Variance Formula

The statistical measure stating the degree of spread or dispersion relative to the mean for a set of data points is called variance. It measures the amount by which individual values in a data set deviate from its mean value; low variance means most data points are close to the mean, suggesting homogeneity while high variance suggests a wider spread in the values, that is, a lot of variation. Variance measures as the average of squared differences of the values from the mean. There exists a formula in populations and samples by which this average can be measured. For instance, this measure is considered very important in a number of fields due to its applicability in risks, prediction, and hypothesis testing, thus offering a better understanding of how data tends to behave.

Formula For Variance

Population Variance (σ2):

Formula:

Where:

N = number of data points in the population

xi= each data point

μ = population mean

Sample Variance (2s):

Formula:

 

Where:

n = number of data points in the sample

xi = each data point

x = sample mean

Note: The1 In the denominator, n−1 is used, known as Bessel's correction to render an unbiased estimate of the population variance.

Formulas for Grouped Data

Formula for Population Variance

Population variance for grouped data is,

Formula for Sample Variance

Sample variance for grouped data is,

Where,

f = frequency of the class

m = midpoint of the class

These two formulas can also be written as:

Population variance:

Here,

σ2 = Variance 

     xi = Mid Value of ith class  

     fi = Frequency of ith class  

    N = Total number of observations (Population size)

Sample variance:

Here,

   s2 = Sample variance  

     xi = Midvalue of ith class  

     fi = Frequency of ith class  

   n = Sample size (or Number of data values in sample)    

Variance Formula Example Question

Calculate the variance for the following data set for the given height of the trees in feet: 3, 21, 98, 203, 17, 9

Solution:

Step 1: Sum up the numbers in your given data set.

3 + 21 + 98 + 203 + 17 + 9 = 351

Step 2: Square your answer:

351 × 351 = 123201

…and divide by the number of items     

We have 6 numbers in our data set so:

123201/6 =20533.5

Step 3: Take your original list of numbers from Step 1 and square them individually this time:

3 × 3 + 21 × 21 + 98 × 98 + 203 × 203 + 17 × 17 + 9 × 9

Square them all at once:

9 + 441 + 9604 + 41209 + 289 + 81 =51,633

Step 4: Subtract the number from Step 2 by the number from Step 3.

51633 – 20533.5 = 31,099.5

Save this number for now.

Step 5: Subtract 1 from the tally of your data set. For our demonstration:

6 – 1 = 5

Step 6: Divide the number from Step 4 by the number from Step 5. You want to determine variance:

31099.5/5 = 6219.9

Step 7: Square root your result from Step 6. This is what you obtain by finding your standard deviation:

√6219.9 = 78.86634

This answer is 78.86.  

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