Meaning of the standard deviation sign
In statistics, the phrase standard deviation sign usually refers to the symbol used to denote standard deviation. The most common symbols are \( \sigma \) and \( s \), and the correct choice depends on whether the data describe a population or a sample.
Key idea: \( \sigma \) is tied to population parameters (true, fixed values). \( s \) is tied to sample statistics (computed from observed data).
| Quantity | Symbol | What it represents | Typical companion symbol |
|---|---|---|---|
| Population standard deviation | \( \sigma \) | Spread of the entire population around the population mean \( \mu \) | Population variance \( \sigma^2 \) |
| Sample standard deviation | \( s \) | Spread of a sample around the sample mean \( \bar{x} \); estimates population spread | Sample variance \( s^2 \) |
| Population mean | \( \mu \) | True population average | Pairs with \( \sigma \) |
| Sample mean | \( \bar{x} \) | Average computed from the sample | Pairs with \( s \) |
Formulas where the symbols appear
Standard deviation is the square root of variance. The variance symbols are the “squared” versions of the standard deviation symbols: \( \sigma^2 \) for a population and \( s^2 \) for a sample.
Population (size \(N\))
\[ \sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2, \qquad \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2}. \]
Sample (size \(n\))
\[ s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2, \qquad s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2}. \]
The \(n-1\) in \(s^2\) (Bessel’s correction) makes \(s^2\) an unbiased estimator of the population variance \( \sigma^2 \) under standard assumptions.
Worked example showing σ and s
Consider the dataset \(2, 4, 4, 4, 5, 5, 7, 9\). Treating it as a complete population uses \( \sigma \); treating it as a sample uses \( s \).
| \(x_i\) | \(x_i-\bar{x}\) | \((x_i-\bar{x})^2\) |
|---|---|---|
| 2 | \(-3\) | 9 |
| 4 | \(-1\) | 1 |
| 4 | \(-1\) | 1 |
| 4 | \(-1\) | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
The mean is \[ \bar{x} = \frac{2+4+4+4+5+5+7+9}{8} = \frac{40}{8} = 5. \] The sum of squared deviations is \[ \sum (x_i-\bar{x})^2 = 9+1+1+1+0+0+4+16 = 32. \]
If treated as a population (use \( \sigma \)):
\[ \sigma^2 = \frac{32}{8} = 4, \qquad \sigma = \sqrt{4} = 2. \]
If treated as a sample (use \( s \)):
\[ s^2 = \frac{32}{7} \approx 4.5714286, \qquad s = \sqrt{\frac{32}{7}} \approx 2.1380899. \]
How the symbol is used in intervals and the “±” confusion
Standard deviation itself is always nonnegative, so the standard deviation sign is not “±”. The “±” appears when describing an interval around a center, such as \( \mu \pm \sigma \) or \( \bar{x} \pm s \). For example, the interval \( \mu \pm 2\sigma \) means values within two standard deviations of the mean.
Quick recognition checklist
Use \( \sigma \) when the question states the data represent an entire population or a population model parameter is given.
Use \( s \) when the data are a sample and the spread is computed from observed values.
Variance symbols: \( \sigma^2 \) and \( s^2 \) are the squared versions of the standard deviation signs.