Margin of Error Calculator
Calculate survey and experiment margin of error ($\text{ME} = z^* \cdot \frac{s}{\sqrt{n}}$) for 90%, 95%, or 99% confidence levels, standard error, and interval width.
Calculate Survey & Sample Margin of Error
Enter standard deviation (s), sample size (n), and confidence level.
Calculation Results
Calculated using margin of error equations: \text{ME} = z^* \cdot \frac{s}{\sqrt{n}}
Quick Summary
The Margin of Error Calculator evaluates survey sampling uncertainty ($\text{ME} = z^* \cdot \frac{s}{\sqrt{n}}$), standard error ($\text{SE}$), critical multiplier ($z^*$), and total interval width.
Formula Explanation
\text{ME} = z^* \cdot \left(\frac{s}{\sqrt{n}}\right)
\text{Total Interval Width} = 2 \cdot \text{ME}How It Works
Margin of error measures how much sample results are expected to differ from true population values due to random sampling variability. The Margin of Error Calculator multiplies critical score $z^*$ by standard error $s / \sqrt{n}$.
Step-by-Step Worked Example
Practical Problem: Standard deviation $s = 15$, sample size $n = 400$, confidence level = 95% ($z^* = 1.960$). Calculate margin of error.
- Step 1: Calculate square root of sample size ($\sqrt{n}$): $\sqrt{400} = \mathbf{20.0}$.
- Step 2: Calculate Standard Error ($\text{SE}$): $\text{SE} = 15 / 20.0 = \mathbf{0.7500}$.
- Step 3: Multiply by critical multiplier $z^* = 1.960$: $\text{ME} = 1.960 \times 0.7500 = \mathbf{1.4700}$.
- Step 4: Calculate total interval width ($2 \times \text{ME}$): $2 \times 1.4700 = \mathbf{2.9400}$.
- Step 5: Interpretation: The sample estimate has a margin of error of $\pm 1.4700$ units at 95% confidence.
Real-World Calculation Examples
Scenario 1: Survey (s = 15, n = 400, 95% CL)
Parameters: s = 15, n = 400
Result: Margin of Error = ±1.4700 (Width = 2.94).
Scenario 2: Election Political Poll (n = 1000)
Parameters: s = 0.50 (proportion), n = 1000
Result: Margin of Error = ±3.10%.
Scenario 3: Factory Quality Inspection Audit
Parameters: s = 2.5g, n = 100
Result: Margin of Error = ±0.49g.
Scenario 4: Customer Satisfaction Survey (n = 2500)
Parameters: Large sample size
Result: Ultra-precise margin of error ±0.60%.
Key Benefits of Using This Calculator
Survey Precision Planning
Helps researchers evaluate sampling precision before conducting surveys.
Supports Means & Proportions
Works for continuous means (using $s$) and proportions (using $s = \sqrt{\hat{p}(1-\hat{p})}$).
Multi-Confidence Level Support
Computes margin of error for 90%, 95%, and 99% confidence levels.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is margin of error?
Margin of error is the radius of a confidence interval, representing the maximum expected difference between sample results and population truth.
How does sample size n affect margin of error?
Larger sample sizes reduce margin of error ($\text{ME} \propto 1/\sqrt{n}$). Quadrupling sample size cuts margin of error in half.
How do I calculate margin of error in Excel?
Use formula =CONFIDENCE.NORM(alpha, standard_dev, size).
What critical Z-values are used?
90% CL $\rightarrow z^* = 1.645$; 95% CL $\rightarrow z^* = 1.960$; 99% CL $\rightarrow z^* = 2.576$.
How do I calculate margin of error for a percentage/proportion?
Set $s = \sqrt{\hat{p}(1 - \hat{p})}$; for worst-case maximum margin of error, use $\hat{p} = 0.50 \rightarrow s = 0.50$.
What is relationship between confidence level and margin of error?
Higher confidence levels require larger critical multipliers $z^*$, resulting in a LARGER margin of error.
Does population size N affect margin of error?
For large populations, population size has negligible impact; for small finite populations ($n/N > 5\%$), apply Finite Population Correction $\sqrt{\frac{N-n}{N-1}}$.
What is standard error?
Standard error ($\text{SE} = s/\sqrt{n}$) is the standard deviation of the sample estimate distribution.
What is confidence interval width?
Interval width is twice the margin of error ($\text{Width} = 2 \times \text{ME}$).
How to reduce margin of error without increasing sample size?
You can lower the confidence level (e.g. from 99% to 90%) or decrease measurement variability $s$.