Confidence Interval Calculator

What is a Confidence Interval?

In statistics, a confidence interval (CI) is a range of values, derived from sample data, that is likely to contain the value of an unknown population parameter. Because it's often impossible to survey an entire population, we take a sample and use its statistics (like the mean) to estimate the true parameter of the population. However, a sample is just a snapshot and is unlikely to perfectly represent the whole population. The confidence interval accounts for this sampling error by providing a range of plausible values for the true parameter, rather than just a single number (a point estimate).

A confidence interval is associated with a confidence level, which is typically expressed as a percentage (e.g., 95%). A 95% confidence level means that if we were to take 100 different samples from the same population and construct a confidence interval for each sample, we would expect about 95 of those intervals to contain the true population parameter.

Key Components of a Confidence Interval

A confidence interval is constructed from three main parts:

• Point Estimate: This is the best guess for the population parameter, calculated from the sample data. For example, the sample mean (x̄) is the point estimate for the population mean (μ).
• Margin of Error: This is the "plus or minus" part of the interval that is added to and subtracted from the point estimate. It quantifies the uncertainty of our estimate and determines the width of the confidence interval. A larger margin of error results in a wider interval, suggesting more uncertainty.
• Confidence Level: This represents the probability that the interval produced will contain the true parameter. Common choices are 90%, 95%, and 99%. The choice of confidence level involves a trade-off: a higher confidence level creates a wider, more certain interval, but it is less precise.

The general formula for a confidence interval is:

Confidence Interval = Point Estimate ± Margin of Error

The margin of error itself is calculated as:

Margin of Error = Critical Value × Standard Error

How to Use the Confidence Interval Calculator

Our calculator simplifies the process of finding the confidence interval for a population mean. Here’s a step-by-step guide:

1. Enter the Sample Mean (x̄): This is the average of your collected data sample.
2. Enter the Standard Deviation (σ or s): Provide the standard deviation of your data. Use the population standard deviation (σ) if it is known. If not, use the sample standard deviation (s). Our calculator handles both cases.
3. Enter the Sample Size (n): Input the total number of observations in your sample. A larger sample size generally leads to a narrower, more precise confidence interval.
4. Choose the Confidence Level: Select your desired confidence level, typically 90%, 95%, or 99%. This determines the critical value (either a z-score or t-score) used in the calculation.

The calculator will then compute the margin of error and display the lower and upper bounds of the confidence interval for you.

Practical Example

Let's say a researcher wants to estimate the average height of a certain species of plant. They take a random sample of 49 plants and find the sample mean height to be 30 cm. From previous studies, the population standard deviation is known to be 3.5 cm. The researcher wants to calculate a 95% confidence interval for the true average height of this plant species.

• Sample Mean (x̄): 30 cm
• Population Standard Deviation (σ): 3.5 cm
• Sample Size (n): 49
• Confidence Level: 95%

For a 95% confidence level, the critical value (z-score) is 1.96.

First, calculate the Standard Error (SE):

SE = σ / √n = 3.5 / √49 = 3.5 / 7 = 0.5

Next, calculate the Margin of Error (ME):

ME = Critical Value × SE = 1.96 × 0.5 = 0.98

Finally, construct the Confidence Interval:

CI = x̄ ± ME = 30 ± 0.98

Lower Bound: 30 - 0.98 = 29.02 cm

Upper Bound: 30 + 0.98 = 30.98 cm

Interpretation: The researcher can be 95% confident that the true average height of this plant species lies somewhere between 29.02 cm and 30.98 cm.