Statistical Analysis of Data

Data obtained through measurement always contain random error. Random error is readily observed by sampling—making repeated measurements while all experimental conditions remain the same. For various reasons, the measurements will not all be the same and a frequency distribution (see Fig. 1) is one way to display the resulting data. The height of each vertical bar indicates the number of times a particular measurement value occurs and the value is indicated by the position of the bar along the horizontal axis. The measurements are referred to as a sample set, the number of measurements N is called the sample size, and the frequency distribution is called a sample distribution. Were new sample sets taken, random errors would cause each new sample distribution to vary. However, as the sample size increases to infinity, the sample distribution stabilizes. As N → ∞, the sample distribution converges to the parent distribution—a distribution containing complete statistical information about the particular measurement. Many of the statistical analysis procedures described here will strictly apply only to data whose parent distributions would have the bell shape of the Gaussian distribution. This restriction need not apply rigorously. Results based on the assumption of a Gaussian parent distribution will often be acceptable even for data with non-Gaussian distributions.