Confidence Interval Calculator
Results:
Enter values and click "Calculate" to see the confidence interval.
Understanding Confidence Levels and Intervals
In statistics, a confidence level expresses the probability that a given confidence interval will contain the true population parameter. It's a measure of how confident we are that our interval estimate contains the population mean (or proportion, etc.). Common confidence levels are 90%, 95%, and 99%.
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 a more informative estimate than a single point estimate, as it also conveys the precision of the estimate.
Key Components for Calculating a Confidence Interval for a Mean:
- Sample Mean (x̄): This is the average value of your observations from the sample. It's your best point estimate for the true population mean.
- Standard Deviation (s or σ): This measures the amount of variation or dispersion of a set of values. If the population standard deviation (σ) is known, it's used. More commonly, especially with larger sample sizes (n > 30), the sample standard deviation (s) is used as an estimate for σ.
- Sample Size (n): The number of individual observations or data points in your sample. A larger sample size generally leads to a narrower (more precise) confidence interval.
- Confidence Level: The desired probability that the interval will contain the true population parameter. This level determines the critical value (Z-score or t-score) used in the calculation.
How the Confidence Interval is Calculated (Z-Interval for Mean):
The formula for a confidence interval for a population mean, especially when the sample size is large (n > 30) or the population standard deviation is known, is:
Confidence Interval = Sample Mean ± (Z-score * Standard Error)
Where:
- Standard Error (SE) = Standard Deviation / √Sample Size
- Z-score is the critical value corresponding to your chosen confidence level. For common confidence levels:
- 90% Confidence Level: Z = 1.645
- 95% Confidence Level: Z = 1.96
- 99% Confidence Level: Z = 2.576
The term (Z-score * Standard Error) is known as the Margin of Error. It represents the maximum expected difference between the sample mean and the true population mean.
Example Scenario:
Imagine a quality control manager wants to estimate the average weight of a new batch of product. They take a random sample of 100 items (Sample Size) and find the average weight to be 500 grams (Sample Mean) with a standard deviation of 15 grams (Standard Deviation). They want to calculate a 95% confidence interval for the true average weight of the entire batch.
Using the calculator above:
- Sample Mean: 500
- Standard Deviation: 15
- Sample Size: 100
- Confidence Level: 95%
The calculator would yield:
- Z-score (for 95%): 1.96
- Standard Error: 15 / √100 = 15 / 10 = 1.5
- Margin of Error: 1.96 * 1.5 = 2.94
- Lower Bound: 500 – 2.94 = 497.06
- Upper Bound: 500 + 2.94 = 502.94
Interpretation: We are 95% confident that the true average weight of the product batch lies between 497.06 grams and 502.94 grams.
Why Use Confidence Intervals?
Confidence intervals are crucial because they acknowledge that a sample mean is only an estimate and will vary from sample to sample. They provide a range that is likely to contain the true population parameter, giving a more complete picture than a single point estimate alone. They are widely used in research, quality control, and policy-making to quantify uncertainty.