Upper and Lower Limit of 95 Confidence Interval Calculator

What is a 95% Confidence Interval?

A 95% confidence interval is a range of values that is likely to contain the true population parameter with 95% probability. For example, if you calculate a 95% confidence interval for the mean height of adults in a country, you can be 95% confident that the true mean height falls within that range.

The width of the confidence interval depends on several factors, including the sample size, the variability of the data, and the desired confidence level. A 95% confidence interval is commonly used because it provides a good balance between precision and reliability.

How to Calculate the Limits

The formula for calculating the upper and lower limits of a 95% confidence interval for a population mean is:

Where:

• Sample Mean is the average of your sample data
• Critical Value is the z-score or t-score that corresponds to your desired confidence level (1.96 for 95% confidence with large samples)
• Standard Error is the standard deviation of your sample divided by the square root of the sample size

The critical value for a 95% confidence interval is typically 1.96 for large samples (n > 30). For smaller samples, you should use the t-distribution critical value.

Interpreting the Results

When you calculate a 95% confidence interval, you can interpret the results as follows:

• If you take many samples from the same population and calculate a 95% confidence interval for each, approximately 95% of these intervals will contain the true population parameter.
• A 95% confidence interval does not mean that there is a 95% probability that the true parameter lies within the interval. Instead, it means that if you were to take many samples and calculate intervals in the same way, 95% of them would contain the true parameter.
• The width of the confidence interval provides information about the precision of your estimate. A narrower interval indicates a more precise estimate.

Worked Example

Let's say you want to estimate the average height of adults in a city. You take a random sample of 50 adults and find that their average height is 170 cm with a standard deviation of 10 cm.

To calculate the 95% confidence interval for the population mean height:

1. Calculate the standard error: 10 / √50 ≈ 1.414 cm
2. Find the critical value: 1.96 (for 95% confidence with large samples)
3. Calculate the margin of error: 1.96 × 1.414 ≈ 2.76 cm
4. Calculate the lower limit: 170 - 2.76 ≈ 167.24 cm
5. Calculate the upper limit: 170 + 2.76 ≈ 172.76 cm