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 X_i is given as below.

σ² = Σ (X_i-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

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.

Do you see the difference between graph 1 and 2?

See you in another post with another interesting concept.