Confidence Interval Calculator — Free Online CI Tool

How to Use the Confidence Interval Calculator

1. Enter the sample mean (x̄): This is the average of your sample data. If you do not know it yet, calculate it from your raw data using our statistics calculator and enter the result here.
2. Enter the standard deviation (σ): Input the population standard deviation if known, or the sample standard deviation as an estimate. This must be a positive number. The standard deviation determines how wide the interval will be relative to sample size.
3. Enter the sample size (n): Input the number of observations in your sample. Larger samples produce narrower intervals. The minimum is 1, but meaningful confidence intervals typically require at least 30 observations for the z-approximation to be reliable.
4. Select the confidence level: Choose from 80% to 99.9%. Higher confidence levels produce wider intervals. The 95% level is most common in research and is the default.
5. Interpret the results: The interval is displayed as (lower, upper). The margin of error shows how far the interval extends from the mean. The z-critical value and standard error are shown for transparency. A plain-language interpretation summarizes what the interval means.

All calculations update in real time. The results panel shows both the mathematical details and a plain-English interpretation to help you communicate your findings effectively.

Confidence Interval Formula

Variables Explained

• x̄ (x-bar): The sample mean, serving as the point estimate of the population mean.
• z*: The critical value from the standard normal distribution corresponding to the chosen confidence level. For 95% confidence, z* = 1.960.
• σ: The population standard deviation (or its estimate from the sample).
• n: The sample size — the number of observations in the sample.
• SE: The standard error, representing the standard deviation of the sampling distribution of the mean.
• E: The margin of error, which is the half-width of the confidence interval.

Step-by-Step Example

A sample of 36 students has a mean test score of 82 with a known population standard deviation of 9. Find the 95% confidence interval for the true mean.

1. Identify values: x̄ = 82, σ = 9, n = 36, confidence = 95%
2. Find z*: For 95% confidence, z* = 1.960
3. Calculate standard error: SE = 9 / √36 = 9 / 6 = 1.5
4. Calculate margin of error: E = 1.960 × 1.5 = 2.94
5. Lower bound: 82 - 2.94 = 79.06
6. Upper bound: 82 + 2.94 = 84.94

We are 95% confident that the true population mean test score falls between 79.06 and 84.94. The margin of error of ±2.94 points means our sample mean of 82 could be at most about 3 points away from the true average.

Practical Examples

Example 1: Tom's Customer Satisfaction Survey

Tom manages a restaurant and surveys 50 customers on satisfaction (1-10 scale). The sample mean is 7.8 with a standard deviation of 1.5. He wants a 95% confidence interval for the true average satisfaction score.

• Standard error: 1.5 / √50 = 0.212
• Margin of error: 1.960 × 0.212 = 0.416
• 95% CI: (7.384, 8.216)

Tom can report that average customer satisfaction is between 7.4 and 8.2 on a 10-point scale with 95% confidence. Since the entire interval is above 7.0, he can confidently say that most customers rate the experience positively. If he wants a narrower interval for more precise marketing claims, he could survey more customers.

Example 2: Lisa's Production Quality Check

Lisa tests 100 widgets from a production line and finds the average weight is 52.3 grams with a standard deviation of 2.1 grams. The target weight is 50 grams. She builds a 99% confidence interval to determine if the line is producing overweight widgets.

• Standard error: 2.1 / √100 = 0.21
• Margin of error: 2.576 × 0.21 = 0.541
• 99% CI: (51.759, 52.841)

The entire 99% confidence interval is above the target weight of 50 grams. Lisa concludes with very high confidence that the production line is consistently producing overweight widgets and recommends recalibration. Even at the lower bound (51.76g), the widgets are 1.76 grams above target — a practically significant difference.

Example 3: Dr. Chen's Clinical Trial

Dr. Chen studies the effect of a new exercise program on blood pressure reduction. In a sample of 45 participants, the mean reduction is 8.5 mmHg with a standard deviation of 6.2 mmHg. She needs a 95% confidence interval for the true mean reduction.

• Standard error: 6.2 / √45 = 0.924
• Margin of error: 1.960 × 0.924 = 1.811
• 95% CI: (6.689, 10.311)

The confidence interval (6.69, 10.31) does not include zero, meaning the exercise program produces a statistically significant reduction in blood pressure. The clinical significance is also notable: even the conservative lower bound of 6.7 mmHg is considered a meaningful blood pressure reduction. Dr. Chen reports that the program reduces systolic blood pressure by approximately 6.7 to 10.3 mmHg. For the related hypothesis test, see our p-value calculator.

Z-Critical Values Reference Table

Tips and Complete Guide

Choosing the Right Confidence Level

The confidence level reflects how much uncertainty you are willing to accept. In most scientific research, 95% is standard. For exploratory analysis or preliminary studies, 90% may suffice. For high-stakes decisions (medical treatments, safety-critical engineering), use 99% or higher. Remember: higher confidence means wider intervals. The optimal choice balances the cost of being wrong (Type I error) against the need for precision in your estimate.

Confidence Intervals vs. Prediction Intervals

A confidence interval estimates where the population mean lies. A prediction interval estimates where a single future observation will fall. Prediction intervals are always wider than confidence intervals because they include both the uncertainty about the mean and the natural variability of individual observations. If someone asks "What score will the next student get?", you need a prediction interval, not a confidence interval.

Reporting Confidence Intervals Effectively

When reporting confidence intervals in papers or presentations, include the confidence level, the point estimate, and the interval bounds. For example: "The mean reduction in blood pressure was 8.5 mmHg (95% CI: 6.7 to 10.3)." This format is endorsed by the American Psychological Association (APA) and most medical journals. It communicates both the best estimate and the precision of that estimate in a single, concise statement.

Common Mistakes to Avoid

• Saying "95% probability the true mean is in this interval": The true mean is a fixed value — it either is or is not in the interval. The 95% refers to the long-run coverage rate of the method, not the probability for this specific interval.
• Using z-intervals with small samples and unknown σ: When n is small (less than 30) and the population standard deviation is unknown, use the t-distribution instead. The t-distribution accounts for the extra uncertainty from estimating σ with s.
• Ignoring assumptions: The z-based confidence interval assumes the sampling distribution of the mean is approximately normal. This holds for large samples (Central Limit Theorem) or when the population is normally distributed. For small samples from non-normal populations, consider bootstrap methods.
• Comparing overlapping intervals and concluding no difference: Two confidence intervals can overlap even when the difference between the groups is statistically significant. To test whether two means differ, construct a confidence interval for the difference, not separate intervals for each mean.
• Reporting without sample size: A confidence interval without the sample size is incomplete. A narrow interval from n = 10,000 means something different than the same interval from n = 30. Always report n alongside the confidence interval.

Frequently Asked Questions