# Margin of Error Formula

The margin of error is a statistic expressing of an amount of random sampling error in a survey’s results. It asserts a likelihood that the result from a sample is close to the number one would get if the whole population had been queried. In simple words, margin of error is the product of critical value and the standard deviation.

The margin of error is denoted by **E** and the formula is given as,

\[\large E=Z\left(\frac{\alpha}{2}\right)\left(\frac{\alpha}{\sqrt{n}}\right)\]

Here,

$z$ $(\frac{\alpha }{2})$ = represents the critical value.

$z$ $(\frac{\sigma }{\sqrt{n}})$ = represents the standard deviation.

### Solved Examples

**Question: **A random sample of 30 students has an average yearly earnings of 2450 and a standard deviation of 587. Find the margin of error if c = 0.95?

**Solution:**

Given

n=30, Standard Deviation= 587

z $\frac{\alpha}{2}$ = 1.96

$\therefore$ by using the formula E= z $(\frac{\alpha }{2})(\frac{\sigma }{\sqrt{n}})$

= 1.96 $\times$ $(\frac{587}{\sqrt{30}})$

= 210.06

= 210 approximately

The margin of error = 210