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Deviation — Definition, Formula & Examples

Deviation is the difference between a single data point and the mean of the data set. It tells you how far that value sits from the average, and in which direction.

For a data value xix_i in a data set with mean xˉ\bar{x}, the deviation of xix_i is defined as di=xi−xˉd_i = x_i - \bar{x}. A positive deviation indicates the value lies above the mean; a negative deviation indicates it lies below.

Key Formula

di=xi−xˉd_i = x_i - \bar{x}
Where:
  • did_i = Deviation of the $i$-th data point
  • xix_i = The $i$-th data value
  • xˉ\bar{x} = Mean (average) of the data set

How It Works

To find a deviation, first calculate the mean of your data set. Then subtract the mean from the data point in question. The result can be positive, negative, or zero. One key property: when you add up all the deviations in a data set, the sum is always zero. This is why standard deviation squares each deviation before averaging — squaring eliminates the cancellation between positive and negative values.

Worked Example

Problem: A data set contains the values 4, 7, 10, 13, 16. Find the deviation of each value.
Find the mean: Add all values and divide by the count.
xˉ=4+7+10+13+165=505=10\bar{x} = \frac{4 + 7 + 10 + 13 + 16}{5} = \frac{50}{5} = 10
Subtract the mean from each value: Compute each deviation using the formula.
d1=4−10=−6,d2=7−10=−3,d3=10−10=0,d4=13−10=3,d5=16−10=6d_1 = 4 - 10 = -6,\quad d_2 = 7 - 10 = -3,\quad d_3 = 10 - 10 = 0,\quad d_4 = 13 - 10 = 3,\quad d_5 = 16 - 10 = 6
Verify the sum: Check that all deviations sum to zero.
−6+(−3)+0+3+6=0✓-6 + (-3) + 0 + 3 + 6 = 0 \checkmark
Answer: The deviations are −6, −3, 0, 3, and 6. Their sum confirms they equal zero.

Why It Matters

Deviation is the building block of standard deviation and variance. Every time you calculate spread in a statistics or AP Stats course, you start by finding individual deviations. Understanding this step makes the rest of the standard deviation formula far less mysterious.

Common Mistakes

Mistake: Taking the absolute value of every deviation before summing, then concluding the sum should not be zero.
Correction: Raw deviations are signed values (positive or negative). Their sum is always zero. Absolute deviations and squared deviations are separate concepts used for different measures of spread.