Variance Formula, Sample Variance, Population Variance

Variance Formula: Prior to discussing the variance formula, it's essential to revisit the concept of variance. Variance (σ2) represents the squared deviation of values (Xi) of a random variable (X) from its mean (μ). This formula enables the quantification of the extent of this distribution from the random variable's mean. It's crucial to note that the variance formula differs for a population and a sample. Below, we'll explore the variance formulas for both cases.

What is the Variance Formula?

Distinct variance formulas exist for ungrouped and grouped data. The specific variance formulas for each are detailed below.

For ungrouped data, variance can be written as:

Population Variance for population of size N = Σ ( X _i − ¯ X ) / 2 N

Sample Variance for sample of size N = Σ ( X _i − ¯ X ) 2 / N − 1

For grouped data, variance can be written as:

Population Variance, for population of size N = Σ f ( M _i − ¯ X ) 2 / N

Sample Variance, for sample of size N = Σ f ( M _i − ¯ X ) 2 / N − 1

where,

¯ X is the mean

M i is the mid-point of the ith interval.

Note: For ungrouped data, ¯ X = Σ x _i N

For grouped data, ¯ X = Σ M _i f Σ f

Variance Formula Solved Example

Example 1: Determine the variance of the given data set using the variance formula: 24, 53, 53, 36, 21, 84, 64, 34, 77, 54

Solution:

Population Size (N) = 10

The calculated variance for the provided data is 408.4 square units.

Example 2: Determine the mean and variance of the following dataset.

Solution:

Assigning d = (M _i - Midpoint)/10, this division by 10 aids in simplification. (where h = 10)

Mean = Σ M _i f / Σ f = 2060 38 = 54.21 units

Using the variance formula, Variance =

[ Σ f d ² − ( Σ ( f d ) ² / n ) ]* h ² / n =

[191 − ( ( − 3 ) 2 / 38 ) ] ) *10 ² / 38 =

( 191 − 0.2368)*100 / 38

The calculated mean for this data is 54.21 units, and the variance of this dataset equals 502 square units.

Understanding the variance formula is pivotal in statistical analysis, serving as a measure of dispersion within a dataset. Variance, denoted by σ^2, showcases the extent of deviation of individual data points from the dataset's mean, crucial for comprehending data distribution.

The variance formula differs for both ungrouped and grouped data, tailored for population and sample scenarios. For ungrouped data, the population and sample variances use distinct formulas. Similarly, for grouped data, separate formulas cater to population and sample calculations, with consideration for midpoints and frequencies within intervals.

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