## Standard Error

**Standard Error**

A Standard Error is a component of the descriptive statistics that measure the accuracy of a population description using a sample distribution’s standard deviation. A sample mean will deviate from the real mean of a population; this deviation is said to be the mean standard error. Analysts often collect data from a small sample of the entire population when conducting the research. As a result, they will end up with slightly different sets of values, each time with slightly different means.

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__What is a Standard Error?__

The standard deviation of sample distribution is known as the standard error in statistics. It is a measure of statistical accuracy of estimates which equals the standard deviation of theoretical distribution. When we estimate the standard deviation of sample distribution from statistical data, the estimates are known as sample errors. If the sample is mean then it is called a sample error of the mean. It is a method of evaluation of standard deviation for sample distribution. By measuring deviation within the means, it serves as a way to test the sample accuracy or the accuracy of multiple samples.

__How it is calculated?__

The standard error is calculated on the basis of these 3 basic steps:

- Estimating the sample mean for the given data.
- Estimating the standard deviation for sample data.
- Dividing the sample standard deviation by the square root of the sample mean.

Thus, the formula for calculating is as follows:

Where,

SE = standard error

σ = standard deviation

n = no of sample

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__Examples:__

- The admin at Oxford University states that the mean point of grades of all students is 2.5 and the standard deviation is 0.25. Determine the standard error of mean if the sample size is 25.

= 0.25/ √25

= 0.25/5

Standard Error = 0.05

- Calculate standard error of mean if x= 10,15,20,25,30

Mean = (10+15+20+25+30)/5

= 100/5

= 20

SD = √(1/N-1) * ((x_{1 }– x _{m })^{2)}) + ((x_{2 }– x _{m})^{2}) + …+((x _{n} – x _{m)}^{2})

= √ (1/ 5-1) * ((10-20)^{2 }+ (15-20)^{2} + (20-20)^{2} + (25-20)^{2} + (30-20)^{2})

= √ (1/4) * ((-10)^{2} + (-5)^{2 }+ (0)^{2} + (5)^{2 }+ (10)^{2})

= √ (1/4) * (100 + 25 + 0 + 25+ 100)

= √ (1/4) * (250)

= √ 0.25 * 250

= √ 62.5

= 7.90

SE = σ / √n

= 7.90 /√5

= 7.90 / 2.23

SE = 3.54

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__Interpretation:__

The most commonly used SE is the standard error of mean and standard error of the estimate. It indicates the preciseness of an estimation. The SE of mean permits the researcher to construct a confidence interval in which the population means will likely fall. It also shows how reliable the mean of any given sample of the population is probably to be correlated with the mean of the true population. The smaller the standard error, the sample is more reflective of the population as a whole. The SE of the estimate is most commonly used with correlation measures. Taken together with such measures as effect size, p-value, and sample size, the effect size can be a useful tool to the researcher who seeks to understand the accuracy of statistics calculated on random samples.

__Author____:__ Urvi Surti

** About the Author: **Urvi is a commerce graduate and has a keen interest in Finance. She has completed her Chartered Wealth Management (CWM) from the American Academy of Financial Management and is currently pursuing a career in Financial Risk Management (FRM).

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