**Confidence intervals** are often used in radiology literature to express the variability of an experimental result. They are usually reported as the upper and lower bound of variability (upper,lower) for your mean value, with x% certainty ^{1}.

If 95%, it means that if the study were redone many times, 95% of the time the experimental result would fall between the upper and lower bounds ^{1,2}. An important, and often misunderstood, feature of a confidence interval, is that it does not purport to establish the certainty of the real value in the population. Rather it expresses upper and lower bounds between which the measured value in a sample of the whole population will lie with a certain confidence (typically 95%) ^{2}.

#### Example

The size of the pancreatic duct is measured in a cohort of volunteers after the administration of secretin. The average (mean) value of pancreatic duct diameter is 6.6 mm. What is the 95% confidence interval?

- the confidence interval will range above and below your experimental mean value, so you need exclude the 2.5% of the values outlying above and 2.5% below.
- next you have to find the z score of that point on a normally distributed curve which would be 47.5% of the data below it and 2.5% of the data above it (and the other way around, as well). For us, 1.96 is the value of our z score. See Figures 1 and 2.
- we also need the standard error of the mean of our data. Let's use 0.5 mm.
- using this information, we can find the lower bound: 6.6 - (1.96 x 0.5) = 5.62
- we can also find the upper bound: 6.6 + (1.96 x 0.5) = 7.58

So our end result is a sample value of "6.6 +/- 0.98 with a 95% confidence interval". It could also be stated as "(5.62,7.58) with a 95% confidence interval".

The formula for constructing a confidence interval is:

###### CI_{95} = (sample mean) +/- (z_{α/2} x (σ / √n))

- z
_{α/2}: the z score which bounds the % confidence interval you want... in our case 0.05 / 2 = 0.025... which with a z transformation is 1.96 - σ: sample standard deviation (same as population standard deviation)
- √n: the square root of your sample size
- (σ / √n): the standard error of the mean, SE(M)