the mean, ensure the result lies within the range of the data. If your data ranges from 10 to 20 and your mean is 5 or 25, a calculation error has occurred.

• Rounding Precision: In multi-step mean calculations (especially with grouped data), keep full calculator precision until the final step to avoid rounding errors that could lead to an incorrect final significant figure.