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Standard deviation and standard error of the mean are important topics in statistical studies of various subjects such as finance, engineering, medicine, etc. The standard error of the mean is a method used to determine the standard deviation of a sampling distribution provided for a population. It measures the extent by which the sample mean or average differs from the true population mean. The standard error of the mean gives an accurate idea of the sample mean by analyzing the various samples from the sampling distribution.

Let's say for a particular instance, we pick a sample space from a population distribution and the estimated mean or sample mean is denoted by x. We pick another sample from the same population distribution and we find that the estimated mean or the sample mean may have a different value like y and not x. Hence, different samples from the same population distribution may end up giving us different variance and mean. Here the standard error of mean gives us the standard deviation of the sample means collected from all possible samples from the sampling distribution. The standard error of mean is sometimes abbreviated as SEM.

Now let us learn how to calculate the standard error of the mean.

Standard deviation represents the dispersion or variation in the set of values for a given data.

Standard deviation \[\sigma\] = \[\sqrt{\frac{\sum_{i=1}^{n}(x_{i} - \bar{x})^{2}}{(n-1)}}\]

Variance = \[\sigma\]\[^{2}\]

The standard error of the mean \[\sigma\]\[_{\bar{x}}\] = \[\frac{\sigma}{\sqrt{n}}\]

(where \[\bar{x}\] = the sample mean and n=the sample size)

Hence, the standard error of mean mathematically is the ratio of the standard deviation to the square root of the sample size.

The standard error of the mean (SEM)=Standard deviation \[\frac{SD}{\sqrt{samplesize(n)}}\]

The steps required to find the standard deviation of a given data is-

The first step involves taking the square root of the difference between the data entries and sample means and further adding them up.

The second step involves calculating the variance by dividing the sum by one less than the sample size.

The final step involves taking the square root of the variance to arrive at the standard deviation.

After the calculation of standard deviation, apply the formula \[\sigma\]\[_{\bar{x}}\] = \[\frac{\sigma}{\sqrt{n}}\].

From the formula, it is evident that the value of the standard error of mean decreases with an increase in the sample size (n). This can be easily explained because if we calculate the sample at each stage consisting of several data entries, a small variation in one of the values will not explicitly affect the mean.

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Given, x=5,10,15,20,25.

Sample size (n) =5

Mean (x\[_{m}\]) = Sum of individual elements/Sample size

Mean = (5+10+15+20+25)/5

Mean = 75/5

Mean (x\[_{m}\]) = 15

We know that,

Standard deviation \[\sigma\] = \[\sqrt{\frac{\sum_{i=1}^{n}(x_{i} - \bar{x})^{2}}{(n-1)}}\]

The standard error of the mean \[\sigma\]\[_{\bar{x}}\] = \[\frac{\sigma}{\sqrt{n}}\]

\[\sigma\] = \[\sqrt{\frac{(x_{1} - x_{m})^{2} + (x_{2} - x_{m})^{2} + (x_{3} - x_{m})^{2} + (x_{4} - x_{m})^{2} + (x_{5} - x_{m})^{2}}{n-1}}\]

\[\sigma\] = \[\sqrt{\frac{(5-15)^{2} + (10 - 15)^{2} + (15 - 15)^{2} + (20 - 15)^{2} + (25 - 15)^{2} }{5-1}}\]

\[\sigma\] = \[\sqrt{\frac{100+25+0+25+100}{4}}\]

\[\sigma\] = \[\sqrt{\frac{250}{4}}\]

\[\sigma\] = 7.91

Now standard error of the mean \[\sigma\]\[_{\bar{x}}\] = \[\frac{7.91}{\sqrt{5}}\] = \[\frac{7.91}{2.24}\] = 3.53

Hence, the standard error of the mean for the given data is 3.53.

2. For a Given Sample: 20, 22, 24, 26, 28, Find Out the Standard Error of the Mean.

Given, x=20,22,24,26,28.

Sample size (n) =5

Mean(x\[_{m}\]) = \[\frac{\text{Sum of individual elements}}{\text{Sample Size}}\]

Mean(x\[_{m}\]) = \[\frac{20+22+24+26+28}{5}\]

Mean(x\[_{m}\]) = \[\frac{120}{5}\]

Mean(x\[_{m}\]) = 24

We know that,

Standard deviation \[\sigma\] = \[\sqrt{\frac{\sum_{i=1}^{n}(x_{i} - \bar{x})^{2}}{(n-1)}}\]

The standard error of the mean \[\sigma\]\[_{\bar{x}}\] = \[\frac{\sigma}{\sqrt{n}}\]

\[\sigma\] = \[\sqrt{\frac{(x_{1} - x_{m})^{2} + (x_{2} - x_{m})^{2} + (x_{3} - x_{m})^{2} + (x_{4} - x_{m})^{2} + (x_{5} - x_{m})^{2}}{n-1}}\]

\[\sigma\] = \[\sqrt{\frac{(20-24)^{2} + (22-24)^{2} + (24-24)^{2} + (26-24)^{2} + (28-24)^{2}}{5-1}}\]

\[\sigma\] = \[\sqrt{\frac{16+4+0+4+16}{4}}\]

\[\sigma\] = \[\sqrt{\frac{40}{4}}\]

\[\sigma\] = 3.16

Now standard error of the mean \[\sigma\]\[_{\bar{x}}\] = \[\frac{3.16}{\sqrt{5}}\] = \[\frac{3.16}{2.24}\] = 1.41

Hence, the standard error of the mean for the given data is 1.41.

FAQ (Frequently Asked Questions)

1. Why Do We Need Sampling Distribution in Case of the Standard Error of Mean?

To understand the importance of sampling distribution, we need to take up an example. Let us consider a Physics experiment where the aim is to calculate the speed of sound through various materials. If we consider the speeds along the three-dimensional coordinate axes and take the mean of the values, we will get an average speed of sound in the particular medium. However, there are numerous kinds of errors that may creep in while taking the measurements. External factors like small deviation in temperature, pressure changes, wind speed variations, and reaction time of stopwatch may affect the calculation of speed of sound in the material. Here is when we may need the sampling distribution so that we may take several measurements and take the mean each time. The error encountered in the calculation of such mean is compensated by the standard error of the mean.

2. What is the Significance of the Standard Error of Mean and What Can We Infer from the Values of the Standard Error of Mean?

In simpler words, the standard error of mean implies the variation in the mean related to different samples taken from the same sampling distribution. The standard error of mean defines the accuracy with which the sample distribution represents the population with the help of standard deviation. If the effect of random errors and external factors is significant, the standard error will have a greater value. The standard error of mean may turn out to be zero if there is absolutely no change in the data points of the samples taken. The value of the standard error is small when the variation is less and the sample distribution is almost representative of the actual population. The standard error of mean diminishes when there are more sample points considered or the sample size increases.