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how to get standard deviation

To get the standard deviation, you measure how far numbers in a dataset tend to be from their mean (average). It’s basically “average distance from the center.”

What standard deviation means

  • Standard deviation tells you how spread out your data is.
  • Small standard deviation → values are tightly clustered around the mean.
  • Large standard deviation → values are widely spread.

For a set of values x1,x2,…,xnx_1,x_2,\dots,x_nx1​,x2​,…,xn​, the standard deviation is the square root of the variance, which is the average of the squared distances from the mean.

Step‑by‑step: sample standard deviation

Use this when your data is a sample from a larger population. Given data: x1,x2,…,xnx_1,x_2,\dots,x_nx1​,x2​,…,xn​

  1. Find the mean

xˉ=1n∑i=1nxi\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_ixˉ=n1​i=1∑n​xi​

  1. Subtract the mean from each value
    Compute deviations: di=xi−xˉd_i=x_i-\bar{x}di​=xi​−xˉ for each i.

  2. Square each deviation
    Compute di2d_i^2di2​ for all i.

  3. Add up all squared deviations

SS=∑i=1ndi2SS=\sum_{i=1}^{n}d_i^2SS=i=1∑n​di2​

This is called the sum of squares.

  1. Divide by n−1n-1n−1 (sample variance)

s2=SSn−1s^2=\frac{SS}{n-1}s2=n−1SS​

  1. Take the square root (sample standard deviation)

s=s2=∑i=1n(xi−xˉ)2n−1s=\sqrt{s^2}=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}s=s2​=n−1∑i=1n​(xi​−xˉ)2​​

That last formula is the usual “sample standard deviation” you see in statistics.

Population vs sample standard deviation

  • Population standard deviation (you have all values in the population):

σ=∑i=1N(xi−μ)2N\sigma =\sqrt{\frac{\sum_{i=1}^{N}(x_i-\mu)^2}{N}}σ=N∑i=1N​(xi​−μ)2​​

where μ\mu μ is the population mean and NNN is the population size.

  • Sample standard deviation (you only have a sample):

s=∑i=1n(xi−xˉ)2n−1s=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}s=n−1∑i=1n​(xi​−xˉ)2​​

The n−1n-1n−1 is called Bessel’s correction and makes the estimate less biased for samples.

Quick numeric example

Say your data is: 4, 7, 10

  1. Mean: xˉ=(4+7+10)/3=7\bar{x}=(4+7+10)/3=7xˉ=(4+7+10)/3=7
  2. Deviations: (4-7)=-3,\(7-7)=0,\(10-7)=3
  3. Squared deviations: 9,\0,\9
  4. Sum of squares: 9+0+9=18
  5. Sample variance: 18/(3-1)=18/2=9
  6. Sample standard deviation: \sqrt{9}=3

So the standard deviation is 3.

How to get it quickly (practice or tools)

You can get standard deviation in most tools:

  • Excel / Google Sheets :
    • Sample: =STDEV.S(A1:A10)
    • Population: =STDEV.P(A1:A10)
  • Calculators / apps : Many “statistics” or “descriptive stats” calculators have a button or field for standard deviation.

If you share a specific dataset, I can walk through the calculation step by step for that exact set.