Sample Standard Deviation

1. Definition

• The sample standard deviation (s) measures how much the values in a sample spread out around the sample mean.
• It is the square root of the sample variance.
• It tells us, on average, how far each data point is from the sample mean.

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2. Formula

For a sample of size $n$:

$s = \sqrt{\frac{\sum_{i=1}^{n} (x_i – \bar{x})^2}{n – 1}}$

Where:

• $x_i$ = each observation
• $\bar{x}$ = sample mean
• $n$ = sample size
• Denominator = $n-1$ (Bessel’s correction, to make $s$ an unbiased estimator of the population standard deviation σ).

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3. Step-by-Step Example

Sample Data: {5, 7, 9}

1. Find the mean:

$\bar{x} = \frac{5 + 7 + 9}{3} = \frac{21}{3} = 7$

2. Subtract the mean (deviations):

• (5 – 7) = –2
• (7 – 7) = 0
• (9 – 7) = +2

3. Square deviations:

• (–2)² = 4
• 0² = 0
• (2)² = 4

4. Sum squared deviations:

$4 + 0 + 4 = 8$

5. Divide by (n – 1):

$\frac{8}{3 – 1} = \frac{8}{2} = 4$

6. Take square root:

$s = \sqrt{4} = 2$

Sample standard deviation = 2

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4. Interpretation

• Small $s$: Data points are close to the mean (low variability).
• Large $s$: Data points are spread out from the mean (high variability).
• If $s = 0$: All values are identical.

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5. Sample vs Population Standard Deviation

Concept	Formula	Denominator	Used for
Population standard deviation (σ)	$\sqrt{\frac{\sum (x_i – μ)^2}{N}}$	$N$	Whole population
Sample standard deviation (s)	$\sqrt{\frac{\sum (x_i – \bar{x})^2}{n-1}}$	$n-1$	A sample (estimate of σ)

n – 1 instead of n corrects for bias → otherwise, sample variance would underestimate σ.

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6. When It’s Used

• To measure spread/variability in sample data.
• In t-tests, ANOVA, regression, etc. (standard error calculations use sss).
• To build confidence intervals around sample means.

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In short:
The sample standard deviation (s) is the square root of the average squared deviation from the sample mean, using denominator n−1n-1n−1. It measures how spread out the sample data is, and it’s the best unbiased estimator of the population standard deviation (σ).