95 Confidence Interval with 248 Degrees of Freedom Calculator

This calculator helps you determine a 95% confidence interval for a population mean when you have 248 degrees of freedom. Confidence intervals provide a range of values that are likely to contain the true population mean, with 95% confidence that the interval will contain the population mean.

What is a 95% Confidence Interval?

A 95% confidence interval is a range of values that is likely to contain the true population mean with 95% confidence. It's calculated based on sample data and provides a measure of the precision of your estimate.

When you have 248 degrees of freedom, you're working with a relatively large sample size, which typically results in a narrower confidence interval because you have more information to estimate the population mean accurately.

Confidence intervals are different from confidence levels. A 95% confidence level means that if you took 100 different samples and calculated 95% confidence intervals for each, you would expect approximately 95 of those intervals to contain the true population mean.

Degrees of Freedom in Statistics

Degrees of freedom refer to the number of independent pieces of information available in your data. For a confidence interval calculation, degrees of freedom are typically calculated as:

df = n - 1

Where n is the sample size. With 248 degrees of freedom, your sample size is 249 (248 + 1).

Degrees of freedom affect the shape of the t-distribution used to calculate confidence intervals. With more degrees of freedom, the t-distribution approaches the normal distribution, resulting in slightly narrower confidence intervals.

How to Calculate a 95% Confidence Interval

The formula for a confidence interval using the t-distribution is:

Confidence Interval = Sample Mean ± (t-value × (Sample Standard Deviation / √n))

Where:

Sample Mean (x̄) is the average of your sample data
t-value is the critical value from the t-distribution table for your degrees of freedom and confidence level
Sample Standard Deviation (s) measures the dispersion of your sample data
n is your sample size

For a 95% confidence interval with 248 degrees of freedom, the t-value is approximately 1.965 (this is close to the z-value for a normal distribution, which is 1.96, because with large degrees of freedom the t-distribution approaches the normal distribution).

Worked Example

Let's say you have a sample of 249 measurements (giving you 248 degrees of freedom) with:

Sample Mean (x̄) = 50
Sample Standard Deviation (s) = 10

Using the formula:

Confidence Interval = 50 ± (1.965 × (10 / √249)) = 50 ± (1.965 × 0.218) = 50 ± 0.428 = (49.572, 50.428)

This means you can be 95% confident that the true population mean falls between approximately 49.57 and 50.43.

Interpreting Your Results

When you calculate a 95% confidence interval:

The interval provides a range of plausible values for the population mean
95% of similar intervals would contain the true population mean if you repeated the sampling process many times
A narrower interval indicates more precise estimation (often from a larger sample size)
A wider interval suggests more uncertainty in your estimate

In practical terms, if your confidence interval is (45, 55) and you're testing a hypothesis that the population mean is 50, you might conclude that there's insufficient evidence to reject the hypothesis at the 95% confidence level.

Frequently Asked Questions

What does a 95% confidence interval mean?: It means that if you took 100 different samples and calculated 95% confidence intervals for each, you would expect approximately 95 of those intervals to contain the true population mean.
How does sample size affect the confidence interval?: Larger sample sizes typically result in narrower confidence intervals because you have more information to estimate the population mean accurately. With 248 degrees of freedom, you're working with a relatively large sample size.
What if my data isn't normally distributed?: The t-distribution provides more accurate confidence intervals when your data isn't perfectly normal, especially with smaller sample sizes. With 248 degrees of freedom, the t-distribution is very close to the normal distribution.
Can I use this calculator for other confidence levels?: This calculator specifically calculates 95% confidence intervals. For other confidence levels, you would need to adjust the t-value accordingly.
What if my sample size is different?: The degrees of freedom would change (df = n - 1), and you would need to look up the appropriate t-value for your specific degrees of freedom and confidence level.