end. We are actually interested in knowing the true value of variable X, and we make replicate measurements in order to decrease the influence of random errors on our estimation of this true value.
Using the term ‘estimation’ does not imply that one is guessing the value. Instead ‘estimation’ refers to the fact that measurements estimate the true value, but measurement errors of some type are
almost always present.
Even if we were interested in the 100 individual values of X, we face the problem that a short or prolonged examination of a list of numbers provides minimal insight, because the human mind cannot easily comprehend a large quantity of numbers simultaneously. What we really care about is usually the essence or basic properties of the dataset, in particular:
- what is the average value?
- what is the scatter?
- are these data consistent with a theoretically predicted value for X?
- are these data related to another variable, Y?
With caution, each of the first three questions can be described with a single number, and that is the subject of this chapter. The engaging question of relationship to other variables is discussed in Chapter 3.
Histograms
The 100 measurements of X are more easily visualized in a histogram than in a tabulation of numbers. A histogram is a simple binning of the data into a suite of adjacent intervals of equal width. Usually one picks a fairly simple histogram range and interval increment. For example, our 100 measurements range from a minimum of -2.41 to a maximum of 2.20, so one might use a plot range of -2.5 to 2.5 and an interval of 0.5 or 1.0, depending on how many data points we have. The choice of interval is arbitrary but important, as it affects our ability to see patterns within the data. For example, Figure 1 shows that for the first 20 values of this dataset:
- an interval of 0.2 is too fine, because almost every data point goes into a separate bin. The eye tends to focus mainly on individual data points rather than on broad patterns, and we cannot easily see the relative frequencies of values.
- an interval of 0.5 or 1.0 is good, because we can see the overall pattern of a bell-shaped distribution without being too distracted by looking at each individual point.
