Understanding the Confidence Interval Calculator

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. Instead of providing a single number estimate, it gives an upper and lower bound. This calculator helps you find that range for either a population mean or a population proportion.

How to Use the Calculator

1. Select the Parameter: Choose whether you want to calculate a confidence interval for a 'Mean' or a 'Proportion'.
2. Enter Your Data:
   • For a Mean, you need the sample mean (x̄), the sample standard deviation (s), and the sample size (n).
   • For a Proportion, you need the number of "successes" (x) in your sample and the total sample size (n).
3. Choose a Confidence Level: Select a confidence level from the dropdown. A 95% confidence level is the most common, meaning that if you were to repeat the experiment many times, 95% of the calculated intervals would contain the true population parameter.
4. Calculate: Click the "Calculate" button to see the results, which include the margin of error and the final confidence interval range.

The Formulas Behind the Calculation

This calculator uses Z-scores from the standard normal distribution to determine the critical value for the selected confidence level. This is a common and robust method, especially for larger sample sizes (n > 30).

Confidence Interval for a Mean

The formula to calculate the confidence interval for a population mean is:

CI = x̄ ± Z * (s / √n)

• CI is the confidence interval.
• x̄ is the sample mean.
• Z is the critical value from the Z-distribution corresponding to the confidence level.
• s is the sample standard deviation. If you need help calculating this, you might use a standard deviation calculator.
• n is the sample size.

The term Z * (s / √n) is known as the Margin of Error.

Confidence Interval for a Proportion

The formula to calculate the confidence interval for a population proportion is:

CI = p̂ ± Z * √[p̂(1 - p̂) / n]

• p̂ (p-hat) is the sample proportion, calculated as x / n.
• x is the number of successes.
• n is the sample size.
• Z is the Z-score for the chosen confidence level.

Similarly, the term Z * √[p̂(1 - p̂) / n] is the Margin of Error for the proportion.

Practical Example

Imagine a researcher wants to estimate the average height of a certain type of plant. They measure 100 plants (n=100) and find a sample mean height of 30 cm (x̄=30) with a sample standard deviation of 4 cm (s=4). They want to find the 95% confidence interval for the true average height of all such plants.

Using the calculator:

1. Select 'Mean'.
2. Enter Sample Mean = 30, Sample SD = 4, and Sample Size = 100.
3. Select a 95% confidence level.

The calculator finds a margin of error of 0.784 and a confidence interval of (29.216, 30.784) cm. This means the researcher can be 95% confident that the true average height of the entire plant population lies between 29.216 cm and 30.784 cm.

Related Calculators

Statistics is a broad field with many interconnected tools. If you found this calculator useful, you might also be interested in other calculators in our Math Calculators section.