When we have 50 or 100 measurements instead of 20, we find that a finer histogram-binning interval is better for visualizing the pattern of the data. Figure 2 shows that an interval of about 0.5 is best for 100 measurements of this data type.
Normal Distribution
The data shown in Figures 1 and 2 have what is called a normal distribution. Such a distribution is formally called a Gaussian distribution or informally called a bell curve. The normal distribution has both a theoretical and an empirical basis. Theoretically, we expect a normal distribution whenever some parameter or variable X has many independent, random causes of variation and several of these so-called ‘sources of variance’ have effects of similar magnitude. Even if an individual type of error is non-normally distributed, groups of such errors are. Empirically, countless types of measurements in all scientific fields exhibit a normal distribution. Yet we must always verify the assumption that our data follow a normal distribution. Failure to test this assumption is scientists’ most frequent statistical pitfall. This mistake is needless, because one can readily examine a dataset to determine whether or not it is normally distributed.
Mean and Standard Deviation
For any dataset that follows a normal distribution, regardless of dataset size, virtually all of the information is captured by only three variables:
N: the number of data points, or measurements;
X: the mean value; and
σ: the standard deviation.
The mean (X), also called the arithmetic mean, is an average appropriate only for normal distributions. The mean is defined as:
