Confidence Interval Calculator: Margin of Error & Z-Value
Free confidence interval calculator: margin of error and interval for the population mean. How to calculate margin of error, Z-value for 99% confidence level, sample mean confidence interval formula, what does 95% confidence level mean. Normal distribution.
What This Calculator Does & Who It's For
Calculator Purpose & Ideal Users
- What You'll Get:Confidence interval: [lower, upper] for the population mean. Margin of error: Plus/minus value and percentage of the mean. Critical value (Z): The multiplier for your chosen confidence level (70% to 99.99%). Calculation step: The formula xฬ ยฑ Z ร (ฯ/โn) with your numbers. Z-value reference table: Toggle in the calculator for common levels.
- Ideal Users:Students & coursework: Homework on confidence intervals, margin of error, or the sample mean confidence interval formula. Researchers & analysts: Quick interval estimates from summary statistics (n, xฬ, ฯ or s). Anyone reporting survey or experiment results: Get the interval and margin of error for reports or presentations.
- Assumptions & Scope:Assumes a normal distribution (or large n). Accepts population (ฯ) or sample (s) standard deviation. For very small samples (e.g. n < 30), a t-interval may be more appropriate; this tool uses Z. All calculations are local; no data is uploaded.
This confidence interval calculator computes a range for the population mean from your sample mean, sample size, and standard deviation, using a chosen confidence level and Z-value.
What Is a Confidence Interval Calculator?
A confidence interval calculator computes a range within which an unknown population parameter (e.g. the mean) is likely to fall, based on sample data and a confidence level (e.g. 95%). You enter n, xฬ, and ฯ or s; the tool returns the confidence interval for the population mean, margin of error, and Z-value. Use it for how to calculate margin of error, the sample mean confidence interval formula, or a Z-value for 99% confidence level. See above for what you'll get and who it's for. Free online; all calculations local.
Defining the Interval: What Does a Confidence Level Mean?
A confidence interval is a range of values that is likely to contain the true population parameter (e.g. the population mean), based on your sample and a specified confidence level. The interval is xฬ ยฑ Z ร (ฯ/โn). What does a 95% confidence level mean? It means that if you repeated the sampling and interval construction many times, about 95% of those intervals would contain the true parameterโit describes the procedureโs reliability, not the probability that your single interval contains the parameter (the parameter is fixed; the interval is random).
The Role of Sample Size: Why Larger n Narrows the Interval
Increasing the sample size (n) typically narrows the confidence interval, giving a more precise estimate. The standard error of the mean is ฯ/โn, so doubling n reduces the standard error by a factor of โ2, and the margin of error (Z ร ฯ/โn) shrinks. Larger n means a smaller margin of error and a tighter range. This calculator lets you change n and see how the interval and margin of error update.
How to Calculate Margin of Error: Sample Mean Confidence Interval Formula
How to calculate margin of error: the margin of error for the mean is Z ร (ฯ/โn). The sample mean confidence interval formula is xฬ ยฑ Z ร (ฯ/โn), or [xฬ โ Zรฯ/โn, xฬ + Zรฯ/โn]. The Z-value comes from the confidence level (e.g. Z-value for 99% confidence level โ 2.576, 95% โ 1.960). This calculator shows the calculation step and a Z-value reference table. Supports population (ฯ) or sample (s) standard deviation; for very small samples a t-interval may be more appropriate.
Confidence Interval Calculator FAQ
? What does a 95% confidence level mean?
A 95% confidence level means that if you repeated the same sampling and interval construction many times, about 95% of those intervals would contain the true population parameter (e.g. the population mean). It describes the reliability of the procedure, not the probability that a single interval contains the parameter. This confidence interval calculator lets you choose 70% to 99.99% and shows the corresponding Z-value and margin of error.
? How do I find the Z-score for a confidence level?
For a given confidence level (e.g. 95%), the Z-value (critical value) is the number of standard errors on each side of the sample mean so that the area under the normal curve between โZ and +Z equals that percentage. Common values: 90% โ Z โ 1.645, 95% โ Z โ 1.960, 99% โ Z โ 2.576. This calculator includes a Z-value reference table (toggle โZ-value referenceโ) for levels from 70% to 99.99%.
? What is the difference between sample and population standard deviation?
Population standard deviation (ฯ) uses the true mean of the entire population and divides by the population size N. Sample standard deviation (s) uses the sample mean and divides by nโ1 (Besselโs correction). For large samples the difference is small; this calculator accepts either ฯ or s in the โStandard deviationโ field and uses it in the formula xฬ ยฑ Z ร (ฯ/โn). For very small samples you may prefer a t-interval; this tool assumes a normal distribution and uses Z.
? How do I calculate margin of error?
The margin of error for a confidence interval for the mean is Z ร (ฯ/โn): the critical value Z (from the confidence level) times the standard error of the mean (ฯ/โn). Enter your sample size n, sample mean xฬ, standard deviation ฯ or s, and confidence level into this confidence interval calculator; it shows the margin of error, the plus/minus range, the percentage margin, and the interval [lower, upper].
? What is the sample mean confidence interval formula?
The confidence interval for the population mean (assuming a normal distribution) is xฬ ยฑ Z ร (ฯ/โn), where xฬ is the sample mean, Z is the critical value for the chosen confidence level, ฯ is the population standard deviation (or sample s), and n is the sample size. The interval is [xฬ โ Zรฯ/โn, xฬ + Zรฯ/โn]. This calculator computes that range and shows the calculation step so you can follow the math.