# Standard Error

Standard Error is the measure of the variability of a sample statistic used to estimate the variability of a population. Standard Error is important in dealing with sample statistics, such as sample mean, sample proportion, etc. Sample Error Formula is used to determine the accuracy of a sample that reflects a population. The standard error formula is the discrepancy between the sample mean and the population mean.

In this article, we will learn about, Standard Error, Standard Error Formula, Standard Error of Mean, Standard Error of Estimate, related Examples, and Error in detail.

## What is Standard Error?

The term “sample” in statistics refers to a specific set of information that is generated. The data we obtained on the height of some people in a locality, for example, maybe the sample. A population is a collection of people from which we draw a sample. There are several ways to define a population, and we must always be clear about what constitutes a population. This collection necessitates a large number of calculations.

Standard error represents how well a given sample represents the population. The Standard Error indicates how well the sample mean predicts the real population mean.

## Standard Error(SE) Formula

The SE formula is used to determine the reliability of a sampling that represents a population. The sample mean that differs from the provided population and is expressed as:

SE = S/âˆš(n)

where,

• S is Standard Deviation of Data
• n is Number of Observations

## Standard Error of Mean (SEM)

Standard Error of Mean(SEM) is also known by the name Standard Deviation of Mean, is the standard deviation of the measure of sample mean of the population. Standard Error is used when we have to find the population mean from the sample of data and we take two samples to get two different mean in that case we use Standard Error Mean.

Standard Error Mean formula is the ratio of standard deviation to the root of sample size,

SEM = S/âˆš(n)

where,

• S is Standard Deviation
• n is Number of Observations

## Standard Error of Estimate (SEE)

Standard Error Estimate is use to find the accuracy of prediction of any event. Its abbreviation is SEE. Standard Error Estimate (SEE) is also called the Sum of Sqaures Error. SEE is the square root of average squared deviation.

Standard Error of Estimate(SEE) formula is discussed below,

SEE = âˆš[Î£(xi – Î¼)/(n – 2)]

where,

• xi is values of Data
• Î¼ is Mean Value of Data
• n is Sample Size

## How to Calculate Standard Error?

Various steps to calculate the standard error are added below,

Step 1: Find the number of measurement(n) and find the sample mean(Î¼).

Step 2: Find the variation of sample values from the mean value.

Step 3: Find the square of the deviation and find their sum. Î£(xi â€“ Î¼)Â²

Step 4: Divide the sum from step 3 by (n-1).

Step 5: Take the square root of the number, i.e. standard deviation (Ïƒ).

Step 6: Divide the standard deviation by the square root of measurement(n).

## Standard Error Solved Examples

Example 1: Find the standard error for the sample data: 1, 2, 3, 4, 5.

Solution:

Mean of Given Data = (1+2+3+4+5)/5

Î¼ = 15/5

Î¼ = 3

Standard Deviation = âˆš((1 – 3)2 + (2 – 3)2 + (3 – 3)2 + (4 – 3)2 + (5 – 3)2)/(5 – 1)

Ïƒ = âˆš((4 + 1 + 0 + 1 + 4)/4)

Ïƒ = âˆš(10/4)

Ïƒ = 1.5

SE = 1.5/âˆš5

SE = 0.67

Example 2: Find the standard error for the sample data: 2, 3, 4, 5, 6.

Solution:

Mean of Given Data = (2+3+4+5+6)/5

Î¼ = 20/5

Î¼ = 4

Standard Deviation(Ïƒ) = âˆš((2 – 4)2 + (3 – 4)2 + (4 – 4)2 + (5 – 4)2 + (6 – 4)2)/(5 – 1)

Ïƒ = âˆš((4 + 1 + 0 + 1 + 4)/4)

Ïƒ = âˆš(10/4)

Ïƒ = 1.58

SE = 1.58/âˆš5

SE = 0.706

Example 3: Find the standard error for the sample data: 10, 20, 30, 40, 45.

Solution:

Mean of the given data = (10+20+30+40+45)/5

Î¼ = 145/5

Î¼ = 29

Standard Deviation(Ïƒ) = âˆš((10 – 29)2 + (20 – 29)2 + (30 – 29)2 + (40 – 29)2 + (45 – 29)2)/(5 – 1)

Ïƒ = âˆš(820/4)

Ïƒ = 14.317

SE = 14.317/âˆš6

SE = 5.84

Example 4: Find the standard error for the sample data: 2, 6, 9, 5.

Solution:

Mean of Given Data = (2+6+9+5)/4

Î¼ = 5.5

Standard Deviation = âˆš((2 – 5.5)2 + (6 – 5.5)2 + (9 – 5.5)2 + (5 – 5.5)2)/(4 – 1)

Ïƒ = âˆš(25/3)

Ïƒ = 2.88

SE = 2.8/âˆš5.5

SE = 1.19

Example 5: Find the standard error for the sample data: 5, 8, 10, 12.

Solution:

Mean of Given Data(Î¼) = (5+8+10+12)/4

Î¼ = 8.75

Standard Deviation = âˆš((5 – 8.75)2 + (8 – 8.75)2 + (10 – 8.75)2 + (12 – 8.75)2)/(4 – 1)

Ïƒ = âˆš(26.75/3)

Ïƒ = 2.98

SE = 2.98/âˆš8.75

SE = 1.0074

## FAQs on Standard Error

### 1. What is Standard Error?

Standard error is the mathematical concept that is used to find the estimate value of variability in the given set of data. It is written as SE

### 2. What is Standard Error Formula?

Standard Error Formula is,

SE = S/âˆš(n)

where,

• S is Standard Deviation of Data
• n is Number of Observations

### 3. Why do we calculate Standard Error?

We calculate standard error to find out the accuracy of a sample mean as compared to the population mean.

### 4. What is the difference between SD and SE?

SD i.e. standard deviation measures the variability within a sample while SE i.e. Standard Error measures the variability across the different samples of a population.

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