Mean, Median, and Mode
Measures of Central Tendency
Essential Understanding: These three measures help us find the “center” or “typical value” of a data set. Each measure has its own purpose and is useful in different situations.
The Three Averages
Mean (Arithmetic Average)
Definition: The sum of all values divided by the number of values.
When to use: When all data values are important and there are no extreme outliers.
Median (Middle Value)
Definition: The middle value when data is arranged in order.
Finding it:
- Odd \(n\): Middle value
- Even \(n\): Average of two middle values
When to use: When data has outliers or is skewed.
Mode (Most Common)
Definition: The value that appears most frequently.
Special cases:
- No mode: All values appear equally
- Bimodal: Two modes
- Multimodal: More than two modes
When to use: For categorical data or finding the most popular item.
Interactive Calculator
Data Analyzer
Enter your data values separated by commas and see the mean, median, and mode calculated instantly!
Mean
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Median
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Mode
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Step-by-Step Examples
Data: The ages of 7 students are: 14, 15, 13, 15, 16, 15, 14
Finding the Mean:
Finding the Median:
Finding the Mode:
Data: Test scores: 72, 85, 63, 91, 78, 84
Mean from a Frequency Table
Formula for Grouped Data
When data is given in a frequency table:
\[ \bar{x} = \frac{\sum fx}{\sum f} \]where \(f\) = frequency and \(x\) = data value (or class midpoint for grouped data)
Question: Find the mean number of goals scored.
| Goals (\(x\)) | Frequency (\(f\)) | \(fx\) |
|---|---|---|
| 0 | 3 | 0 |
| 1 | 7 | 7 |
| 2 | 5 | 10 |
| 3 | 4 | 12 |
| 4 | 1 | 4 |
| Total | \(\sum f = 20\) | \(\sum fx = 33\) |
Solution:
\[ \bar{x} = \frac{\sum fx}{\sum f} = \frac{33}{20} = 1.65 \text{ goals} \]When to Use Each Measure
| Measure | Best Used When | Avoid When |
|---|---|---|
| Mean | Data is symmetrical with no outliers | Data has extreme values (outliers) |
| Median | Data is skewed or has outliers | You need to use all data values in calculations |
| Mode | Data is categorical or you want the most common value | All values appear with equal frequency |
The Effect of Outliers
Example: Salaries at a small company: $30,000, $35,000, $32,000, $28,000, $500,000
- Mean: \(\frac{625,000}{5} = \$125,000\) – Heavily affected by the CEO’s salary!
- Median: $32,000 – A much better representation of a “typical” salary
This is why median household income is often reported instead of mean income.
CSEC Practice Questions
Test Your Understanding
Sorted: 3, 5, 7, 8, 9, 12, 15
7 values, so median is the 4th value = 8
Sum of all 5 numbers = Mean × Count = 12 × 5 = 60
Sum of known numbers = 10 + 15 + 8 + 14 = 47
Fifth number = 60 – 47 = 13
Key Points to Remember
- Mean = \(\frac{\text{Sum of values}}{\text{Number of values}} = \frac{\sum x}{n}\)
- Median = Middle value when data is ordered (average of two middle values if even count)
- Mode = Most frequently occurring value (can have no mode, or multiple modes)
- For frequency tables: Mean = \(\frac{\sum fx}{\sum f}\)
- Outliers affect the mean but not the median
- Always show your working – write out the sum and division for mean
- For median: First sort the data, then find the middle position
- For grouped data: Use class midpoints and the \(\frac{\sum fx}{\sum f}\) formula
- Check your answer: The mean should be between the smallest and largest values
- Read carefully: Questions may ask for the “most appropriate” average – consider outliers!