Confidence Interval Calculator
Use this calculator to determine the confidence interval for a population mean, given your sample data and desired confidence level. A confidence interval provides a range of values that is likely to contain the true population mean with a certain level of confidence.
Results:
Enter your sample data and desired confidence level, then click "Calculate" to see the results.
Understanding Confidence Levels and Intervals
In statistics, a confidence interval is a type of interval estimate (of a population parameter) that is computed from the observed data. The confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data.
What is a Confidence Level?
The confidence level represents the long-run frequency of confidence intervals that contain the true value of the population parameter. For example, if you construct 100 confidence intervals with a 95% confidence level, you would expect approximately 95 of those intervals to contain the true population mean.
Key Components of the Calculation:
- Sample Mean (x̄): This is the average of your sample data. It's your best point estimate for the true population mean.
- Sample Standard Deviation (s): This measures the amount of variation or dispersion of your sample data. A larger standard deviation indicates that the data points are spread out over a wider range of values.
- Sample Size (n): This is the number of observations or data points in your sample. Generally, a larger sample size leads to a narrower (more precise) confidence interval.
- Desired Confidence Level: This is the probability that the confidence interval will contain the true population parameter. Common choices are 90%, 95%, and 99%.
How the Calculator Works:
This calculator uses the following formula to determine the confidence interval for a population mean, assuming a sufficiently large sample size (typically n > 30) where the Z-distribution can approximate the t-distribution:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
- Standard Error (SE): Calculated as
s / sqrt(n). It estimates the standard deviation of the sample mean. - Critical Value (Z-score): This value corresponds to your chosen confidence level. For example:
- 90% Confidence Level: Z ≈ 1.645
- 95% Confidence Level: Z ≈ 1.960
- 99% Confidence Level: Z ≈ 2.576
- Margin of Error (ME): This is the product of the Critical Value and the Standard Error. It represents the "plus or minus" amount around the sample mean.
Interpreting the Results:
The calculator will provide a lower bound and an upper bound for your confidence interval. For instance, if you calculate a 95% confidence interval for the average height of students to be [165 cm, 175 cm], you can be 95% confident that the true average height of all students in the population falls within this range.
Example Scenario:
Imagine a researcher wants to estimate the average score of all high school students on a new standardized test. They take a random sample of 150 students and find the following:
- Sample Mean (x̄): 75 points
- Sample Standard Deviation (s): 12 points
- Sample Size (n): 150 students
- Desired Confidence Level: 95%
Using the calculator:
- The critical Z-value for 95% confidence is 1.960.
- Standard Error = 12 / sqrt(150) ≈ 0.9798
- Margin of Error = 1.960 * 0.9798 ≈ 1.9204
- Lower Bound = 75 – 1.9204 = 73.0796
- Upper Bound = 75 + 1.9204 = 76.9204
The 95% confidence interval for the average test score is approximately [73.08, 76.92]. This means the researcher is 95% confident that the true average score of all high school students on this test lies between 73.08 and 76.92 points.