Spread: Variance & Standard Deviation
βStandard deviation is 'how far from the mean, on average'.β
The formula
ΟΒ² = { (xββxΜ)Β² + β― + (xββxΜ)Β² } / nHow to read it: the variance is the average of the squared deviations of each value from the mean, and its square root is the standard deviation
- xα΅’ β xΜ
- β the deviation β how far a value sits from the mean
- (xα΅’ β xΜ)Β²
- β the squared deviation (squaring removes the sign)
- ΟΒ²
- β the variance β the average of the squared deviations
- Ο
- β the standard deviation β the (positive) square root of the variance
The hook
Two datasets can share the same mean yet be worlds apart in 'how spread out' they are β variance and standard deviation are the rulers for that spread.
In plain words
The average of the squared deviations of each value from the mean is the variance; its square root is the standard deviation.
The intuition
The mean is the bullseye, and the standard deviation is how far the arrows land from center on average. Small means tightly clustered; large means widely scattered.
How it's built
β Find the mean, β‘ find each deviation (value β mean), β’ the deviations just add to 0, so square them to kill the sign and average β variance, β£ squaring inflated the units, so take the square root to get back β standard deviation.
Example
Data 3, 4, 5, 6, 7 β mean 5. Deviations β2, β1, 0, 1, 2; squares 4, 1, 0, 1, 4 sum to 10. Variance = 10/5 = 2, standard deviation = β2 β 1.41.
Common mistake
Averaging the raw deviations always gives 0, so it can't measure spread β that's why we square. Also, variance is in squared units (e.g. pointsΒ²), so to speak in the original units you use the standard deviation.
Where it's used
How evenly test scores fall, how stable a product's quality is, investment risk (volatility) β measuring the 'stability the mean alone hides'.
Where it came from
The name and symbol for standard deviation came from Karl Pearson in the late 1800s; this tool for capturing spread in one number became a pillar of modern statistics.
Prerequisites
Quick check
If a dataset has a variance of 9, what is its standard deviation?
- 3β
- 4.5
- 9
- 81
Practice
What is the variance of the data 4, 4, 4, 4?
Answer: 0
- Every value equals the mean 4
- All deviations are 0 β variance 0
Takeaway: If all values are equal, there's no spread, so the variance is 0.
What is the variance of the data 2, 4, 6, 8, 10?
Answer: 8
- Mean = 30/5 = 6
- Deviations β4, β2, 0, 2, 4; squares 16, 4, 0, 4, 16 sum to 40
- 40 / 5 = 8
Takeaway: Mean β deviation β square β average, in that order.
Find the variance and standard deviation of the data 1, 2, 3, 4, 5.
Answer: undefined
- Mean = 3
- Deviations β2, β1, 0, 1, 2; squares sum to 10
- Variance = 10/5 = 2, standard deviation = β2 β 1.41
Takeaway: The standard deviation is the variance's square root, restoring the original units.