The variance of a sample is the average amount by which the observations in a sample differ from each other. The standard deviation of a sample is a statistical measure that describes how far, on average, the observations in a sample differ from the sample’s mean.
The variance of a sample can be thought of as variability. Variability can be either positive or negative, meaning that some observations may be higher or lower than the mean observation by a certain amount.
Standard deviation is often misunderstood and misused. Many people think that the standard deviation of any population is 64, which is not true. The truth about the standard deviation is that it varies based on what you are measuring and how you are measuring it.
This article will explain what the standard deviation of any population really is and when it equals 64.
Another measure of dispersion is the sample variance. The sample variance is calculated by taking the squared difference between each observation and the mean, then dividing by how many observations there are.
For example, let’s say we have a dataset where the values are 2, 3, 4, 5, 6, 7. The mean is 4.5, so the difference between each observation and the mean is:
2 – 4.5 = –2.5
3 – 4.5 = 0.5
There are five observations here, so we divide 0.5 by 5 to get an average difference of 0 between each observation and the mean .
Calculating the sample variance
The second method of calculating the variance of a population is by calculating the variance of a sample. This is done by using what is called the sample variance.
The formula for the sample variance is:
Where: s = The sample mean, n = The number of observations (100 in this case), and ∑xi = The sum of all x values.
Just like in the case of the population mean, we can calculate the standard deviation of the sample mean using this formula. The only difference is that instead of having one value, we have several values that make up our sample mean.
Sample variances are more practical when working with samples rather than populations because we can more easily get access to samples than to populations. Therefore, this method is more practical than calculating the population variance due to lack of access to the population.
Standard deviation of a population
A population is the total number of items or events that you are studying. For example, if you were studying the height of all people, the population would be all people.
To calculate the standard deviation of a population, you first need to calculate the variance of a population. The variance of a population is similar to the variance of a sample, except you are using the total number of individuals in the population instead of a sample size of n individuals.
Variance is denoted by σ², where σ stands for sigma and 2 stands for two variables being averaged (x and y in this case).
Calculating the standard deviation of a population is then done by dividing the variance by n-1, where n is the number of individuals in the population. This gets rounded to 64 according to Wolfram|Alpha.
Variance of a population
A more complicated, yet just as important parameter is the variance of a population. The variance of a population is the average amount that all values in a population vary from the population’s mean.
Variance is typically represented by the Greek letter sigma, Σ. The standard deviation of a population is typically represented by σ. For example, the standard deviation of height for adults aged 18–25 is 1.76 meters. This means that half of all adults in this age range have a height between 1.75 and 2 meters, or 1.76 meters on average.
To calculate the variance of a population, you would have to calculate the variance of each value in the population, then take the average of those variances. Doing this gives you what is known as averaged variance or just variance for short.
Example of calculating the sample variance
The formula for the sample variance is s = √n(x-mean)2 where n is the number of observations, x is the value of each observation, and mean is the average observation.
To understand this formula, consider a sample of 100 observations with an average value of 64. To calculate the sample variance, you would first need to find the mean observation by averaging all 100 observations.
Then, find the difference between each observation and the mean observation and square each difference. Add these squared differences together and then take the square root of that total to find the sample variance.
This process can be confusing so let’s look at an example. Consider 5 observations with values of 65, 66, 67, 68, and 69. The average of these observations is 66 so we will use that as our mean value. Next we need to find the difference between each observation and the mean observation.
Why is the sample standard deviation different from the population standard deviation?
The standard deviation of a population is the average amount that all values in the population vary from the population average.
So if we say that the average height of people in America is 5 feet 9 inches, then we are saying that 95% of people in America are between 5 feet 9 inches and 6 feet 0 inches.
The difference between the average height of people in America and someone who is exactly 5 feet 9 inches is 0. The person is not shorter or taller than the average – they are exactly the same.
However, if we took a sample of 100 people in America, then our sample standard deviation would be different from the population standard deviation. This is because our sample would not include all people in America – it would only include 100 people.
The sample standard deviation accounts for variability among samples, while the population standard deviation accounts for variability among populations.
Why is the sample variance different from the population variance?
There is a difference between the sample variance and population variance because the sample is taken from a specific population.
As mentioned before, the population variance is the average of the squared differences of all pairs of values in a distribution. So, how can we find this average if we do not have an entire population to take samples from?
We can use our sample as a representation of the entire population. By using your sample data, you can calculate the variance of your population using the formula for variance.
What if my sample size is not 100?
If your sample size is not 100, you need to adjust the standard deviation formula. You will have to determine how much of an effect the smaller sample size will have on the standard deviation.
To do this, you first have to know what proportion of the total sample size your 100 observations represent. Then, you have to find out how much the standard deviation would change due to the lower number of observations.
The proportion of 100 observations relative to the total number of observations is called the coefficient of variance. It is usually around 1, but it can be any number between 0 and 1.
If the coefficient of variance is not 1, then you need to multiply the standard deviation formula with that number.