What is a Standard Error Formula?

The standard error is the error that arises in the sampling distribution while performing statistical analysis. It is a standard deviation variant as both concepts correspond to the spread measures. A high standard error corresponds to the higher spreading of data for the undertaken sample. Calculating the standard error formula is done for a sample. At the same time, the standard deviation determines the population.

Therefore, a standard error on mean one would express and determine as per the relationship described as follows: –

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Here,

  • The standard error expressed as σ͞x.The standard deviation of the population is expressed as σ.The number of variables in the sample expressed as n.

In statistical analysis, mean, median, and mode are central tendencyCentral TendencyCentral Tendency is a statistical measure that displays the centre point of the entire Data Distribution & you can find it using 3 different measures, i.e., Mean, Median, & Mode.read more measures. The standard deviation, variance, and standard error on mean classifies as the variability measures. The standard error on mean for sample data is directly related to the standard deviation of the larger population and inversely proportional or related to the square rootSquare RootThe Square Root function is an arithmetic function built into Excel that is used to determine the square root of a given number. To use this function, type the term =SQRT and hit the tab key, which will bring up the SQRT function. Moreover, this function accepts a single argument.read more of several variables taken up for making a sample. Hence, if the sample sizeSample SizeThe sample size formula depicts the relevant population range on which an experiment or survey is conducted. It is measured using the population size, the critical value of normal distribution at the required confidence level, sample proportion and margin of error.read more is small, then there could be an equal probability that the standard error would also be large.

Explanation

Example of Standard error formula

Below are the formula examples for calculating standard error.

  • Identify and organize the sample and determine the number of variables. Next, the average means of the sample corresponds to the number of variables present in the sample. Next, determine the standard deviation of the sample. Next, determine the square root of the number of variables taken up in the sample. Now, divide the standard deviation computed in step 3 by the resulting value in step 4 to arrive at the standard error.

Example #1

Let us take the example of stock ABC. For 30 years, the stock delivered a mean dollar return of $45. In addition, one observed that the stock delivered returns with a standard deviation of $2. Help the investor to calculate the overall standard error on the mean returns offered by the stock ABC.

Solution:

  • Standard Deviation (σ) = $2Number of Years (n) = 30Mean Dollar Return = $45

The calculation of standard error is as follows:

  • σ͞x = σ/√n= $2/√30= $2/ 5.4773

The standard error is,

  • σ͞x =$0.3651

Therefore, the investment offers a dollar standard error on the mean of $0.36515 to the investor when holding the stock ABC position for 30 years. However, if the stock held for a higher investment horizon, then the standard error on the dollar means would reduce significantly.

Example #2

Let us take the example of an investor who has received the following returns on stock XYZ: –

Help the investor calculate the overall standard error on the mean returns offered by the stock XYZ.

First, determine the average mean of the returns as displayed below: –

  • ͞X = (x1+x2+x3+x4)/number of years= (20+25+5+10)/4=15%

Now, determine the standard deviation of the returns as displayed below: –

  • σ = √ ((x1-͞X)2 + (x2-͞X)2 + (x3-͞X)2 + (x4-͞X)2) / √ (number of years -1)= √ ((20-15) 2 + (25-15) 2 + (5-15) 2 + (10-15) 2) / √ (4-1)= (√ (5) 2 + (10) 2 + (-10) 2 + (-5) 2 ) / √ (3)= (√25+100+100+25)/ √ (3)=√250 /√ 3=√83.3333= 9.1287%

Now, the calculation of standard error is as follows,

  • σ͞x = σ/√n= 9.128709/√4= 9.128709/ 2

The standard Error is,

  • σ͞x = 4.56%

Therefore, the investment offers a dollar standard error on the mean of 4.56% to the investor when holding the stock XYZ position for 4 years.

Standard Error Calculator

You can use the following calculator.

Relevance and Use

The standard error tends to be high if the sample size for the analysis is small. Therefore, a sample is always taken from a larger population, which comprises a larger size of variables. It always helps the statistician determine the sample mean’s credibility concerning the population mean.

A large standard error tells the statistician that the sample is not uniform concerning the population mean. There is a large variation in the sample concerning the population. Similarly, a small standard error tells the statistician that the sample is uniform concerning the population mean. There is no or small variation in the sample concerning the population.

One should not mix it with the standard deviation. Instead, one should calculate the standard deviation for the entire population. The standard errorStandard ErrorStandard Error (SE) is a metric that measures the accuracy of a sample distribution that signifies a population by using standard deviation. In other words, it is a measure to the dispersion of a sample mean concerned with the population mean and is not standard deviation.read more, on the other hand, is determined for the sample mean.

Standard Error Formula in Excel

Now, let us take the excel example to illustrate the concept of the standard error formula in the Excel template below. Suppose the school administration wants to determine the standard error on mean on the height of the football players.

The sample comprises of following values: –

Help the administration assess standard error on mean.

Step 1: Determine the mean as displayed below: –

Step 2: Determine the standard deviation as displayed below: –

Step 3: Determine the standard error on mean as displayed below: –

Therefore, the standard error on the mean for football players is 1.846 inches. The management should observe that it is significantly large. Therefore, the sample data taken up for the analysis is not uniform and displays a large variance.

The management should either omit smaller players or add significantly taller players to balance the football team’s average height by replacing them with individuals with smaller heights compared to their peers.

This article has been a guide to Standard Error Formula. Here, we discuss the formula for calculating the mean, standard error, examples, and downloadable Excel sheet. You can learn more from the following articles: –

  • EBITDA Margin FormulaGross Margin FormulaFormula of Relative Standard DeviationFormula of Margin of Error