The sum of deviations refers to the total of the differences between each data point in a set and a central value, typically the mean. The "exact answer" to calculating the sum of deviations depends on whether you are referring to the raw (signed) deviations, absolute deviations, or squared deviations. Each has distinct applications and calculation methods.
Understanding Deviations
A deviation measures how far a single data point strays from a central or average value. It is calculated by subtracting the reference point (most commonly the mean) from the individual data value.
The Fundamental Property: Sum of Raw Deviations from the Mean
One of the most fundamental properties in statistics is that the sum of the raw (signed) deviations of a dataset from its arithmetic mean is always zero. This is a mathematical certainty, not a calculation you perform to get a non-zero result. The positive deviations perfectly balance out the negative deviations.
- Formula: $\sum_{i=1}^{n} (x_i - \bar{x}) = 0$
- $x_i$: Each individual data point
- $\bar{x}$: The arithmetic mean of the dataset
- $n$: The total number of data points
This property makes the mean a unique measure of central tendency because it's the only point from which the sum of deviations is zero.
Calculating the Sum of Absolute Deviations
While the sum of raw deviations is zero, it doesn't tell us about the overall spread or variability of the data. For that, we often look at the sum of absolute deviations. This measures the total "distance" of all data points from the mean, without regard to direction (positive or negative).
To calculate the sum of absolute deviations:
- Calculate the Mean: Find the arithmetic mean ($\bar{x}$) of your dataset. This is done by summing all the data points and dividing by the count of data points.
- Find Individual Absolute Deviations: For each data point ($x_i$), subtract the mean ($\bar{x}$) and then take the absolute value (modulus) of the result. This ensures all differences are treated as positive magnitudes.
- Absolute deviation for each point: $|x_i - \bar{x}|$
- Sum All Absolute Deviations: Add up all the individual absolute deviations calculated in the previous step.
This method gives you the total "spread" from the mean when negative and positive deviations are considered equally impactful.
Example: Sum of Absolute Deviations
Let's calculate the sum of absolute deviations for the dataset: [2, 4, 6, 8, 10]
-
Calculate the Mean:
$\bar{x} = (2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6$ -
Find Individual Absolute Deviations:
| Data Point ($x_i$) | Deviation ($x_i - \bar{x}$) | Absolute Deviation ($|x_i - \bar{x}|$) |
| :----------------- | :-------------------------- | :------------------------------------- |
| 2 | $2 - 6 = -4$ | $|-4| = 4$ |
| 4 | $4 - 6 = -2$ | $|-2| = 2$ |
| 6 | $6 - 6 = 0$ | $|0| = 0$ |
| 8 | $8 - 6 = 2$ | $|2| = 2$ |
| 10 | $10 - 6 = 4$ | $|4| = 4$ | -
Sum All Absolute Deviations:
Sum of Absolute Deviations = $4 + 2 + 0 + 2 + 4 = 12$
Sum of Squared Deviations
Another common way to quantify the total deviation is by summing the squared deviations. This approach is fundamental to calculating variance and standard deviation. Squaring the deviations also ensures positive values, and it gives more weight to larger deviations.
- Formula: $\sum_{i=1}^{n} (x_i - \bar{x})^2$
To calculate the sum of squared deviations, you follow similar steps: calculate the mean, subtract the mean from each data point, then square each result, and finally sum these squared differences.
