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Variance: Definition, Types, Application and Steps to Calculate Variance

GMAT VarianceVariance plays a major role in interpreting data in statistics. The most common application of variance is in polls. For opinion polls, the data gathering agencies cannot invest in collecting data from the entire population. They set criteria for sampling the population based on ethnicity, income group, regions, education level, salary and religion, so that the population is completely represented by the samples.

With samples, the data can vary from the mean by a large extent. Variance is used to find the variation of the data from the mean. Interestingly, the variance exaggerates the spread, and thus standard deviation was introduced.

Standard Deviation = Square Root (Variance)

Variance Definition

Variance is represented by (Sigma)^2
. In simple mathematical terms, it is the average of the square difference between each data from the mean.

Steps to Calculate Variance

1) Find the mean of the data set
2) For each data, find the difference between the number and the mean
3) Square the resulting number and
divide by (n-1) where n is the number of data collected

Let us look at the average height of men in six countries:

Argentina    1.73
Germany    1.810
India    1.647
Japan    1.70
South Africa    1.69
Australia    1.74

Steps to Calculate Variance

1) Find the mean of the data set
= (1.73+1.81+1.64+1.7+1.69+1.74)/6
= 1.71

2) For each data, find the difference between the number and the mean

Mean = 1.71

Argentina    1.73
Diff(Argentina) = (1.73-1.71) = .02

Germany    1.810
Diff(Germany) = 1.81 – 1.71 = 0.1

India    1.647
Diff(India)  = 1.647 – 1.71 = -0.063

Japan    1.70
Diff(Japan)  = 1.70 – 1.71 = -.01

South Africa    1.69
Diff(South Africa)  = 1.69 – 1.71 = -.02

Australia    1.74
Diff(Australia) = 1.74 – 1.71 = .03

4) Square the resulting number and divide by (n-1) where n is the number of data collected

(Diff(Argentina)^2 + Diff(Germany)^2+ Diff(India)^2+ Diff(Japan)^2+ Diff(South Africa)^2+ Diff(Australia)^2)/5
= (0.004 + 0.01 + 0.003969 + 0.0001 + .0004 + .0009)/6 = 0.00322

Variance (
Sigma^2) = 0.003864
            
If we plot the heights of men from the six countries, it would look like:
          
Variance GMAT

Variance (
Sigma^2) = 0.003864

The range of the heights is 0.163 while the difference from the mean is in the 0.01 to 0.10 range. The variance is underrepresenting the difference. Therefore, we find the square root of the variance, and calculate the standard deviation.

Standard Deviation (
Sigma) = Sqrt (Variance) = 0.0621

This falls within the .01 to .10 range, which is the range of the difference from the mean. With the standard deviation data of .062, statisticians can find how spread out the height is with respect to the mean.

Sample Variance vs. Population Variance

Since we assume that the average height of the men from each country is not a complete data set but a sample from the most representative population in each country, the sum of square of the difference between each number and the mean is divided by (n-1)
If the question clearly specifies that the data is from the entire population set, then divide the sum of square of the difference between each number and the mean, by the number of data set (n)

Sample Variance = 1/ (n-1) * (Sum ((xi – mean) ^2)), where i = 1..n

Population Variance = 1/ (n) * (Sum ((xi – mean) ^2)),
where i = 1..n

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