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A Comprehensive Guide to Variance: Definition, Formulas, and Calculations

By   /  March 25, 2024  /  Comments Off on A Comprehensive Guide to Variance: Definition, Formulas, and Calculations

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A Comprehensive Guide to Variance: Definition, Formulas, and Calculations

The concept of measures of dispersion such as variance of ungrouped data is used to show the data collecting and representation processes in statistical research. This variance in the data gathered from the study will help in a more thorough understanding of this idea.

We will guide you through the process of calculating variance for ungrouped data in this in-depth discussion. We will go into profound detail about their relationships with sample and population variance. To help with comprehension, we will also solve a few examples.

What is a Variance?

A variance is a measure of dispersion or variation that describes the level of variability among the values in a set of data. It’s used to determine the separation between each number in the collection and the mean.  Market security and volatility are terms that traders and analysts use to characterize variance. 

The variance is used to observe and comprehend the interpretation and representation of the data under examination. Variance can only ever be zero or positive. It can never be negative. 

The variance may be zero if the values of the data being considered are equal. Several measures can be used to measure the magnitude of the variance.

Types of Variance

Sample variance and population variance are the two types of variance.

Population Variance

Population variance is a type of variance that involves gathering data from every member of the population to ensure an accurate estimation. It is calculated to measure the variance across an entire population. The following formula can be used to calculate the population variance of ungrouped data:

σ2 = ∑ (yi – μ)2 / M

For ungrouped data, the population variance is represented by σ2. It is essential to remember that the relevant observation or value in the formula above is denoted by yi. The population variance for problems with the raw distribution data can be determined with this calculation.

Sample Variance

To assess the variability within a specific sample, we use the concept of sample variance. This measure not only quantifies the spread of sample data but also helps in estimating the population variance. The following formula can be used to calculate the sample variance of ungrouped data:

S2 = ∑ (yi – ȳ)2 / M – 1

The sample variance is denoted by S2. It is essential to keep in mind that in the mathematical formula above M represents the number of data values. Sample variance for problems involving the raw distribution data can be found with the help of this formula.

How to calculate variance?

Understanding the types of variance helps in calculating variance for sample and population data according to specific formulas. Here we’ll solve some examples to understand how to find variance manually. 

Example 1:

Determine what will be the value of the variance for the values given in the below table.

7 13 17 25 21 24 19 18

Solution: 

Step 1: First of all, we are to compute the average of the given data values in the above table.

x̅ = μ = (7 + 13 + 17 + 25 + 21 + 24 + 19 + 18) / 8

x̅ = μ = (144) /8

x̅ = μ = 18

Step 2: Now we will determine the below necessary calculations in the table to proceed further to the next for determining the variance.

x (y kȳ) (y k ȳ)2
7 -11 121
13 -5 25
17 -1 1
25 7 49
21 3 9
24 6 36
19 1 1
18 0 0
Total Σ (y – μ)2 = Σ (y – ȳ)2 = 242

Step 3: We will employ the relative formula corresponding to the calculations that we do in the table.

The formula for population variance

σ2 = ∑ (yi – μ)2 / M

Putting the relative values:

σ2 = (242) / 8 

σ2 = 30.25

The formula for sample variance:

S2 = ∑ (yi – ȳ)2 / M – 1

S2 = (242) / (8 – 1)

S2 = (242) / 7

S2 = 34.57

Example 2:

We have also another formula to determine the variance for the given data values. Let us explore that one as well.

Calculate what will be the variance and the standard deviation for the following given scores of the candidates in the table.

Candidates Saeed Maham Moeen Aham Siam Syed Fiaz Sami Umar
Score (yi) 81 69 87 53 34 96 45 22 29

Solution:

Step 1: Now we will perform the following necessary computations as given in the table:

yi 81 69 87 53 34 96 45 22 29 ∑y = 516
yi 2 6561 4761 7569 2809 1156 9216 2025 484 841 ∑y2 = 35422

Step 2: We will employ the relative formula corresponding to the calculations that we perform in the table.

The formula for variance

Sk2 = (∑ y2 / n) – (∑y / m)2

Putting the relative values:

Sk2 = (35422 / 9) – (516 / 9)2

Sk2 = 3935.78 – (266256 / 81)

Sk2 = 3935.78 – 3287.11

Sk2 = 648.67 scores2

Final Words

In this article, we have explored the concept of the variance accurately. We have explained the population variance and the sample variance with their formulae for ungrouped data values. In the example section, we have solved some examples that will help us to understand the variance in a better way.

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