Normal distribution

The normal distribution (or bell curve or Gaussian distribution) is a type of data spread that is encountered frequently in radiology and in other sciences.

Data that are normally distributed can be evaluated using parametric statistics. When data are not normally distributed (e.g. skewed, or multimodal), then nonparametric statistics are appropriate. Skewed data sets are also often encountered in radiology.

With normally distributed data:

• the mean and variance do not depend on each other
• mean, median, and mode are the same
• skew = 0

Experimental data may approach a normal distribution without satisfying these ideal conditions.

Standard deviation

The standard deviation (σ) is an expression of how a population or sample data is spread about the mean value (its "variance")

σ = sqrt[Σ(X - Xbar) / N]

• X: individual data point
• Xbar: arithmetic mean of the data
• Σ(X - Xbar): the sum of all the differences between individual data points and the mean value
• N: total number of data points
• sqrt: square root

Although the formula looks complex, what it's trying to express is relatively straightforward... it's trying to express all the variation around the mean by adding the differences together, and the more data points there are, the less deviation there is. The squaring and square root operation accounts for the positive and negative variation around the mean, trying to keep the total non-negative (the sum of all variance would be zero).

Technically, "σ" refers to a sample's standard deviation, while "s" refers to a population's standard deviation.

Central limit theorem

The normal distribution is also useful when sampling data out of a non-normal data set. If the number in the sample is large enough (n = 25-30), then the sample will be normally distributed, even though the population is not.