Normal Distribution Two Thirds Confidence Interval Calculator

A two-thirds confidence interval for a normal distribution is a range of values that is likely to contain the true population mean with 66.7% probability. This calculator helps you determine this interval based on your sample data.

What is a Two-Thirds Confidence Interval?

A two-thirds confidence interval is a statistical range that provides an estimate of the true population mean. When we say we have a 66.7% confidence interval, it means that if we were to take many samples and calculate the interval for each, approximately 66.7% of those intervals would contain the true population mean.

This type of interval is less common than 95% or 99% confidence intervals, but it can be useful in certain situations where a higher level of confidence isn't necessary or practical.

Note: The two-thirds confidence level corresponds to a z-score of approximately ±0.47 in a standard normal distribution.

How to Calculate the Two-Thirds Confidence Interval

The formula for calculating a two-thirds confidence interval for a normal distribution is:

Confidence Interval = Sample Mean ± (z × (Sample Standard Deviation / √Sample Size))

Where z is the z-score for the two-thirds confidence level (approximately 0.47)

To calculate the interval:

Calculate the sample mean (average of your sample data)
Calculate the sample standard deviation (measure of how spread out the data is)
Determine the sample size (number of data points in your sample)
Use the formula above to calculate the margin of error
Subtract and add the margin of error to your sample mean to get the confidence interval

The result will be a range of values that represents the two-thirds confidence interval for your sample data.

Interpreting the Results

When you calculate a two-thirds confidence interval, you're essentially saying that there's a 66.7% probability that the true population mean falls within this range. This means:

If you took many samples and calculated the interval for each, about 66.7% of those intervals would contain the true population mean
The remaining 33.3% of intervals would not contain the true population mean
The width of the interval depends on your sample size and the variability in your data

It's important to note that this doesn't mean there's a 66.7% chance that the true mean is within your specific interval. Instead, it's a statement about the method's reliability over many repetitions.

For practical purposes, a two-thirds confidence interval is less conservative than a 95% interval but more conservative than a 90% interval.

Worked Example

Let's say you have a sample of 30 measurements with a mean of 50 and a standard deviation of 5. To calculate the two-thirds confidence interval:

Sample Mean = 50
Sample Standard Deviation = 5
Sample Size = 30
z-score for two-thirds confidence = 0.47
Margin of Error = 0.47 × (5 / √30) ≈ 0.47 × 0.91 ≈ 0.43
Lower Bound = 50 - 0.43 ≈ 49.57
Upper Bound = 50 + 0.43 ≈ 50.43

The two-thirds confidence interval would be approximately 49.57 to 50.43. This means we're 66.7% confident that the true population mean falls within this range.

Example Calculation Details
Statistic	Value
Sample Mean	50
Sample Standard Deviation	5
Sample Size	30
z-score (two-thirds confidence)	0.47
Margin of Error	0.43
Lower Bound	49.57
Upper Bound	50.43

Frequently Asked Questions

What does a two-thirds confidence interval mean?

A two-thirds confidence interval means that if you were to take many samples and calculate the interval for each, approximately 66.7% of those intervals would contain the true population mean.

How is the two-thirds confidence interval different from other confidence intervals?

The two-thirds confidence interval is less conservative than a 95% interval but more conservative than a 90% interval. It provides a middle ground between these two common confidence levels.

When would I use a two-thirds confidence interval instead of a 95% interval?

You might use a two-thirds confidence interval when you need a less conservative estimate than 95% but more conservative than 90%. This could be appropriate in situations where you want to balance precision with confidence.

What factors affect the width of the two-thirds confidence interval?

The width of the interval is affected by the sample size and the variability in your data. Larger samples with less variability will result in narrower intervals.