Confidence Intervals

Whenever we estimate a value, there is room for error. This results in an interval or range of values in which we believe the true value lies.

In statistics, these ranges are known as Confidence Intervals.

For instance, we may estimate that the temperature will be between 70 and 80 degrees tomorrow.

All confidence intervals have the following formula:

Statistic +/- Margin of Error

Every confidence interval has a low end and a high end. In our example, the low end is 70 and the high end is 80.

The statistic you are estimating is exactly in the middle. In our temperature example, our statistic aka point estimate would be 75.

The error that we are leaving room for in our estimate is known as our Margin of Error. The margin of error in our example would be 5 degrees.

Statistic/Point Estimate

The statistic or point estimate is the specific value that you are trying to predict. This is often an average or mean, but can also be a proportion, or even standard deviation.

You can find a list of confidence interval formulas for different types of statistics here.

Margin of Error

The margin of error is added to the estimate to find the high end of the interval.

The margin of error is subtracted from the estimate to find the low end of the interval.

The Margin of Error is calculated using the following formula:

Margin of Error = Confidence Level * Standard Error

The two components of Margin of Error are Confidence Level and Standard Error.

Confidence Level

We must choose how confident we want to be about our results when calculating our margin of error.

As the confidence level increases, our margin of error increases (making our interval wider and therefore less precise/reliable).

For example, we can say that we are 75% confident that the temperature will be between 70 and 80 degrees tomorrow. However, we can say with more certainty that we are 99% confident that the temperature will be between 0 and 100 degrees.

The larger our confidence, the larger our margin of error. The larger our margin of error, the less accurate our results will be.

This relationship is also evident in the formula for Confidence Intervals.

Most of the time, people use a confidence level of 95%. However, often confidence levels of 80%, 90%, or 99%, and others are used.

The value plugged in for Z (or sometimes T) in the equation is not equal to the confidence level percentage. Rather, it is the equivalent "critical value" or probability for obtaining the confidence level percentage.

The critical values for Z*, or confidence levels, can be found on a Z table or using a Statistics Calculator.

The most common critical values for the Z distribution are listed here:

The critical values for T-distribution can be found using a Student's T-Distribution Table or a statistics calculator.

To use the table you may need to know your alpha value which is 1 - Confidence Level. For example, the alpha for a 95% confidence level is .05.

Standard Error

The Standard Error is calculated by dividing the standard deviation (s) by the sample size (n).

Sample Size

As the sample size increases, the standard error becomes smaller (the bigger value in the denominator of the fraction makes the term smaller).

Therefore, as the sample size increases, the margin of error becomes smaller, allowing for more a more precise confidence interval.

The makes sense, as the larger your sample, the more accurate your results will be.

Summary

Confidence intervals are an important estimation tool used in statistics. There is a tradeoff between confidence (reliability) and error (accuracy/precision). The more confident you are about your results, the larger and less accurate your estimate range will be.

If you want to narrow down on a smaller and more accurate range of possible values, you'll have to decrease your confidence level or increase your sample size.