Mathematical Foundation
The mean, or arithmetic average, is a fundamental measure of central tendency that represents the typical value in a dataset. It provides a single representative value that summarizes the entire collection of data points by distributing the total sum equally across all observations. The mean is sensitive to extreme values and serves as the foundation for many statistical calculations and data analysis techniques.
- For individual data values: $\overline{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + \cdots + x_n}{n}$
- For grouped data: $\overline{x} = \frac{\sum (f_i \cdot x_i)}{\sum f_i}$ where $f_i$ is frequency
- For data with weights: $\overline{x} = \frac{\sum (w_i \cdot x_i)}{\sum w_i}$ where $w_i$ is weight
- $\overline{x}$ = sample mean
- $n$ = number of data points
- $x_i$ = individual data values
- $\sum$ = summation symbol
Interactive Mean Calculator
Data Visualization
Sum: 41
Count: 6
Mean: 6.83
Worked Example 1: Calculating Mean of Test Scores
A student scored 85, 92, 78, 96, and 88 on five math tests. Find the mean score.
Solution:
Sum = 85 + 92 + 78 + 96 + 88 = 439
Mean = 439 ÷ 5 = 87.8
The student's average test score is 87.8
Worked Example 2: Mean with Frequencies (Grouped Data)
The table shows the number of hours students spent studying:
| Hours | Frequency |
|---|---|
| 2 | 5 |
| 4 | 8 |
| 6 | 6 |
| 8 | 3 |
Solution:
Mean = (2×5 + 4×8 + 6×6 + 8×3) ÷ (5+8+6+3) = (10 + 32 + 36 + 24) ÷ 22 = 102 ÷ 22 = 4.64
The average study time is 4.64 hours
Worked Example 3: Weighted Mean
A student's grades are: Quiz (20% weight): 85, Midterm (30%): 78, Final (50%): 92. Find the weighted mean.
Solution:
Weighted Mean = (85×0.2 + 78×0.3 + 92×0.5) ÷ (0.2 + 0.3 + 0.5) = (17 + 23.4 + 46) ÷ 1 = 86.4
The weighted average grade is 86.4
- Balance Point: The mean acts as a balance point for the data - the sum of deviations from the mean equals zero.
- Least Squares Property: The mean minimizes the sum of squared deviations from any point.
- Effect of Extreme Values: Outliers can significantly affect the mean, making it less representative of the data.
- Additive Property: The mean of combined datasets can be calculated from individual means and sample sizes.
Mean (Average)
- Uses all data points
- Affected by outliers
- Mathematically useful
- Balance point of data
Median
- Middle value when ordered
- Resistant to outliers
- Position-based measure
- Not affected by extreme values
Mode
- Most frequent value
- Can have multiple modes
- Resistant to outliers
- May not exist
Practice Problems
Sum = 12 + 15 + 18 + 22 + 25 + 30 = 122
Mean = 122 ÷ 6 = 20.33
Original sum = 8 × 25 = 200
New sum = 7 × 22 = 154
Removed number = 200 - 154 = 46
| Score | Frequency |
|---|---|
| 60 | 3 |
| 70 | 5 |
| 80 | 4 |
| 90 | 2 |
Sum = (60×3) + (70×5) + (80×4) + (90×2) = 180 + 350 + 320 + 180 = 1030
Total frequency = 3 + 5 + 4 + 2 = 14
Mean = 1030 ÷ 14 = 73.57
Original sum = 3 × 15 = 45
New sum = 4 × 17 = 68
Fourth number = 68 - 45 = 23
Sum = 5 + 8 + 12 + x + 15 = 40 + x
Mean = (40 + x) ÷ 5 = 10
40 + x = 50
x = 10
Key Takeaways
- Definition: The mean is the arithmetic average, calculated by summing all values and dividing by the number of values.
- Formula: $\overline{x} = \frac{\sum x_i}{n}$ for individual data, or $\overline{x} = \frac{\sum (f_i \cdot x_i)}{\sum f_i}$ for grouped data.
- Properties: The mean serves as the balance point of the data; sum of deviations from mean equals zero.
- Sensitivity: The mean is affected by extreme values (outliers), unlike the median.
- Applications: Used in quality control, performance evaluation, financial analysis, and scientific research.
- Weighted Mean: When data points have different importance, use weights: $\overline{x} = \frac{\sum (w_i \cdot x_i)}{\sum w_i}$.
- Comparison: Mean uses all data points but is sensitive to outliers; median is robust but ignores magnitude of values.