How to Calculate Standard Deviation

Answer: The standard deviation is the square root of the variance — it measures how spread out numbers are. Compute it by finding the mean, squaring each deviation from the mean, averaging those squares (use $N$ for a full population or $n-1$ for a sample), then taking the square root.

Explanation

• Use the population formulas when you have data for the entire population.
• Use the sample formulas (divide by $n-1$) when your data is a sample and you want an unbiased estimator of the population variance.

Formulas

Population variance and standard deviation:
$$\sigma^2 = \frac{1}{N}\sum_{i=1}^N (x_i-\mu)^2$$
$$\sigma = \sqrt{\sigma^2}$$

Sample variance and standard deviation:
$$s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i-\bar{x})^2$$
$$s = \sqrt{s^2}$$

Steps (practical)

1. Compute the mean: $ \mu = \frac{1}{N}\sum x_i$ (or $\bar{x}$ for a sample).
2. For each data point compute the deviation: $x_i – \mu$.
3. Square each deviation: $(x_i-\mu)^2$.
4. Sum the squared deviations: $\sum (x_i-\mu)^2$.
5. Divide by $N$ (population) or $n-1$ (sample) to get variance.
6. Take the square root to get standard deviation.

Worked example

Data: 2, 4, 4, 4, 5 (here treat as population)

1. Mean: $\mu = \frac{2+4+4+4+5}{5} = 3.8$
2. Squared deviations: $(2-3.8)^2=3.24$, $(4-3.8)^2=0.04$ (three times), $(5-3.8)^2=1.44$
3. Sum: $3.24+0.04+0.04+0.04+1.44=4.8$
4. Population variance: $\sigma^2 = \frac{4.8}{5}=0.96$
5. Population standard deviation: $\sigma=\sqrt{0.96}\approx0.98$

If treated as a sample:

• Sample variance: $s^2=\frac{4.8}{5-1}=1.20$
• Sample standard deviation: $s=\sqrt{1.20}\approx1.10$

Quick notes

• Units: SD has same units as the original data.
• Use $n-1$ (Bessel’s correction) for samples to correct bias.
• Outliers strongly affect SD because of squaring deviations.

Calculator / software

• Excel: STDEV.P(range) for population, STDEV.S(range) for sample.
• Python (numpy): np.std(data, ddof=0) for population, np.std(data, ddof=1) for sample.

Example Python:

import numpy as np
data = np.array([2,4,4,4,5])
pop_sd = np.std(data, ddof=0)   # population
sample_sd = np.std(data, ddof=1) # sample

If you want, I can show this with your specific dataset or demonstrate step-by-step on a new example.