Where is the individual in statistics?

January/February 2003, Volume 57, Number 1 If there is any single statistic that we practitioners should know so deeply that it is integral to our view of the world, it is the standard deviation. This statistic is a beautiful one for evidence-based practice and, well, just beautiful in general. The reason for its beauty is that it depicts variation among individuals. Our practice is with individuals. We design our intervention plans around the individual. When we use research findings to inform practice, we need to know how individuals vary in their responses during evaluation procedures, and to know how they vary in their responses to intervention. The standard deviation, as well as other measures of variation such as the range of scores, provides us direct and useful information about individuality and uniqueness against the backdrop of general human tendencies. Although the standard deviation is commonly reported in research reports, my experience is that readers often overlook it and fail to recognize its importance. The standard deviation can validate our humane and individualized approach to service and can transform that service. Before I describe how this statistic can do these things, let me refresh your knowledge about what it is. You can find the mathematical equation for the standard deviation in any introductory statistics or quantitative research methods textbook (e.g., Portney & Watkins, 1999; Rosenthal & Rosnow, 1991). The look of this equation is perhaps the primary reason that people do not immediately see the beauty of the statistic so it is not shown here. Instead I describe its logic as it is calculated by hand in its most simple and intuitively understandable form. In essence, the standard deviation is an average of difference scores. The first in the four steps of the calculation of the standard deviation is to calculate a difference score for each individual in the group: the individual’s score on a test or assessment minus the group’s mean score. This calculation describes how far the score of the single individual deviates (or differs or varies) from the group as a whole. The group’s mean score is a measure of central tendency in that it is the central score around which each individual’s score differs to some degree. Other measures of central tendency are the median (middle score) and mode (most frequent score) but these are not used in the calculation of the standard deviation. The second step in the calculation of the standard deviation is to square each individual’s difference score. This step makes all of these scores positive in sign. A difference score that is positive, because the individual’s original score was higher than the mean of the group, remains positive when squared, while a difference score that is negative, because of an original score lower than the mean, becomes positive. The reason for making all difference scores positive becomes clear in the third step of the calculation. At this third step, the variance is calculated. It is calculated by adding together the individuals’ squared difference scores and then dividing by the number of individuals.1 If we had not squared the difference scores in the second step of the calculation, the sum of the difference scores would be equal or nearly equal to 0. It so happens that the scores for many client attributes are distributed in the clinical population symmetrically around the group mean: half of the scores fall below the mean, and half above. The positive and negative sign difference scores would cancel each other out when added together and we would have no useful measure of the average deviation of individuals’ scores from the mean. The fourth and final step is to take the square root of the variance. This final step creates the standard deviation, which is on a numerical scale that is the same as the individual’s original score on the test or assessment. A standard deviation of 5 tells us that on the average, individuals’ scores differ from the group mean by 5 points. If the group mean is 25, and the standard deviation is 5, then individual scores tend to fall between 20 and 30. The standard deviation becomes a useful tool when we understand its relationship to what is called the normal distribuEVIDENCE-BASED PRACTICE FORUM Where is the Individual in Statistics?