Demystifying Standard Deviation and Its Relationship to Variance
Variance and standard deviation are closely related concepts that are used to measure the dispersion of a dataset. While variance is standard for mathematical derivations, standard deviation is the go-to metric for reporting and interpreting statistical spread.
In this article, we’ll demystify standard deviation and look at its close relationship to variance.
The Mathematical Connection
The relationship between the two metrics is simple:
- Standard Deviation is the square root of the Variance.
- Variance is the square of the Standard Deviation.
Mathematically:
- $\sigma = \sqrt{\sigma^2}$ (for populations)
- $s = \sqrt{s^2}$ (for samples)
Why Do We Need Both?
If they are so closely related, why do statisticians use both?
The answer lies in units of measurement.
When calculating variance, you square the differences from the mean (e.g., $x_i - \mu$). This means the units of variance are also squared. For example, if you are measuring heights in inches, the variance is expressed in squared inches ($\text{in}^2$). Squaring the units makes the variance difficult to interpret intuitively—what does a height spread of “45 squared inches” actually mean?
Taking the square root of the variance brings the metric back to the original units of measurement. In our height example, the standard deviation is in inches, making it much easier to understand.
Interpreting Standard Deviation in the Real World
Standard deviation tells us, on average, how much individual values deviate from the mean.
The Empirical Rule (68-95-99.7 Rule)
For data that follows a normal distribution (bell curve), the standard deviation provides highly predictable information about how data is spread:
- Approximately 68% of the data falls within $\pm 1$ standard deviation of the mean.
- Approximately 95% of the data falls within $\pm 2$ standard deviations of the mean.
- Approximately 99.7% of the data falls within $\pm 3$ standard deviations of the mean.
Practical Applications
- Investment Volatility: Stocks with high standard deviations are riskier because their daily returns deviate wildly from their average performance.
- Academic Grading: A teacher can use standard deviation to see if student scores are tightly packed or spread out, which helps in grading on a curve.
- Manufacturing Tolerance: In high-precision manufacturing, engineers use standard deviation to verify that components are built to extremely tight specs (low standard deviation).
Using our online Variance Calculator, you get both the variance and the standard deviation calculated instantly from your data, along with a visual distribution chart showing the mean and the $\pm 1$ standard deviation boundaries.