50 ± 1.96 → [48.04, 51.96]
±1.96
1
1.96
Our confidence interval Calculator estimates a range of plausible values for a population mean using sample data, standard deviation, sample size, confidence level, and margin of error.
A confidence interval gives a range of plausible values for a population parameter, calculated from sample data at a selected confidence level.
Our confidence interval calculator instantly calculates the confidence interval, margin of error, standard error, and Z-score from your sample data.
Enter the sample mean, sample size, and standard deviation.
Choose a 90%, 95%, or 99% confidence level.
Instantly view and copy the confidence interval, margin of error, standard error, and Z-score.
Enter the sample mean, sample size, and standard deviation.
Choose a 90%, 95%, or 99% confidence level.
Instantly view and copy the confidence interval, margin of error, standard error, and Z-score.
You can calculate a confidence interval manually by combining a sample mean with a critical value and standard error. The exact formula depends on whether you know the population standard deviation and whether you use a z-distribution or Student's t-distribution.
Formula:
When you know the population standard deviation , calculate the confidence interval for a population mean with:
When you do not know , use the sample standard deviation and a t-critical value:
The standard error measures the sampling variability of the sample mean:
or:
Calculate the margin of error as:
Then calculate the lower and upper bounds:
Step by step Guide:
Practical Example:
Suppose a sample has a mean of 50, a known population standard deviation of 10, and a sample size of 100. We want a 95% confidence interval.
First, calculate the standard error:
For a confidence level:
Calculate the margin of error:
Now calculate both bounds:
Therefore:
The interval estimates plausible values for the true population mean under the assumptions of the selected estimation procedure.
Confidence intervals combine sample statistics, standard error, critical values, and confidence levels to estimate a population parameter.
The sample mean $\bar{x}$ is the arithmetic average of observed sample data. It acts as the point estimate for the population mean $\mu$.
The population mean $\mu$ represents the true mean of an entire population. A confidence interval uses sample statistics to estimate its plausible range.
The confidence level, such as 95% or 99%, describes the long-run success rate of the interval estimation procedure under repeated sampling.
The margin of error shows how far each confidence interval bound extends from the point estimate. It depends on critical value and standard error.
Standard error measures how much a sample statistic varies across repeated samples. For a mean, it commonly equals $s/\sqrt{n}$ or $\sigma/\sqrt{n}$.
A z-score expresses a value in standard deviation units. A z-critical value, such as $1.96$ for a 95% CI, determines interval width.
A t-score comes from Student's t-distribution. Use its critical value when estimating a mean with an unknown population standard deviation.
From a dataset, calculate the sample mean , sample standard deviation , and sample size . Then find and apply the appropriate t-critical value to obtain the lower and upper bounds.
In time series analysis, a confidence interval can form a confidence band around an estimation output, forecast, or Impulse Response Function, helping assess uncertainty, variability, and whether a response appears significant.
Confidence interval width depends mainly on sample size, confidence level, standard deviation, standard error, critical value, and data variability.
A larger sample size generally produces a smaller standard error because $SE=s/\sqrt{n}$. With other factors unchanged, increasing $n$ narrows the confidence interval and improves precision around the population parameter estimate.
A higher confidence level requires a larger critical value. Moving from a 95% to a 99% confidence level therefore increases the margin of error and creates a wider confidence interval when sample size and standard deviation remain unchanged.
Standard deviation measures the spread of observed data. A larger $s$ or $\sigma$ increases standard error and margin of error, producing a wider interval. Lower standard deviation generally creates a narrower confidence interval.
Greater data variability increases uncertainty in the sample mean and usually widens the confidence interval. More concentrated observations reduce variability, standard error, and interval width when the sample size and confidence level stay constant.
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These answers clarify confidence intervals, z-distribution and t-distribution selection, interval width, probability, standard error, and interpretation.