Confidence Interval Calculator
Calculate CI for Sample Mean ($\mu$)
[Image of bell curve with confidence interval]
$$ \bar{x} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}} $$
Distribution Statistic
Sample Mean ($\bar{x}$)
Standard Deviation ($s$)
Sample Size ($n$)
Confidence Level (%)
1
2
3
+
–
.
4
5
6
*
/
C
7
8
9
0
CLEAR
Confidence Interval
Detailed Solution
👨🏫
By Prof. David Anderson
Statistics Professor | 20+ Years Exp.
"In statistical inference, a single number (like an average) is never enough. It's like trying to hit a bullseye with a single dart while blindfolded. A Confidence Interval (CI) is like throwing a net instead of a dart—it gives you a range where the truth likely hides. Today, I'll walk you through how to build this net, whether you are measuring the average height of students (Mean) or the winning chance of a politician (Proportion)."
Confidence Interval Calculator: Mean, Proportion & Margin of Error
The Complete Guide to Interval Estimation, Z-Intervals, and T-Intervals
The Confidence Interval Calculator transforms a "Point Estimate" (a single number) into an "Interval Estimate" (a range). It quantifies the uncertainty in your data by adding a Margin of Error (MOE) to your sample statistic.
This tool features a Dual-Mode Input system, allowing you to calculate intervals for continuous data (Mean) or categorical data (Proportion), automatically selecting between Z-statistics and T-statistics based on your sample size.
1. Dual-Mode Input: Which Interval Do You Need?
The mathematical formula changes entirely based on your data type. Select the mode that matches your data:
CI for Mean ($\mu$)
Use for Continuous Data (e.g., Height, Salary, Time, Weight).
Inputs: Sample Mean ($\bar{x}$), Standard Deviation ($s$), Sample Size ($n$).
CI for Proportion ($p$)
Use for Binary/Categorical Data (e.g., % of Votes, Success Rate, Yes/No).
Inputs: Number of Successes ($X$), Total Trials ($n$).
2. Visualizing: Factors Affecting Interval Width
The "Width" of your confidence interval represents Precision. A narrower interval is more precise. Three factors control this width:
Sample Size ($n$)
Increase $n$ ➜ Narrower Width (More Precise)
Variability ($s$ or $\sigma$)
Increase Variability ➜ Wider Width (Less Precise)
Confidence Level
95% to 99% ➜ Wider Width (To capture more certainty)
3. The Math: Margin of Error & Formulas
A Confidence Interval is constructed using a simple logic: Point Estimate $\pm$ Margin of Error.
A. For Population Mean ($\mu$)
The calculator automatically decides between Z and T:
• Z-Interval: Used if $n \ge 30$ (Large Sample) or $\sigma$ is known.
• T-Interval: Used if $n < 30$ (Small Sample) and $\sigma$ is unknown.
$$ CI = \bar{x} \pm Z \left( \frac{s}{\sqrt{n}} \right) $$
Where $\frac{s}{\sqrt{n}}$ is the Standard Error and $Z \times SE$ is the Margin of Error.
B. For Population Proportion ($p$)
We use the Wald Interval formula for proportions (assuming large enough sample size).
$$ CI = \hat{p} \pm Z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$
Where $\hat{p}$ is the sample proportion ($X/n$).
4. Deep Dive: Margin of Error (MOE)
The Margin of Error is one of the most searched terms in statistics (especially during election seasons). It represents the "radius" of your interval.
Formula: $MOE = \text{Critical Value} \times \text{Standard Error}$
If a poll says "40% support Candidate A with a margin of error of 3%", it means the Confidence Interval is $40 \pm 3$, or [37%, 43%].
Formula: $MOE = \text{Critical Value} \times \text{Standard Error}$
If a poll says "40% support Candidate A with a margin of error of 3%", it means the Confidence Interval is $40 \pm 3$, or [37%, 43%].
5. Step-by-Step Calculation Guide
Let's calculate the CI for a sample of student heights.
Step 1 Collect Data
Sample Mean ($\bar{x}$) = 170 cm.
Sample Standard Deviation ($s$) = 10 cm.
Sample Size ($n$) = 100.
Sample Standard Deviation ($s$) = 10 cm.
Sample Size ($n$) = 100.
Step 2 Find Standard Error
$$ SE = \frac{s}{\sqrt{n}} = \frac{10}{\sqrt{100}} = \frac{10}{10} = 1.0 $$
Step 3 Find Z-Score
For 95% Confidence Level, the Critical Z-value is 1.96.
Step 4 Calculate MOE
$$ MOE = 1.96 \times 1.0 = 1.96 $$
Step 5 Build Interval
Lower Limit: $170 - 1.96 = 168.04$
Upper Limit: $170 + 1.96 = 171.96$
Result: [168.04, 171.96]
Upper Limit: $170 + 1.96 = 171.96$
Result: [168.04, 171.96]
6. How to Report Confidence Intervals (APA Style)
In academic writing, you must report the Interval alongside the Mean and SD.
"The average height was 170 cm (95% CI [168.04, 171.96])."
OR for Proportions:
"40% of participants agreed (95% CI [37%, 43%])."
OR for Proportions:
"40% of participants agreed (95% CI [37%, 43%])."
7. Professor's FAQ Corner
Does 95% Confidence mean a 95% probability the true mean is inside?
NO. The true mean is a fixed number; it doesn't move. "95% Confidence" means that if we repeated this experiment 100 times and calculated 100 intervals, about 95 of those intervals would capture the true mean. It is a probability of the method, not the specific interval.
Why do we use n-1 for T-intervals?
When sample sizes are small, we underestimate the population variability. Using Degrees of Freedom ($df = n-1$) in the T-distribution makes the interval slightly wider to account for this extra uncertainty.
Should I use 95% or 99% Confidence Level?
There is a trade-off. 99% gives you more certainty but creates a wider interval (less precise). 95% is the industry standard balance between precision and certainty.
How can I reduce my Margin of Error?
The most effective way is to increase your sample size ($n$). Since MOE involves dividing by $\sqrt{n}$, you need to quadruple your sample size to cut the error in half.
What if my sample size is really small (n < 30)?
You must use the T-Distribution. However, this assumes your original data follows a normal distribution. If your small sample is heavily skewed, even the T-interval might be inaccurate.
References
- Neyman, J. (1937). "Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability".
- Cumming, G. (2014). "The New Statistics: Why and How". Psychological Science.
- Altman, D. G. (2000). Statistics with Confidence. BMJ Books.
Estimate Your Range
Choose Mean or Proportion mode above to calculate your 95% Confidence Interval.
Calculate Interval