 # Sampling Error Formula

The sampling error formula, as the name suggests, is used to calculate the overall sampling error in statistical analysis. To recall, statistical error arising out of nature of sampling is known as sampling error. The error in statistical analysis arises because of the unrepresentativeness of the observation in the samples taken.

For example, the weight of two thousand citizens from a country of two million is taken and the average weight is taken out from that, then it’s not the same as the average weight of the two million people in the country.

Since to determine the characteristics of a whole population the sampling is done, the difference between the sample values and population is called sampling error. We should note that the exact value of sampling cannot be done since the population value if not known although often sampling error can be found out by probabilistic modelling of a sample.

Thus, the sampling error formula is given by:

$$\begin{array}{l}Sampling\ Error = Z\times \frac{\sigma}{\sqrt{n}}\end{array}$$

Where,

Z is the Z score value based on the confidence interval (approx = 1.96)
σ is the population standard deviation
n is the size of the sample

## Solved Examples

Example 1: Suppose that the population standard deviation is 0.40 and the size of the sample is 2500 then find the sampling error at 95% confidence level.

Solution:
From the given data,
σ = 0.40
Sample size = n = 2500
Value of z at 95% of confidence level = 1.96
Sampling error = z × σ/√n
= 1.96 × 0.40/√(2500)
= 1.96 × 0.40/50
= 0.01568

Example 2: Find the sampling error of the sample size 100 of population with standard deviation 0.5 at 90% confidence level.

Solution:
From the given data,
From the given data,
σ = 0.5
Sample size = n = 100
Value of z at 90% of confidence level = 1.645
Sampling error = z × σ/√n
= 1.645 × 0.5/√(100)
= 1.645 × 0.5/10
= 0.08225

Note:
Z-value at 90% confidence level = 1.645
Z-value at 95% confidence level = 1.96
Z-value at 99% confidence level = 2.58