Standard Error

Standard Error Definition

Standard Error or SE is used to measure the accurateness with the help of a sample distribution that signifies a population taking standard deviation into use, or in other words, it can be understood as a measure with respect to the dispersion of a sample mean concerned with the population mean. It not be confused with standard deviation. This is higher because of the fact that standard errors use sample data or statistics while standard deviations use parameters or population data.

Standard Error Formula

It is represented as below –

Standard Error Formula

Here, “σM ” represents the S.E. of the mean, which is also the S.D. (standard deviation) of the sample data of the mean, “N” represents the sample size while “σ” signifies the S.D. of the original distribution. S.E formula will not assume N.D. (normal distribution). However, few uses of the formula do assume a normal distribution. This equation for standard error signifies that the size of the sample will have an inverse effect on the S.D. of the mean, i.e., the larger the size of the sample mean, the smaller shall be the S.E. of the same and vice-versa. This is why the size of the S.E. of the mean is shown as inversely proportional to the square root of N (sample size).

Standard-Error

Steps to Find Standard Error

  • In the first step, the mean must be calculated by summing all the samples and then dividing them by the total number of samples.
  • In the second step, the deviation for each measurement must be calculated from the mean, i.e., subtracting the individual measurement.
  • In the third step, one must square every single deviation from the mean. In this way, squared negatives will become positive.
  • In the fourth step, the squared deviations must be summed up, and for this purpose, all the numbers obtained from Step 3 must be added up.
  • In the fifth step, the sum obtained from the fourth step must be divided by one digit less than the sample size.
  • In the sixth step, the square root of the number obtained in the fifth step must be taken. The result shall be S.D. or standard deviation.
  • In the second last step, a
  • S.E needs to be calculated by dividing the standard deviation by the square root of the N (sample size).
  • In the last step, the S.E. from the mean must be subtracted, and accordingly, that number must be recorded. The S.E. must be added to the mean, and the result must be recorded.

Examples of Standard Error

Below are examples of standard errors.

You can download this Standard Error Excel Template here – Standard Error Excel Template

Example #1

Cancer mortality in a sample of 100 is 20 percent, and in the second sample of 100 is 30 percent. Evaluate the significance of the contrast in the mortality rate.

Solution

Use the below-given data.

Example 1

Standard Error Example 1.2

  • =SQRT(20*80/(100)+(30*70/(100)))
  • =6.08

Example 1.3

  • Z= 20-30/6.08
  • Z= -1.64

Example #2

A random sample of 5 male basketball players is chosen. Their heights are 175, 170, 177, 183, and 169 (in cm). Find the S.E. of the mean of this height (in cm) measurements.

Solution

  • = (175+170+177+183+169)/5
  • Sample Mean = 174.8

Calculation of Sample Standard Deviation

Standard Error Example 2

  • =SQRT(128.80)
  • Sample Standard Deviation =5.67450438

Example 2.1

  • =5.67450438/SQRT(5)
  •  = 2.538

Example #3

The mean profit earning for a sample of 41 businesses is 19, and the S.D. of the customers is 6.6. Find the S.E. of the mean.

Solution

Use the below-given data.

Standard Error Example 1

Calculation of Standard Error

Example 1.1

  • = 6.6/SQRT(41)
  •  = 1.03

Interpretation of Standard Error

Standard error functions very similar to descriptive statistics as it permits the researcher to develop confidence intervals with respect to the sample statistics that are already obtained. This helps in estimating the intervals in which the parameters are supposed to fall. S.E. of the mean and S.E. of the estimate are the two commonly used S.E. statistics.

The S.E. of the mean allows the researcher to develop a confidence interval in which the population means will fall. 1-P is used as the formula that signifies the probability for the population mean that will fall in the confidence interval.

The S.E. of the estimate is mostly taken into use by various researchers, and it is used along with the correlation measure. It allows the researchers to construct a confidence interval underneath the actual population correlation that shall fall. The S.E. of the estimate is used for determining the preciseness of an estimate with respect to population correlation.

S.E. is helpful in indicating the preciseness of an estimate of population parameters the sample statistics actually are.

Difference Between Standard Error And Standard Deviation

Standard error and standard deviation are two different topics, and these must not be confused with one other. The short form for standard error is S.E. while the abbreviation for standard deviation is S.D. S.E. of a sample mean truly an estimate of the distance of the sample mean from the population mean, and it helps in gauging the accurateness of an estimate while S.D. measures the amount of dispersion or variability and it is generally the extent to which individuals belonging to the same sample differs from the sample mean.

Conclusion

Standard Error is the measure of the accuracy of a mean and an estimate. It offers a useful way for the quantification of a sampling error. S.E. is useful since it represents the total amount of sampling errors that are associated with the sampling processes. The standard error of the estimate and standard error of the mean are two commonly used SE statistics.

The standard error of the estimate allows in making predictions but doesn’t really indicate the accurateness of the prediction. It measures the precision of the regression, whereas the Standard error of the mean helps the researcher in developing a confidence interval in which the population mean is most likely to fall. SEM can also be understood as the statistic or parameter of the mean.

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