Confidence Interval Calculator

Quickly calculate confidence intervals for any dataset with our fast & accurate Confidence Interval Calculator.

Sample Information

The total number of observations in your sample (n)
The arithmetic average of your sample data (x̄)
The standard deviation of your sample (σ or s)

Confidence Level

Common values: 90%, 95%, 99% (higher values create wider intervals)
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What is Confidence Interval?

A confidence interval is a range of values, derived from sample data, that is likely to contain the value of an unknown population parameter. It provides an estimate of the uncertainty associated with a sample statistic. For example, if you calculate a 95% confidence interval for the mean, it means you are 95% confident that the true population mean lies within that range.

How to Calculate Confidence Interval

To calculate a confidence interval, you need three key pieces of information: the sample mean (\(\bar{x}\)), the standard deviation (\(s\)), and the sample size (\(n\)). The formula for a confidence interval for the mean is:

\[ \text{CI} = \bar{x} \pm z \left( \frac{s}{\sqrt{n}} \right) \]

Here, \(z\) is the z-score corresponding to your desired confidence level (e.g., 1.96 for 95% confidence). If the sample size is small or the population standard deviation is unknown, you might use the t-distribution instead of the z-score.

Confidence Interval & Standard Deviation

The standard deviation (\(s\)) plays a crucial role in calculating the confidence interval. It measures the spread of the data points around the mean. A larger standard deviation indicates more variability, which results in a wider confidence interval. Conversely, a smaller standard deviation leads to a narrower interval, indicating more precise estimates.

For example, if the sample mean is 50, the standard deviation is 10, and the sample size is 100, the 95% confidence interval would be:

\[ \text{CI} = 50 \pm 1.96 \left( \frac{10}{\sqrt{100}} \right) = 50 \pm 1.96 \times 1 = [48.04, 51.96] \]

Confidence Interval Calculator (Upper and Lower Bound Calculator)

While manual calculations are possible, using a confidence interval calculator can save time and reduce errors. These tools typically require you to input the sample mean, standard deviation, sample size, and desired confidence level. The calculator then computes the interval for you, ensuring accuracy and efficiency.