Hi,
My professor gave us the formulas for sample variance and population variance, as well as the shortcut formulas for each.
However, he said that on some of the problems, we will HAVE to use the long formula, which for population variance would be: E(x-mean)^2/N
Can someone give me an example of a problem in which I would be unable to use the shortcut formulas so that I will know when it's not appropriate to use it?
Sure! Let's consider a scenario where you have a population of 10 individuals, and you know their ages. You want to calculate the population variance based on these ages. Here's how to determine whether you can use the shortcut formula or not:
1. Shortcut formula for population variance: This formula assumes that you have the complete data set of the population. It simply requires you to calculate the mean of the population (let's call it μ) and then compute the sum of the squared differences between each data point and the mean. The formula is:
Population Variance = Σ(x - μ)^2 / N
where Σ represents the sum, x represents each data point, μ represents the mean, and N is the size of the population.
2. Long formula for population variance: The long formula does not require you to have the entire data set; instead, it allows you to calculate the variance based on the available data and the mean. The formula is:
Population Variance = Σ(x - μ)^2 / N
where Σ represents the sum, x represents each available data point, μ represents the mean, and N is the size of the available data.
In this scenario, if you have the ages of all 10 individuals in the population, you can use the shortcut formula because you have the complete data set (i.e., N = 10). However, if you only have the ages of a subset of individuals (let's say 5 out of 10), you won't be able to use the shortcut formula because you don't have the complete data set. Instead, you would need to use the long formula and calculate the variance based on the available data.
So, to summarize, if you have the complete data set for the population, you can use the shortcut formula for population variance. If you only have a subset of the data, you need to use the long formula.