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Standard Error Formula

The standard error is an important statistical measure and it is related to the standard deviation. The accuracy of a sample that represents a population is known through this formula. The sample mean deviates from the population and that deviation is known as standard error formula. In this article, we will discuss the standard error formula in various cases. Let us learn the concept!

Standard Error Formula

What is the standard error?

In statistics, the word sample refers to the specific group of data that is collected. For example, the sample may be the data we collected on the height of players on the school’s team. A population is an entire group from which we take the sample. There are many ways to define a population, and we always need to be very clear about what is the population. Many computations are required for this collection.

In order to determine how well the sample is representing the population, we need to go out and measure the standard error in the specific measurements. Standard Error is telling how well the sample mean estimates the true population means. A large standard error will indicate that there is a lot of variability in the population. A small standard error would mean that the population is in a uniform shape.

Standard Error Formula

Source: en.wikipedia.org

The standard error i.e. SE is very similar to standard deviation. Both are measures of the spread of data. The higher the number, the more spread out our data is. Although the two terms are essentially equal, there is one important difference. While the standard error uses sample data, standard deviation uses population data.

SE Calculation

How we find the standard error depends on what statistical measure we need. For example, the calculation is different for the mean value or proportion value. When we are asked to find the sampling error, you’re probably finding the standard error. We might be required to find standard errors for other statistical measures also.

The Formula for Standard Error

Depending upon the statistical measure in the corresponding data, relevant methods will be used to measure the standard error. The following are the popular measures for data collection.

  • Sample mean.
  • Sample proportion.
  • Difference between means
  • Difference between proportions.

1) Standard Error in the Sample Mean:

S.E. = \(\frac {S}{\sqrt n} \)

2) Standard Error in the Sample Proportion:

S.E. = \(\sqrt(\frac {p(1-p)}{n}) \)

3) Standard Error in the Difference between means:

S.E. = \(\sqrt {\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \)

4) Standard Error in the Difference between proportions:

S.E. = \(\sqrt {\frac{p_1 \times (1-p_1)}{n_1} + \frac{ p_2 \times (1-p_2)}{n_2}} \)

Where,

S.E. Standard Error
S Standard Deviation
n Number of observations in the sample
p Proportion of successes
n_1 Number of observations in Sample 1
n_2 Number of observations in Sample 2

Solved Examples

Q.1: Calculate the standard error of the given data, with respect to mean value.

X:   10, 12, 16, 21, 25

Solution: As given n =5

First let us compute Mean,

\(\bar x = \frac{ \sigma X}{n} \)

= \(\frac {10+12+16+21+25} {5} \)

=16.8

Now, Standard Deviation is:

S= \( \sqrt{\frac{1}{4}}\left(10-16.8\right)^{2}+\left(12-16.8\right)^{2}+\left(16-16.8\right)^{2}+\left(21-16.8\right)^{2}+\left(25-16.8\right)^{2}\)

= \(\sqrt{\frac{2167.2}{4}} \)

= 5.418

Standard Error in the Sample Mean:

S.E. = \(\frac {S}{\sqrt n}\)

= \(\frac {5.418}{\sqrt 5}\)

= 2.42

Therefore, Standard Error in mean will be 2.42.

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