Measures of Variance – Standard Deviation

I’m writing about variance in this blog post series. I have discussed about Mean Deviation or Average Deviation in my previous post.

Dispersion or variance in observations is what we need to explain. Data researchers wants to know why some samples are above/below the average for a given variable.

A variance measures the degree of spread (dispersion) in a variable’s values.

Population variance for variable Xi is given as below.

σ2 = Σ (Xi-X̄)2/n

Standard Deviation

This is a method to find the distance between each sample and a center point, like mean. This distance is called deviations from the measures of central tendency. It is also the square root of the variance.

Steps to calculate the standard deviation

  1. Calculate the mean of the series
  2. Find the deviations for various items from the mean d = x – x̄
  3. Square the deviations d2
  4. Multiply the respective frequencies f * d2
  5. Total the product Σ f * d2
  6. Apply the formula

Example – 1

Let’s calculate the standard deviation for the following data set.

Village code 4 wheeler count
10 8
20 12
30 20
40 10
50 7
60 3

Solution

Village code (x) 4 wheeler count (f) f * X d = x – x̄ d2 f * d2
10 8 80 -20.83 434.03 3472.22
20 12 240 -10.83 117.36 1408.33
30 20 600 -0.83 0.69 13.89
40 10 400 9.17 84.03 840.28
50 7 350 19.17 367.36 2571.53
60 3 180 29.17 850.69 2552.08
X = 210 N = 60 Σ f * X = 1850 Σ f * d2 = 10858.33

Mean = Σ f*X / N

Mean = 1850/60

Mean = 30.8 (Corrected to single decimal place)

Standard deviation σ2 = √ (f * d2)/N

σ2 =√ (10858.33 / 60)

σ2 =√180.97

σ2 =13.45

Example – 2

5 samples of a spare parts manufacturing plant is given below.

64, 68, 74, 76, 78

Mean x̄ = (64 + 68 + 74 + 76 + 78)/5 = 72

Samples 64 68 74 76 78
Mean x̄ 72 72 72 72 72
(Xi – x̄)2 64 16 4 16 36
Σ (Xi – x̄)2 136
S2 = (Σ (Xi – x̄)2)/N 27.2
σ2 = √S2 5.21536192416212

Standard deviation

Example – 3

Let’s use the similar data set. All the samples are of same size 50. Let’s see how it differs.

Samples 50 50 50 50 50
Mean x̄ 50 50 50 50 50
(Xi – x̄)2 0 0 0 0 0
Σ (Xi – x̄)2 0
S2 = (Σ (Xi – x̄)2)/N 0
σ2 = √S2 0

The deviation is 0.

Example – 4

Let’s consider the same example with a high variation.

Samples 0 30 60 90 120
Mean x̄ 60 60 60 60 60
(Xi – x̄)2 3600 900 0 900 3600
Σ (Xi – x̄)2 9000
S2 = (Σ (Xi – x̄)2)/N 1800
σ2 = √S2 42.4264068711929

Let’s try to visualize it.

Standard deviation 2

Do you see the difference between graph 1 and 2?

See you in another post with another interesting concept.

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2 thoughts on “Measures of Variance – Standard Deviation

  1. Pingback: Measures of shape – Skew & kurtosis | JavaShine

  2. Pingback: Working with data types of R | JavaShine

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