What is the variance of a sample?

Hello there! I'm an expert in statistical analysis and I'd be delighted to explain the concept of variance in detail for you. Variance is a statistical measure that gives us an idea of how much the values in a set of data vary from the mean (average) value. It's a crucial concept in statistics because it helps us understand the spread or dispersion of the data. In other words, variance tells us how much the individual data points deviate from the average. When we talk about the variance of a sample, we're referring to a subset of a larger population. The formula for calculating the sample variance is slightly different from that of the population variance to account for the fact that we're not looking at the entire population. Here's how you calculate the sample variance, step by step: 1. Find the Mean: The first step is to calculate the mean (average) of the sample data. You do this by adding up all the numbers in the sample and then dividing by the number of values in the sample. \[ \text{Mean} = \frac{\sum_{i=1}^{n} x_i}{n} \] where \( x_i \) represents each value in the sample, and \( n \) is the number of values in the sample. 2. Find the Differences: Next, for each number in the sample, you find the difference between that number and the mean you just calculated. \[ \text{Difference} = x_i - \text{Mean} \] 3. Square the Differences: To ensure that all the differences are positive (since squaring a negative number will give a positive result), you square each of these differences. \[ \text{Squared Difference} = (x_i - \text{Mean})^2 \] 4. Sum the Squared Differences: Add up all the squared differences to get the total sum of squared differences. \[ \text{Sum of Squared Differences} = \sum_{i=1}^{n} (x_i - \text{Mean})^2 \] 5. Divide by (n-1): This step is unique to sample variance and is known as Bessel's correction. It adjusts for the fact that we're using a sample rather than the entire population. We divide the sum of squared differences by \( n - 1 \) (the number of degrees of freedom), rather than \( n \), to get the unbiased estimate of the population variance from our sample. \[ \text{Sample Variance} = \frac{\sum_{i=1}^{n} (x_i - \text{Mean})^2}{n - 1} \] 6. Interpretation: A high variance indicates that the data points are spread out over a large range of values, while a low variance indicates that the data points are closer to the mean. It's important to note that variance is measured in the squared units of the data. If you want a measure that's in the same units as the data, you would take the square root of the variance, which gives you the standard deviation. Now, let's move on to the translation.

Definition of Sample Variance. The variance is mathematically defined as the average of the squared differences from the mean. ... Step 2: Subtract the mean and square the result. Step 3: Work out the average of those differences.Mar 22, 2018