Statistics

Population and Sample Standard Deviation

To measure the spread of a dataset we use standard deviation (sd). A higher standard deviation means numbers are more spread out in the data set while a low standard deviation indicates closer to the mean.

Two types –

  • Sample standard deviation.
  • Population standard deviation.

The greek letter sigma σ indicates population standard deviation and “s” denotes sample standard deviation.

Formula

Population and Sample Standard Deviation

Let’s Calculate:

In the real world, we may not need to calculate it manually by hand. However, It is always good to know how it really works.

  1. First, calculate the mean of the dataset.
  2. Secondly, find out the variance of a dataset.
  3. Finally, square root of variance will provide Standard Deviation.

Example:

We have a list of 5 different people’s heights in cm. 174,180,190,195,170. Let’s find the standard deviation for this dataset.

Mean:

The average value of a dataset.

  • Sum all data points
  • Divide the sum by the size of the dataset.
$$μ = {{174 + 180 + 190 + 195 + 170}\over{5}}= {{909}\over{5}}= {181.8}$$

Our population mean is 181.8 cm

Variance:

Variance means the difference between numbers in a dataset.

  • Subtract each datapoints from the mean and sqaure it and get the sum
  • Divide the squared sum by the size of the dataset.
$$σ^{2} = {{(174 -181.8)^{2} + (180 -181.8)^{2} + (190 -181.8)^{2} + (195 -181.8)^{2} + (170 -181.8)^{2}}\over{5}}$$
$$= {{-7.8^{2} + -1.8^{2} + 8.2^{2} + 13.2^{2} + -11.8^{2}}\over{5}}$$ $$σ^{2} = {{15.6 + 3.6 + 16.4 + 26.4 + 23.6}\over{5}} = {{85.6}\over{5}}= {17.12}$$

The variance for this data set is 17.12 cm

Standard Deviation:

$$σ = √17.12$$
$$σ = 4.14$$

Finally, the answer is 4.41 cm

In the above example, population standard deviation has been calculated. But, If we consider this as a sample data set we need to minus 1 from the sample size. hence the denominator would be 5-1 = 4 and the result would be like below.

$$σ^{2} = {{15.6 + 3.6 + 16.4 + 26.4 + 23.6}\over{5-1}} = {{85.6}\over{4}}= {21.4}$$
$$σ = √21.4$$
$$σ = 4.63$$