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Standard Error

The statistics is a wide subject in which we study about terms like data, sample and population, dependent and independent variables, mean, median, mode, standard deviation, variance and many many more. For standard deviation, one more term is used - called standard error

The standard error actually refers to the standard deviation of a statistic. The sample mean of a data is usually deviated from actual population mean. The amount of this deviation is termed as the standard error.
Standard error is used to measure the amount of accuracy by which the given sample represents its population.

For the sample to be more representative over the given population, the standard error must be smaller. Also, the standard error and sample size are inversely proportional to each other. It means that the smaller the sample size, the bigger the standard error and the bigger the sample size, the smaller the standard error.

Let us go ahead in this article and learn details about standard error as well as standard error of the mean, methods of their computations with solved examples.

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Standard Error of the Mean

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The term "standard erroris also utilized for the estimate of the standard deviation which is derived from a given sampleStandard error of the mean is defined as the standard deviation of the estimate of sample mean of the mean of population.

For Instance: Usually, the estimator of population mean is sample mean is the usual estimator of a population mean. But if we draw another sample from the same population, it may provide different value. 

Thus, there would be a 
distribution of the sampled means having its different variance and mean. Therefore, standard error of the mean (abbreviated as SEM) may be defined as the standard deviation of such sample means of all the possible samples taken from the same given population. The standard error of mean defines an estimate of standard deviation which has been computed from the sample.

Standard error of mean is calculated by dividing sample standard deviation by the square root of sample size; i.e. the formula for SEM is given by -
$SEM$ = $\frac{s}{\sqrt{n}}$
In which,
SEM stands for standard error of the mean,
s represents sample standard deviation

n denotes the sample size.

This value estimated by above formula is sometimes compared with true standard deviation of sample mean given by the following formula -
$SE$ = $\frac{\sigma}{\sqrt{n}}$
Where, $\sigma$ is population standard deviation.

Standard Error of the Estimate

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There is one more concept related to standard error is "standard error of the estimate". This term refers to deviation of some estimate from intended values.
The formula for the calculation of standard error of the estimate of a sample is written below:
$SEE$ = $\sqrt{\frac{\sum (x_{i} - \bar{x})^{2}}{n-2}}$
In this formula, SEE stands for the standard error of the estimate, $x_{i}$ denotes data values, $\bar{x}$ is the mean where Y refers to individual data sets, Y' is mean of the given sample and N is the sample size.
The students can remember this formula very easily as it similar to the formula of standard deviation formula with a difference of n - 2 for n -1 in the formula for sample standard deviation.

Difference Between SE and SEM

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We usually get confused between the standard error and the standard error of mean (SEM). The differences between them are given below:

i) The standard error quantifies the scatter of the data. It is means how far the data values disperse from one another. While standard error of mean quantifies precisely the true value of mean of given population.

ii) Standard error is standard deviation. On the other hand, SEM takes standard deviation and sample size both in consideration.

iii)
 The value of standard error of the mean is smaller than of standard error.

iv) As the sample gets bigger, the value of SEM gets smaller. Thus, SEM is inversely proportional to size of the sample. But sample size does not impact of the standard error. It is not changed 
predictably as more data is available. It can be be predicted even if sample size is big or small.

How to Calculate Standard Error of the Mean

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The student are required to follow the steps mentioned below for the calculation of standard error of the mean:

Let us consider the example of an exam in which the scores of 5 students are given as 12, 74, 55, 79, 90.

Step 1:
 Find the mean of given data by dividing the sum of samples by total number of samples.

Mean of 12, 74, 55, 79, 90

$\bar{x}$ = $\frac{12 + 74 + 55 + 79 + 90}{5}$ = 62

Step 2: Construct a table for standard deviation of the given data. This table would be 3 columns - first for given data value, second for the deviation of each data from the mean and third one for square of these deviations, as shown in the example below.

$x_{i}$  $x_{i}-\bar{x}$ $(x_{i}-\bar{x})^{2}$ 
12  -50  2500
74  12  144
55  -7  49
79  17  289
90  28  784 
    $\sum (x_{i}-\bar{x})^{2}$ = 3766

Step 3: By using the formula:

$\sigma$ = $\sqrt{\frac{\sum (x_{i} - \bar{x})^{2}}{n}}$

calculate the standard deviation.

$\sigma$ = $\sqrt{\frac{3766}{5}}$ = 27.44

Step 4: Now, the standard deviation is to be divided by the square root of sample size.
That gives you the

“standard error”.

Step 5: Subtract the standard error from the mean and record that number. Then add the standard

error to the mean and record that number. You have plotted mean± 1 standard error (S. E.),

the distance from 1 standard error below the mean to 1 standard error above the mean

$SEM$ = $\frac{\sigma}{\sqrt{n}}$

$SEM$ = $\frac{27.44}{\sqrt{5}}$

= $\frac{27.44}{2.24}$

= 12.25
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