Arithmetic Mean
Description
The Arithmetic Mean, commonly known simply as the "average," is the most fundamental measure of central tendency in statistics and mathematics. It represents the theoretical "balance point" or center of gravity of a dataset. If you were to place weights on a weightless lever corresponding to each data point's value, the fulcrum would need to be placed exactly at the mean to balance the beam.
Mathematically, it is defined as the sum of all values divided by the total count of values. While intuitive and easy to calculate, the mean has a specific characteristic: it is sensitive to extreme outliers. For example, if ten people in a room earn $50,000, and one billionaire enters earning $1,000,000,000, the mean income will be heavily skewed upward, possibly not reflecting the "typical" person (for which the Median might be a better measure).
The mean is denoted by $\bar{x}$ (read "x-bar") when referring to a sample of data, and by the Greek letter $\mu$ (mu) when referring to an entire population.
History & Origins
The concept of the "mean" or "middle" has been studied since antiquity. Pythagoreans (c. 500 BC): The Greek mathematician Archytas of Tarentum, a contemporary of Plato, distinguished between three types of means: the Arithmetic Mean, the Geometric Mean, and the Harmonic Mean. For the Pythagoreans, these ratios were connected to music theory and the harmony of the universe. Adolphe Quetelet (19th Century): A Belgian statistician who introduced the concept of the "Average Man" (l'homme moyen), applying the arithmetic mean to human physical and social characteristics, laying the groundwork for modern sociology.
Proof: Sum of Deviations is Zero
A key property of the mean is that the sum of the distances (deviations) of all data points from the mean is always zero. This confirms it is the center of gravity.
Let the deviation of a point $x_i$ from the mean $\bar{x}$ be $d_i = x_i - \bar{x}$.
We want to find the sum of all deviations: $\sum_{i=1}^{n} (x_i - \bar{x})$.
Distribute the summation: $\sum_{i=1}^{n} x_i - \sum_{i=1}^{n} \bar{x}$.
Since $\bar{x}$ is a constant (it does not change with i), $\sum_{i=1}^{n} \bar{x} = \bar{x} + \bar{x} + ... = n\bar{x}$.
So the expression becomes: $\sum_{i=1}^{n} x_i - n\bar{x}$.
Recall the definition of the mean: $\bar{x} = \frac{\sum x_i}{n}$. Multiply both sides by n to get $\sum x_i = n\bar{x}$.
Substitute $n\bar{x}$ for $\sum x_i$: $n\bar{x} - n\bar{x} = 0$.
Conclusion: The sum of deviations from the mean is always zero.
Variables
| Symbol | Meaning |
|---|---|
x̄ | Sample Mean (read "x-bar") |
μ | Population Mean (read "mu") |
n | Number of values (count) |
Σ | Summation (Sigma) - add them all up |
xᵢ | Individual values in the dataset |
Examples
Basic Calculation
Problem: Find the mean of the test scores: 70, 80, 80, 90, 100
Solution:
Grade Point Average (GPA)
Problem: A student has the following grades: Math (95), History (85), Science (70), Art (90). What is the average grade?
Solution: 85
- List the values: $x_1=95, x_2=85, x_3=70, x_4=90$.
- Count the items: $n = 4$.
- Calculate the Sum ($\Sigma x$): $95 + 85 + 70 + 90 = 340$.
- Divide by n: $\bar{x} = 340 / 4$.
- Result: 85.
Sports: Points Per Game
Problem: A basketball player scores 12, 25, 18, and 15 points in 4 games. What is their scoring average?
Solution: 17.5 PPG
- Sum: $12 + 25 + 18 + 15 = 70$.
- Count: 4 games.
- Mean: $70 / 4 = 17.5$.
- The player averages 17.5 points per game.
Common Mistakes
Confusing Mean and Median
The Mean is the sum divided by count. The Median is the middle number when sorted. They are different, especially if there are outliers.
Averaging Averages
If Class A has 10 students with average 90, and Class B has 100 students with average 80, you cannot just say (90+80)/2 = 85. You must use a "Weighted Mean" based on class size.
Real-World Applications
Finance & Economics
The mean is used to calculate stock returns over time or the "Moving Average" to smooth out price fluctuations. In economics, GDP per Capita is the arithmetic mean of a country's total economic output divided by its population.
Signal Processing
In electronics, the "DC offset" of a signal is essentially its mean voltage. Engineers often calculate the mean to remove the DC component and focus on the AC component (fluctuations).
Frequently Asked Questions
When should I use Median instead of Mean?
Use the Median when your data has extreme outliers (like house prices or salaries), as the Mean can be distorted by very high or low values.
What is the difference between x-bar and mu?
$\bar{x}$ is for a SAMPLE (a portion of data). $\mu$ is for the POPULATION (all possible data). The calculation is the same, but the notation indicates scope.