The Margin of Error (MOE) Calculator is a free online tool that calculates the MOE of the surveys with and without Finite Population Correction (FPC) for the given parameter values.
This calculator evaluates whether the surveyed sample size is sufficient to ensure the accuracy of the collected data.
What is the Margin of Error?
In statistics, the margin of error expresses the amount of random sampling error in the results of a survey. It reflects the uncertainty or variability that can arise from taking a sample rather than surveying an entire population.
The margin of error is usually calculated based on the sample size and the desired level of confidence for the survey. A lower confidence level or a larger sample size will result in a lower margin of error. Finding a margin of error is very helpful in measuring the accuracy of the survey.
Formula of Margin of Error
There are two types of margin of error formulas: finite population correction (FPC) and no FPC. The FPC is a statistical adjustment used when the sample size is a significant proportion of the population.
Formula of Margin of Error with FPC
The margin of error with Finite Population Correction - FPC can be calculated by using the following formula:
MOE = [z √P̂(1 - P̂)]/[√(P – 1) (n/P – n)]
Where,
- P̂ (P-hat) is the sample proportion (Max = 1)
- P is the population size
- n is the sample size.
- z is the z-score for the desired confidence level. To locate it, refer to the z-score Table.
Formula of Margin of Error (without Finite Population Correction - No FPC)
Use the following formula to find the MOE without FPC.
MOE = z √ [P̂(1 - P̂) / n]
Where,
- P̂ is sample proportion (Max = 1)
- n is the sample size.
- z is the z score for the desired confidence level. To locate it, refer to the z-score Table.
Z-Score Table
The following table shows the z-scores for some common confidence levels:
Confidence level | Z-score |
80% | 1.282 |
85% | 1.440 |
90% | 1.645 |
95% | 1.960 |
98% | 2.326 |
99% | 2.576 |
99.5% | 2.807 |
99.9% | 3.291 |
How to Calculate Margin of Error?
Let’s learn how to find the margin of error with the help of examples.
Example 1: (No FPC)
A university conducts a poll to estimate the percentage of students in favor of a new grading system. Out of 1,200 students surveyed, 720 expressed support for the new system. Determine the margin of error at a 95% confidence level.
Solution:
Step 1: Find sample proportion (P̂).
Number of students in favor = 720
Total number of surveyed students = 1200
P̂ = Number of students in favor / Total number of surveyed students
P̂ = 720/1200
P̂ = 0.6 or 60%
Step 2: Determine the Z score associated with the confidence level. Look up the corresponding z score in a z-table.
The Z-Score for a 95% confidence interval is 1.960.
Step 3: Identify the sample size (n). In this example, the sample size is 1200.
Step 4: Substitute the values of z, P̂, and n into the formula for the margin of error and simplify.
Since, MOE with No FPC = z √ [P̂(1 - P̂) / n]
= (1.960) √ [0.6(1 – 0.6) /1200]
= (1.960) √ [0.6(0.4) /1200]
= 1.960 √ (0.0002)
= 1.960 (0.014)
MOE = 0.02744
Step 4: Multiply Step 3 by 100 to convert MOE to a percentage.
MOE = 0.02744 × 100 % = 2.744 %
Thus, the margin of error is approximately 0.02744 or 2.32% at a 95% confidence level.
Example 2: (With FPC)
A company has 1,000 employees. They surveyed 200 employees to determine their satisfaction with the new benefits package. The survey finds that 65% of the employees are satisfied with these new benefits. Calculate the margin of error at a 99% confidence level.
Solution:
Step 1: Identify the Values needed for the formula.
z = 2.576 (z-score for a 99% confidence level)
Sample proportion P̂ (P-hat) = 0.65
n = 200 (sample size)
P = 1000 (population size)
Step 2: Substitute the values into the formula and then simplify it.
MOE = [z √P̂(1 - P̂)]/[√(P – 1) (n/P – n)]
= [2.576 √0.65 (1 – 0.65)]/[√(1000 – 1) (200/1000 – 200)]
= [2.576 √0.65 (0.35)]/[√(999) (0.25)]
= [2.576 √(0.2275)]/[√(999) (0.25)]
= [2.576 √(0.2275)]/[√(249.75)]
= [2.576 (0.4769)]/[15.803]
MOE = 0.0777
Step 3: Multiply Step 2 by 100 to convert MOE to a percentage.
MOE = 0.0777 × 100 % = 7.77 %
Therefore, the margin of error at a 99% confidence level is approximately 7.77 %.