Letter σ. The standard deviation is represented in the Greek letter σ (sigma). This term refers to the amount of variability in a given set of data.
The population standard deviation, denoted by σ, is a measure of how spread out the values of a population are from the population mean. It is calculated using the following formula:
σ = √(∑(x_i - μ)^2 / N)
where:
- σ is the population standard deviation
- x_i is each value in the population
- μ is the population mean
- N is the number of values in the population
To calculate the population standard deviation, you would first need to calculate the population mean. Then, you would subtract the mean from each value in the population to get the deviations. Next, you would square each deviation to make it positive. Finally, you would add up all of the squared deviations and divide by the number of values in the population. The square root of this result is the population standard deviation.
For example, let’s say we have the following population of heights:
150, 155, 160, 165, 170
To calculate the population standard deviation, we would first need to calculate the population mean. The population mean is calculated by adding up all of the values in the population and dividing by the number of values. In this case, the population mean is 160.
Next, we would subtract the mean from each value in the population to get the deviations. The deviations are:
-10, -5, 0, 5, 10
Now, we would square each deviation to make it positive:
100, 25, 0, 25, 100
Next, we would add up all of the squared deviations:
250
Finally, we would divide the sum of the squared deviations by the number of values in the population and take the square root to get the population standard deviation. The population standard deviation is 10.
A population standard deviation of 10 means that the average height in the population is 160, and that the typical height is within 10 units of the mean.
It is important to note that the population standard deviation is a population parameter, meaning that it is a measure of the variability of the entire population. In practice, it is often difficult or impossible to calculate the population standard deviation, because we do not have data for the entire population. Instead, we often estimate the population standard deviation by calculating the sample standard deviation.
