Confidence Intervals

The 60-second read.

A confidence interval is the range within which we can expect a true population value (such as the mean) to lie, at a given confidence level, based on a dataset of samples.   The confidence limits are the two endpoints of the interval.  The lower limit is called the Lower Confidence Limit (LCL), and the upper limit is called the Upper Confidence Limit (UCL).  For example, you may have a mean of 101 with a 95% LCL of 95 and a 95% UCL of 106.  So we are 95% sure the true mean is between 95 and 106 based on our sample.

We often use the 95% UCL of the mean to show that we are 95% confident that the site is below an agreed-upon limit.  The 95% UCL is also commonly used in risk assessments as a conservative estimate of the mean concentration of a chemical within an exposure area.  For example, if it has been agreed that soils at a site will be remediated to be below 100 mg/Kg and we sample the site and the 95% UCL of the mean of our samples were 98 mg/Kg, we would be 95% confident that the site mean was below 98 mg/Kg. Therefore, we are below our agreed to cleanup goal.

Note: the 95% UCL is not the same as the 95th percentile of the data, which is the value that 95% of the data fall below, or the 95% prediction limit, which is the limit we can expect the next measurement to be below.  These will be discussed in another article.

The three-minute read.

You might see a report stating:  “The mean concentration of ethyl methyl death of the soils is 100 micrograms per kilogram.”

But what does that really mean? How accurate is that estimate? Is it based on 5, 50, or 500 samples? How much variance was there in the results?

Remember, we almost never know the actual value for the population—it’s too large to measure.  Instead, we take samples of the population and use that data to estimate the population values. 

The graph below shows the distribution of a population.  This population was sampled five times with 10 samples in each sample group and confidence intervals were generated from the five samples.  As you can see:

What Impacts the Confidence Interval the Most?

Two key factors determine how wide the confidence interval will be:

1. The spread (variance) of the dataset (most commonly measured as the standard deviation)

2. The number of observations (the confidence interval decreases with more observations)

We can control the number of observations, but we cannot control the spread or variance in the data.

Let’s look at two examples.

Example 1: Effect of Variance

The following datasets both have a mean of 100, but one (MW-02) has a standard deviation twice as large as the other:

• MW01: 91, 93, 95, 98, 102, 102, 103, 105, 105, 106

• MW02: 84, 86, 89, 100, 101, 102, 105, 107, 111, 113

The 95% confidence intervals for the means are:

• MW01: 97 to 103 (100 ± 3)

• MW02: 94 to 106 (100 ± 6)

The higher variance in MW02 leads to a wider confidence interval.  This is the same as the B and C confidence intervals plotted on the graph.

Example 2: Effect of Sample Size

Now consider two datasets, one with 10 observations and the other with 20.  Both have a mean of 100 and similar standard deviations:

1. MW03: 88, 88, 91, 92, 100, 101, 103, 104, 114, 120

2. MW04: 82, 84, 88, 88, 89, 90, 94, 97, 98, 99, 103, 103, 104, 104, 105, 106, 111, 115, 117, 120

The 95% confidence intervals for the means are:

1. MW03: 94 to 106 (100 ± 6)

2. MW04: 96 to 104 (100 ± 4)

The larger sample size in MW04 results in a narrower confidence interval.

It is important to remember that there are different formulas for the confidence interval depending on the distribution of the sample data.