Hello there, I'm a data science enthusiast with a strong background in statistics. I'm here to help you understand the concept of variance and how to calculate it. Variance is a crucial measure in statistics that describes the spread of a set of numbers or how much the numbers deviate from the mean (average) value. It's a key concept in understanding the variability and uncertainty in data sets. Let's dive into the variance formula and how to calculate it step by step: ### Variance Formula The variance (denoted as \( \sigma^2 \) for a population and \( s^2 \) for a sample) is defined as the average of the squared differences from the mean. It's a measure of how much each number in the set differs from the mean and thus from every other number in the set. ### Steps to Calculate Variance 1. Calculate the Mean: The first step is to find the mean (average) of the data set. The mean is calculated by adding up all the numbers in the data set and then dividing by the count of numbers. \[ \text{Mean} (\bar{x}) = \frac{\sum_{i=1}^{n} x_i}{n} \] where \( x_i \) represents each value in the data set, and \( n \) is the total number of values. 2. Find the Differences from the Mean: Next, for each number in the data set, subtract the mean and calculate the square of the result. This gives you the squared difference for each number. \[ \text{Squared Difference} = (x_i - \text{Mean})^2 \] 3. Sum the Squared Differences: Add up all the squared differences to get the total. \[ \text{Sum of Squared Differences} = \sum_{i=1}^{n} (x_i - \text{Mean})^2 \] 4. Calculate the Variance: For a sample, divide the sum of squared differences by the number of values minus one (\( n - 1 \)) to get the sample variance. For a population, you would divide by the number of values (\( n \)). \[ \text{Sample Variance} (s^2) = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1} \] \[ \text{Population Variance} (\sigma^2) = \frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n} \] where \( \mu \) is the population mean. ### Interpretation of Variance - A high variance indicates that the data points are spread out over a large range of values. - A low variance indicates that the data points are closer to the mean and to each other. - Variance is always a non-negative number because the squaring of differences ensures that all results are positive. ### Properties of Variance - Scale-Dependent: Variance is sensitive to the scale of measurement. If all measurements in the data set are multiplied by a constant, the variance will be multiplied by the square of that constant. - Additivity: The variance of the sum of independent random variables is the sum of their variances. This is known as the additivity property. ### Example Let's consider an example to illustrate the calculation: Suppose we have a sample of test scores: 2, 3, 7, 9, 13. 1. Calculate the Mean: \( \bar{x} = \frac{2 + 3 + 7 + 9 + 13}{5} = \frac{34}{5} = 6.8 \) 2. Find the Squared Differences: - \( (2 - 6.8)^2 = 16.24 \) - \( (3 - 6.8)^2 = 7.84 \) - \( (7 - 6.8)^2 = 0.04 \) - \( (9 - 6.8)^2 = 4.84 \) - \( (13 - 6.8)^2 = 36.4 \) 3. Sum the Squared Differences: \( 16.24 + 7.84 + 0.04 + 4.84 + 36.4 = 65.32 \) 4. Calculate the Sample Variance: \( s^2 = \frac{65.32}{5 - 1} = \frac{65.32}{4} = 16.33 \) So, the variance of this sample is 16.33. Understanding variance is fundamental to many statistical analyses and is often used in conjunction with other measures like standard deviation to describe the distribution of data. It's also a key component in more complex statistical models and hypothesis testing. Now, let's move on to the next step. read more >>

The Variance is defined as: The average of the squared differences from the Mean. To calculate the variance follow these steps: Work out the Mean (the simple average of the numbers) Then for each number: subtract the Mean and square the result (the squared difference).read more >>