Choosing the Right Measure of Central Tendency
We know the three main measures of central tendency: Mean (average), Median (middle value), and Mode (most frequent value). All three tell us about the "center" or tendency of the data, but they have different properties and are suitable for different situations.
Choosing the wrong measure can give a misleading picture of the data. So, let's understand when it's best to use each measure.
Key Characteristics
Mean (Average)
- How to Calculate: Sum all data values, then divide by the number of data points.
- Property: Uses all data values in the calculation. Because of this, the Mean is very sensitive to extreme values (outliers). A single very large or small value can significantly change the Mean.
- Sensitivity Example: If your test scores are (Mean = ), and then one additional score of 1 (an outlier) is added, the new Mean becomes . The Mean drops drastically due to the outlier.
Median (Middle Value)
- How to Calculate: Sort the data, then find the value exactly in the middle position.
- Property: Only considers the middle position of the data. Therefore, the Median is not sensitive (robust) to extreme values (outliers).
- Robustness Example: For data (Sorted: 7, 8, 8, 9, 10), the Median=8. If a score of 1 is added (Sorted: 1, 7, 8, 8, 9, 10), the Median= . The Median hardly changes despite the outlier.
Mode (Most Frequent Value)
- How to Calculate: Find the data value with the highest frequency.
- Property: Focuses on the most popular value. Like the Median, the Mode is not sensitive (robust) to extreme values (outliers). Also, the Mode is the only measure of central tendency that can be used for qualitative data (non-numerical).
- Robustness & Qualitative Example: For data 7, 8, 9, 8, 10, the Mode=8. If a score of 1 is added, the Mode remains 8. If the data is favorite colors (Red, Blue, Red, Green), the Mode=Red.
Usage Guide
Here's a concise guide on when to use each measure:
| Measure | Best Data Type | Outlier Influence | When to Use? |
|---|---|---|---|
| Mean | Quantitative (Numbers) | Highly Influenced | When data has no significant outliers & you want a representation involving all data values. |
| Median | Quantitative (Numbers) | Not Influenced | When data has outliers or the distribution is skewed (not symmetrical). Provides a robust middle value. |
| Mode | Qualitative & Quantitative | Not Influenced | When you want to know the most popular value/category or the most frequent occurrence. |
Example Scenarios:
- Calculating class average score: If student scores are relatively evenly spread without extreme highs or lows, the Mean is suitable.
- Looking at average income in an area: If there are a few individuals with very high incomes (outliers), the Median will give a better representation of the typical income than the Mean.
- Determining the best-selling shoe size in a store: We want to know the size most frequently purchased, so the Mode is the right choice.
- Finding the favorite color among students: Since the data is categorical (colors), the only applicable measure is the Mode.