Population vs Sample Standard Deviation

Population vs Sample Standard Deviation

The difference between population standard deviation and sample standard deviation boils down to who you’re measuring and how you correct for bias when estimating variability.

📊 Definitions

Type What It Measures Formula Difference
Population Standard Deviation (\(\sigma\)) Variability of all data points in a population Divide by \(N\) (total number of data points)
Sample Standard Deviation (\(s\)) Variability in a subset (sample) of the population Divide by (\(N - 1\)) (Bessel’s correction)

🧠 Why the Difference?

  • Population SD assumes you have access to every data point in the population. No need to correct for bias.
  • Sample SD uses Bessel’s correction (dividing by \(N - 1\)) to avoid underestimating the true variability of the population. This correction compensates for the fact that a sample tends to be less variable than the full population.

🧮 Formulas

  • Population SD:
\[\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}\]
  • Sample SD:
\[s = \sqrt{\frac{1}{N - 1} \sum_{i=1}^{N} (x_i - \bar{x})^2}\]

Where:

  • \(x_i\) = each data point
  • \(\mu\) = population mean
  • \(\bar{x}\) = sample mean
  • \(N\) = number of data points

🎯 When to Use Which?

  • Use population SD when you have data for the entire group you’re studying (e.g., all students in a school).
  • Use sample SD when you’re working with a subset and want to generalize to the whole population (e.g., survey results from 100 out of 10,000 customers).



Note:

Current version of this post is generated partially using generative AI.

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