What is the difference between the population standard deviation and the standard error of the mean?

When it comes to statistics, it's important to be able to distinguish between different types of data. In particular, you need to be able to understand the difference between standard deviation and standard error. Standard deviation is a measure of how spread out data is, while standard error is a measure of how accurately a sample represents the population. Here's a closer look at the difference between standard deviation and standard error.

What is Standard Deviation?

Standard deviation is a measure of how spread out data is. It is calculated by taking the square root of the variance. The variance is calculated by taking the sum of the squared differences between each data point and the mean, and then dividing by the number of data points. The standard deviation is represented by the Greek letter sigma ().

What is Standard Error?

Standard error is a measure of how accurately a sample represents the population. It is calculated by taking the square root of the variance of the sample. The variance of the sample is calculated by taking the sum of the squared differences between each data point and the mean of the sample, and then dividing by the number of data points in the sample. The standard error is represented by the Greek letter epsilon ().

How to Calculate Standard Deviation

To calculate the standard deviation of a population, you need to know the mean and the variance. The mean is simply the sum of all the data points divided by the number of data points. The variance is calculated by taking the sum of the squared differences between each data point and the mean, and then dividing by the number of data points. The standard deviation is then calculated by taking the square root of the variance.

How to Calculate Standard Error

To calculate the standard error of a sample, you need to know the mean and the variance of the sample. The mean is simply the sum of all the data points in the sample divided by the number of data points in the sample. The variance of the sample is calculated by taking the sum of the squared differences between each data point and the mean of the sample, and then dividing by the number of data points in the sample. The standard error is then calculated by taking the square root of the variance of the sample.

What is the Difference Between Standard Deviation and Standard Error?

The main difference between standard deviation and standard error is that standard deviation is a measure of how spread out data is, while standard error is a measure of how accurately a sample represents the population. Standard deviation is calculated by taking the square root of the variance, while standard error is calculated by taking the square root of the variance of the sample.

Have you ever wondered what the difference is between standard deviation and standard error?

• Standard Deviation: This measures the variability of the data in relation to the mean. The closer it is to zero, the closer to the mean the values are in the data set.
• Standard Error: This measures the precision of the estimate of the sample mean. 

If you haven’t, here’s why you should care. 

What IsStandard Deviation?

Standard deviation measures the dispersion —variability — of the data in relation to the mean. In other words, the closer to zero the standard deviation is, the closer to the mean the values are in the studied data set. The standard distribution gives us valuable information in terms of the percentage of data within one, two and three standard deviations from the mean.

Let’s use R to generate some random data:

#generating some random data
set.seed(20151204)
#computing the standard deviation
x<-rnorm(10)
sd(x)
# result: 1.14415

Now, let’s generate a normally distributed graph:

#generating the normally distributed graph with description of segments
plot(seq(-3.2,3.2,length=50),dnorm(seq(-3,3,length=50),0,1),type="l",xlab="",ylab="",ylim=c(0,0.5))
segments(x0 = c(-3,3),y0 = c(-1,-1),x1 = c(-3,3),y1=c(1,1))
text(x=0,y=0.45,labels = expression("99.7% of the data within 3" ~ sigma))
arrows(x0=c(-2,2),y0=c(0.45,0.45),x1=c(-3,3),y1=c(0.45,0.45))
segments(x0 = c(-2,2),y0 = c(-1,-1),x1 = c(-2,2),y1=c(0.4,0.4))
text(x=0,y=0.3,labels = expression("95% of the data within 2" ~ sigma))
arrows(x0=c(-1.5,1.5),y0=c(0.3,0.3),x1=c(-2,2),y1=c(0.3,0.3))
segments(x0 = c(-1,1),y0 = c(-1,-1),x1 = c(-1,1),y1=c(0.25,0.25))
text(x=0,y=0.15,labels = expression("68% of the data within 1" * sigma),cex=0.9)

When we calculate the mean of a particular sample, we’re not interested in the mean of that sample. Instead, we want to draw conclusions about the population from which the sample comes. We usually collect representative sample data because we’re limited in terms of resources for collecting information about the whole population. So, we’ll use it as an estimate of the whole population mean.

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What Is Standard Error?

Of course, there will be different means for different samples from the same population This is called the sampling distribution of the mean. You can use the standard deviation of the sampling distribution to estimate the variance between the means of different samples. This is the standard error of the estimate of the mean.This is where everybody gets confused. The standard error is a type of standard deviation for the distribution of the means.

In short, standard error measures the precision of the estimate of the sample mean.

The standard error is strictly dependent on the sample size. As a result, the standard error falls as the sample size increases. If you think about it, the bigger the sample, the closer the sample mean is to the population mean, and thus, the closer the estimate is to the actual value.

R code for computing standard error below:

#computation of the standard error of the mean
sem<-sd(x)/sqrt(length(x))

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Standard Deviation vs. Standard Error

If you need to draw conclusions about the spread and variability of the data, use standard deviation.

If you’re interested in finding how precise the sample mean is or you’re testing the differences between two means, then standard error is your metric.

What is the difference between standard deviation of means and standard error of means?

Is the standard error the same as population standard deviation?

What is the difference between standard deviation and population standard deviation?

What is the difference between standard error and standard error of mean?