95% Confidence Interval Calculator

A confidence interval gives an estimated range of values which is likely to include an unknown population parameter. For a 95% confidence level, the calculation relies on the critical z-score ($z = 1.96$), which corresponds to the central 95% area under the standard normal distribution curve. First, the standard error (SE) is determined by dividing the standard deviation ($\sigma$) by the square root of the sample size ($n$): $\text{SE} = \sigma / \sqrt{n}$. Next, the margin of error (MOE) is calculated by multiplying the standard error by the z-score: $\text{MOE} = \text{SE} \times z$. Finally, the lower and upper bounds are established by subtracting and adding the margin of error to the sample mean ($\bar{x}$): $\text{Lower Bound} = \bar{x} - \text{MOE}$ and $\text{Upper Bound} = \bar{x} + \text{MOE}$.

Imagine you are studying the average daily study time of university students. You collect a random sample of $n = 100$ students. Your sample reveals a mean study time of $\bar{x} = 4.5$ hours with a standard deviation of $\sigma = 1.2$ hours. To find the 95% confidence interval, follow these steps: First, calculate the standard error: $\text{SE} = 1.2 / \sqrt{100} = 1.2 / 10 = 0.12$. Second, compute the margin of error using the critical z-score of $1.96$: $\text{MOE} = 0.12 \times 1.96 = 0.2352$. Third, subtract and add this margin of error to your sample mean. The lower bound is $4.5 - 0.2352 = 4.26$ hours, and the upper bound is $4.5 + 0.2352 = 4.73$ hours. You can be 95% confident that the true average study time for all students falls between 4.26 and 4.73 hours.

Always verify your sample size before running calculations; samples below 30 often require a Student's t-distribution rather than a standard normal z-score for accurate results. Ensure your data collection method minimizes selection bias, as a flawed sample will yield an unrepresentative confidence interval regardless of the math. Finally, remember that a 95% confidence interval does not mean there is a 95% probability that the true population mean falls within that specific interval; rather, it means that if you repeated the study 100 times, roughly 95 of those calculated intervals would contain the true population parameter.