Confidence Interval Calculator

What This Confidence Interval Calculator Does

This page calculates a confidence interval for a population mean based on a sample average, the sample standard deviation, and the sample size. It is designed for situations where your outcome is numeric (for example, test scores, average rating, response time, weight, or revenue per user), not for yes/no proportions.

You type in three main inputs:

• Sample mean – the average value you observed in your data.
• Sample standard deviation – how spread out the observations are.
• Sample size – how many observations are in your sample.

The calculator then combines these inputs with your chosen confidence level (for example, 90%, 95%, or 99%) and returns a lower and upper bound. This interval represents a range of plausible values for the true population mean, given the uncertainty that comes from working with a sample instead of measuring the entire population.

What Is a Confidence Interval for a Mean?

A confidence interval for a mean is a range of values that likely contains the true average of a population. Instead of reporting just a single point estimate (your sample mean), you show an interval that accounts for sampling variation. This makes your conclusions more honest and informative, especially when the sample is not very large or the data are noisy.

For example, imagine you survey 100 customers and find an average satisfaction score of 4.2 out of 5. A confidence interval might be from 4.0 to 4.4. You would then say something like:

With 95% confidence, the true average satisfaction score for all customers lies between 4.0 and 4.4.

This does not mean there is a 95% probability that the true mean is in this specific interval. Instead, it means that if you repeated the whole sampling and interval-building procedure many times, about 95% of those intervals would contain the true mean. Your single interval is one draw from that long-run process.

Formulas Used by the Calculator

The calculator uses the standard z-based confidence interval for a population mean when the population standard deviation is unknown but approximated by the sample standard deviation, and the sample size is reasonably large.

Define:

• x̄ = sample mean
• s = sample standard deviation
• n = sample size
• z = z-score corresponding to the chosen confidence level

The standard error of the mean (SE) is:

SE = s / √n

The margin of error (ME) is:

ME = z × SE = z × (s / √n)

The confidence interval is then:

CI = x̄ ± ME

In MathML, the core relationship can be written as:

Common z-scores for two-sided confidence intervals are approximately:

• 90% confidence: z ≈ 1.645
• 95% confidence: z ≈ 1.96
• 99% confidence: z ≈ 2.576

Inputs You Need

To use the calculator effectively, be clear on what each input represents:

• Sample Mean – The arithmetic average of your data. Add up all observations and divide by the number of observations.
• Sample Standard Deviation – A measure of spread that reflects how far typical observations deviate from the mean. Many spreadsheet and stats tools can compute this for you.
• Sample Size (n) – The number of independent observations used to compute the mean. Larger samples generally give narrower, more precise intervals.
• Confidence Level – How confident you want to be in the long-run procedure (for example, 90%, 95%, or 99%). Higher confidence produces a wider interval because it uses a larger z-score.

How This Calculator Works Step by Step

1. You enter the sample mean, sample standard deviation, and sample size.
2. You choose a confidence level (or enter a custom percentage).
3. The script looks up the corresponding z-score for the confidence level.
4. It calculates the standard error SE = s / √n.
5. It multiplies the standard error by the z-score to get the margin of error.
6. It subtracts the margin of error from the mean to get the lower bound, and adds it to the mean to get the upper bound.
7. The final result is displayed as a range, along with the chosen confidence level.

Interpreting the Results

When the calculator returns an interval like [77.2, 82.8] at 95% confidence, interpret it as follows:

• You used a method that, in the long run, produces intervals that include the true mean 95% of the time.
• This specific sample produced an interval from 77.2 to 82.8.
• Values near the center of the interval (around the sample mean) are more consistent with your data than extreme values far outside it.

Useful practical tips:

• Narrow intervals indicate a more precise estimate of the mean, often due to a large sample size or low variability.
• Wide intervals signal high uncertainty. You may need more data, better measurement, or a different study design.
• If two groups have intervals that heavily overlap, the difference between their means may be small or not statistically clear.

Worked Example: Test Scores

Suppose you collect exam scores from a sample of students and want to estimate the average score for all students who might take the exam.

• Sample mean (x̄): 80
• Sample standard deviation (s): 10
• Sample size (n): 50
• Confidence level: 95%

Steps:

1. Compute the standard error: SE = 10 / √50 ≈ 10 / 7.071 ≈ 1.414.
2. Use z ≈ 1.96 for a 95% confidence level.
3. Margin of error: ME = 1.96 × 1.414 ≈ 2.77.
4. Lower bound: 80 − 2.77 ≈ 77.23.
5. Upper bound: 80 + 2.77 ≈ 82.77.

Interpretation in plain language:

Based on this sample, the average exam score for the full population of students is estimated to be between about 77.2 and 82.8, using a 95% confidence interval.

Additional Scenarios

Customer Satisfaction Survey

Imagine you survey 120 customers and ask them to rate their satisfaction from 1 to 10.

• Sample mean: 7.4
• Sample standard deviation: 1.8
• Sample size: 120
• Confidence level: 95%

The calculator might return a 95% confidence interval of roughly [7.1, 7.7]. You could report:

We estimate the true average customer satisfaction to be between 7.1 and 7.7 out of 10, using a 95% confidence interval.

Product Measurement Example

A quality engineer measures the weight of 40 random units of a product.

• Sample mean: 502.5 g
• Sample standard deviation: 4.0 g
• Sample size: 40
• Confidence level: 99%

The calculator might yield a 99% confidence interval of about [500.7 g, 504.3 g]. The engineer might say:

With 99% confidence, the true average product weight lies between 500.7 g and 504.3 g.

Comparison: Different Confidence Levels and Related Tools

Choosing a confidence level involves a trade-off between certainty and precision. Higher confidence makes the interval wider; lower confidence makes it narrower.

Assumptions and Limitations

The numbers from this calculator are only as reliable as the assumptions behind the method. Keep these points in mind:

• Normality or sufficiently large sample size – The formula assumes either that the population distribution is approximately normal or that the sample size is large enough for the Central Limit Theorem to make the sampling distribution of the mean close to normal.
• Independent observations – The data points in your sample should be reasonably independent of each other. Strong dependence (for example, repeated measurements on the same unit without proper modeling) can invalidate the interval.
• Using sample standard deviation as an estimate – The population standard deviation is usually unknown, so this method uses the sample standard deviation as an estimate. For small samples, especially with non-normal data, a t-distribution based interval may be more appropriate than a z-based one.
• Numeric means only – This tool is for averages of numeric variables. It is not suitable for binary outcomes (for example, yes/no, success/failure) where a proportion interval is more appropriate.
• Not a substitute for expert review – For regulatory, medical, safety-critical, or other high-stakes decisions, results from this calculator should be reviewed by a qualified statistician or domain expert.

Frequently Asked Questions

What formula does this confidence interval calculator use?

It uses a two-sided z-based confidence interval for a population mean: mean ± z × (standard deviation / √n). The z-score is chosen to match your selected confidence level.

When should I not use this calculator?

Avoid using it when your sample size is very small, your data are highly skewed or heavy-tailed, or your outcome is binary or categorical. In those cases, a t-interval for the mean or a proportion-based interval may be more suitable.

What is the difference between a 90% and 95% confidence interval?

A 95% interval has higher long-run coverage but is wider, because it uses a larger z-score. A 90% interval is slightly narrower but less conservative. Many studies use 95% as a standard compromise between certainty and precision.

Should I use a t-distribution instead of a z-distribution?

If your sample size is small and the population standard deviation is unknown, many textbooks recommend using a t-distribution based interval. This calculator uses a z-based approach, which is more appropriate when the sample is moderately large or the population is approximately normal.

Why is my confidence interval so wide?

Wide intervals usually result from small sample sizes, high variability (large standard deviation), or very high confidence levels such as 99%. Collecting more data, reducing measurement noise, or choosing a slightly lower confidence level can help narrow the interval.