Calculate 95 Confidence Interval

Understanding the 95% Confidence Interval

A confidence interval (CI) is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. The 95% confidence interval is one of the most commonly used intervals, indicating that if we were to take 100 different samples from the same population and calculate a 95% CI for each sample, we would expect approximately 95 of those intervals to contain the true population parameter.

The Math Behind the 95% CI

The formula for calculating a confidence interval for a population mean, when the population standard deviation is unknown and the sample size is reasonably large (or the population is normally distributed), is:

CI = &overline;x ± t* (s / √n)

Where:

• &overline;x (Sample Mean): The average of the data points in your sample.
• s (Sample Standard Deviation): A measure of the dispersion or spread of the data points in your sample around the mean.
• n (Sample Size): The number of observations in your sample.
• t* (Critical t-value): This value depends on the desired confidence level (95% in this case) and the degrees of freedom (df = n – 1). For a 95% confidence interval, the t-value is determined from a t-distribution table or statistical software. For larger sample sizes (generally n > 30), the t-distribution approaches the normal distribution, and the z-score of approximately 1.96 is often used as an approximation. However, using the t-value is more accurate, especially for smaller sample sizes.
• s / √n (Standard Error of the Mean – SEM): This represents the standard deviation of the sampling distribution of the mean. It estimates how much the sample mean is likely to vary from the true population mean.
• ± t* (s / √n) (Margin of Error): This is the "plus or minus" part of the confidence interval. It quantifies the uncertainty in our estimate of the population mean based on the sample.

For a 95% Confidence Interval specifically:

The critical t-value (t*) for a 95% CI can vary slightly with degrees of freedom (n-1). For simplicity and general use with larger samples, a common approximation for the margin of error uses a z-score of 1.96. So, the interval becomes:

CI ≈ &overline;x ± 1.96 * (s / √n)

This calculator uses the approximation with z = 1.96 for wider applicability without requiring a t-table lookup.

Why Use a Confidence Interval?

In statistical inference, we often use a sample to make generalizations about a larger population. It's rare for a sample statistic (like the sample mean) to exactly equal the population parameter (the true population mean). A confidence interval acknowledges this uncertainty by providing a range.

Key use cases include:

• Estimating Population Parameters: Providing a plausible range for unknown population means, proportions, or other metrics.
• Hypothesis Testing: If a hypothesized population value falls outside the confidence interval, it provides evidence against that hypothesis.
• Interpreting Survey Results: Understanding the precision of survey estimates (e.g., the average age of users, the proportion of satisfied customers).
• Scientific Research: Reporting findings with a measure of uncertainty, allowing for better comparison and interpretation across studies.

Interpreting the Result

When you get a result like (Lower Bound, Upper Bound), it means you can be 95% confident that the true population mean lies somewhere within this calculated range. It does *not* mean there's a 95% probability that the true mean falls within this specific interval after it's calculated. Instead, it refers to the reliability of the method used to construct the interval.