3. Mean, Median, Mode, and Range

These four measures describe a dataset in different ways.

Mean (Average):
The mean is the sum of all values divided by the number of values.

\[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \]

Example: Find the mean of 3, 7, 5, 9, and 6.
\[ \text{Mean} = \frac{3 + 7 + 5 + 9 + 6}{5} = \frac{30}{5} = 6 \]

Median:
The median is the middle value when data is arranged in order. It divides the dataset in half.

Example: For 3, 5, 6, 7, 9 (already sorted), the median is 6 (the middle value).
Example: For 3, 5, 6, 7, 9, 12 (six values), the median is the average of the two middle values: \(\frac{6+7}{2} = 6.5\).

Mode:
The mode is the value that appears most frequently.

Example: In the dataset 2, 3, 3, 3, 5, 7, 7, the mode is 3 (appears three times).
A dataset can have no mode (all values appear once), one mode (unimodal), or multiple modes (bimodal, multimodal).

Range:
The range measures spread. It is the difference between the largest and smallest values.

\[ \text{Range} = \text{Largest value} - \text{Smallest value} \]

Example: For 3, 5, 6, 7, 9:
\[ \text{Range} = 9 - 3 = 6 \]

When to use each:

• Mean: Most common. Use when you want a single representative value. Affected by outliers (very large or small values).
• Median: Use when data has outliers. It is more "resistant" to extreme values.
• Mode: Use for categorical data or to identify the most popular value.
• Range: Use to understand spread and variability.

Real-world context: A manager reviews employee salaries: \$30k, \$35k, \$40k, \$45k, \$500k. The mean is \(\frac{30+35+40+45+500}{5} = \$130k\), but the median is \$40k. The median better represents "typical" salary because the CEO's \$500k salary is an outlier.

Common mistake: calculating mean wrong. Always add all values first, then divide by the count.

💡 Tip: Remember the order: "Mean is the average, Median is the middle, Mode is the most frequent, Range is the difference."