Variance and Standard Deviation

 

Variance and Standard Deviation

Variance and standard deviation are statistical measures that help us understand the spread or dispersion of a set of data points. They provide insights into how closely the data points are clustered around the mean or average value.

Variance

 Variance measures the average of the squared differences between each data point and the mean of the data set. It gives us an idea of how much the individual data points deviate from the mean. A higher variance indicates a greater spread or dispersion of the data points, while a lower variance suggests they are more closely clustered around the mean.

For ungrouped data, the formula for variance is as follows: Variance = Σ((x - μ)²) / N

Where:

• Σ represents the sum of the values.
• x is each individual data point.
• μ (mu) is the mean of the data set.
• N is the total number of data points.

Standard Deviation

 Standard deviation is the square root of the variance. It provides a measure of the dispersion in the same units as the original data set. Standard deviation is a widely used measure of variability and is often preferred over variance because it is on the same scale as the data.

The formula for standard deviation, for ungrouped data, is as follows: Standard Deviation = √(Σ((x - μ)²) / N)

Now let's take an example to demonstrate how to calculate the variance and standard deviation for a set of ungrouped data:

Consider the following data set: {12, 15, 18, 20, 22}

Step 1: Find the mean (μ) of the data set. μ = (12 + 15 + 18 + 20 + 22) / 5 = 17.4

Step 2: Calculate the squared differences between each data point and the mean.

(12 - 17.4)² = 31.36

(15 - 17.4)² = 5.76

(18 - 17.4)² = 0.36

(20 - 17.4)² = 6.76

(22 - 17.4)² = 21.16

Step 3: Sum up the squared differences. Σ((x - μ)²) = 31.36 + 5.76 + 0.36 + 6.76 + 21.16 = 65.4

 Step 4: Calculate the variance. Variance = 65.4 / 5 = 13.08

Step 5: Calculate the standard deviation. Standard Deviation = √13.08 ≈ 3.62

Therefore, for the given data set, the variance is approximately 13.08, and the standard deviation is approximately 3.62