Restriction of range
Restriction of range It is important that the researcher sample from the full range of possible values of both variables. If the range of possible values is restricted, the magnitude of the correlation coefficient is reduced. For example, if the range of seating pattern scores is restricted to the first two rows, you will not get an accurate picture of the relationship between seating pattern and exam score. In fact, when only scores of students sitting in the first two rows are considered, the correlation between the two variables is exactly 0.00. With a restricted range comes restricted variability in the scores and thus less variability that can be explained. Figure 12.9 illustrates a scatterplot with the entire range of X values represented and with a portion of those values missing because of restriction of range.
The problem of restriction of range occurs when the individuals in your sample are very similar on the variable you are studying. If you are studying age as a variable, for instance, testing only 6- and 7-year-olds will reduce your chances of finding age effects. Likewise, trying to study the correlates of intelligence will be almost impossible if everyone in your sample is very similar in intelligence (e.g., the senior class of a prestigious private college).
Left scatterplot—positive correlation with entire range of values. Right scatterplot—no correlation with restricted range of values
Curvilinear relationship The Pearson product-moment correlation coefficient (r) is designed to detect only linear relationships. If the relationship is curvilinear, as in the scatterplot shown in Figure 12.10, the correlation coefficient will not indicate the existence of a relationship. The Pearson r correlation coefficient calculated from these data is exactly 0.00, even though the two variables clearly are related.
Because a relationship may be curvilinear, it is important to construct a scatterplot in addition to looking at the magnitude of the correlation coefficient. The scatterplot is valuable because it gives a visual indication of the shape of the relationship. Computer programs for statistical analysis will usually display scatterplots and can show you how well the data fit to a linear or curvilinear relationship. When the relationship is curvilinear, another type of correlation coefficient must be used to determine the strength of the relationship.
Scatterplot of a curvilinear relationship (Pearson product-moment correlation coefficient = 0.00)