A Brief Introduction To Variance With Calculations

In statistics, measure that quantifies the dispersion or spread of data points around their mean. It provides a numerical value that indicates how much individual data points deviate from the average.

The concept of variance was first introduced by the French mathematician Abraham de Moivre in the early 18^th century. However, it was the renowned English statistician Ronald A. Fisher defined and popularized the concept of variance in the early 20^th century.

He developed the mathematical formula for calculating variance, providing researchers with a strong tool to analyze data and make inferences. The concept of variance quickly gained prominence in various fields, including economics, social sciences, and engineering.

In this article, we will discuss the basic definition of Variance, the formula of sample Variance and Variance population, and the procedure to evaluate the population Variance with the help of examples in detail.

What is Variance?

In statistics, variance calculates quantifies the banquet or diffusion of a set of data points about their mean. It offers a mathematical value that shows how much the distinct data points separate from the average.

Today, variance played a crucial role in hypothesis testing, experimental design, and decision-making processes with the start of computers, variance computations became more accessible, and efficient, and further contributed to its widespread usage.

Variance formula

The formula for variance depends on whether you are calculating the variance for a population or a sample.

Note: The only difference between the population and sample variance formulas is the denominator “N “for population variance and (n - 1) for sample variance. The (n - 1) denominator in the sample variance formula accounts for the reduced degrees of freedom when working with a sample instead of the entire population.

Techniques to determine the Population Variance

Some techniques to determine the population variance in detail.

Calculate Dataset:

Accumulate the numbers ideas for which you want to analyze the population variance.

Mean evaluation:

Evaluate the mean of the dataset by adding up whole the data points and dividing by the overall number of data points. This gives you the population mean (µ).

Subtract the Mean:

Subtract the population mean from each data point. The divergence of every point of data from the average is determined in this stage.

Square the Differences:

Square the deviations obtained in the previous step. This ensures that all values are positive and preserves the magnitude of the differences.

Sum the Squared Differences:

Both the squared deviations collected in the step prior should be added together.

Determine the Population Size:

Determine the total number of data points in your dataset. Let's denote this as N.

Calculate the Population Variance:

Gap the adding of squared variances by the whole number of statistics points (N) to attain the population variance (s²). This phase accounts for working with the whole population.

Interpret the Variance:

The resulting value represents the variance of the population. It computes the inconsistency or banquet of the statistics points about the mean.

A population variance calculator could be used to find the variance of the given population set of data according to above techniques.

Calculations of Variance

In this section, we have discussed the example of variance with the help of population and sample variance.

Example 1: (For sample)

Find the variance of the sample data 9,25,30,38,67,80.

Solution:

Step 1:

Step 2:

We write the formula for sample variance

s²=S (x? - x¯)²/(n - 1)

Step 3:

Putting the value of step 2 and simplifying the variance sample

s²=721.4

Step 4:

Therefore, the value of sample variance is 721.4

Example 2: (For population)

Find the variance of the population data 0,2,4,7,10,12,15.

Solution:

Step 1:

Step 2:

We write the formula of variance population

s²=S (x? - µ)²/N

Step 3:

Putting the value of step 2 and simplifying the variance population.

s²=25.85

Step 4:

Therefore, the value of population variance is 25.85

Conclusion

In this article, we have discussed the basic definition of Variance, the formula of Sample Variance and population Variance, and the procedure to evaluate population Variance in detail. Moreover, discussed the example of population and sample Variance explained. To better understand the concept of Variance solved different examples using the variance.