----

##  Normalizing Data Transformations. ERIC Digest.                               

Data transformations are the application of a mathematical                      modification to the values of a variable. There are a great variety             of possible data transformations, from adding constants to                      multiplying, squaring, or raising to a power, converting to                     logarithmic scales, inverting and reflecting, taking the square root            of the values, and even applying trigonometric transformations such             as sine wave transformations. 
The goal of this Digest is to                   present some of the issues involved in data transformation, with                particular focus on the use of data transformation for normalization            of variables. The Digest is intended to serve as an aid to                      researchers who do not have extensive mathematical backgrounds or               who have not had extensive exposure to this issue. It will focus on             three of the most common data transformations utilized for improving            normality as discussed in texts and the literature: square root,                logarithmic, and inverse transformations. Readers looking for more              information on data transformations might refer to Hartwig and                  Dearing (1979) or Micceri (1989). DATA TRANSFORMATION AND                     NORMALITYMany statistical procedures assume that the variables              are normally distributed. A significant violation of the assumption             of normality can seriously increase the chances of the researcher               committing either a Type I (overestimation) or Type                             II (underestimation) error, depending on the nature of the analysis             and the non-normality.. However, Micceri (1989) points out that true            normality is exceedingly rare in education and psychology. Thus, one            reason (although not the only reason) that researchers utilize data             transformations is to improve the normality of variables.                       Additionally, authors such as Zimmerman (1998) have pointed out that            nonparametric tests (where no explicit assumption of normality is               made) can suffer as much, or more, than parametric tests when                   normality assumptions are violated, confirming the importance of                normality in all statistical analyses, not just parametric                      analyses. 
There are multiple options for dealing with non-normal              data. First, the researcher must make certain that the non-normality            is due to a valid reason (real observed data points). Invalid                   reasons for non-normality include mistakes in data entry and missing            data values not declared missing. 
Not all non-normality is due to             data entry error or nondeclared missing values. Two other reasons               for non-normality are the presence of outliers (scores that are                 extreme relative to the rest of the sample) and the nature of the               variable itself. There is great debate in the literature about                  whether outliers should be removed or not. Judd and                             McClelland (1989) argue that outlier removal is desirable, honest,              and important; Orr, Sackett, and DuBois (1991) are among those                  holding the opposing view. Should a researcher remove outliers and              find substantial non-normality, or choose not to remove outliers,               data transformationis a viable option for improving normality of a              variable. IDENTIFYING A NORMALITY ASSUMPTION VIOLATIONThere               are several ways to tell whether a variable is substantially                    non-normal. While researchers tend to report favoring "eyeballing               the data," or visual inspection (Orr, Sackett, and DuBois, 1991),               researchers and reviewers are often more comfortable with a more                objective assessment of normality, which can range from simple                  examination of skew and kurtosis to examination of P-P                          plots (available through most statistical software packages) and                inferential tests of normality, such as the Kolmorogov-Smirnov test             and adaptations of this test. THREE DATA TRANSFORMATIONWhile              many researchers in the social sciences are well trained in                     statistical methods, not many of us have had significant                        mathematical training, or if we have, it has often been long                    forgotten. This section is intended to give a brief refresher on                what really happens when one applies a data transformation. 
Square            root transformation. Most readers will be familiar with this                    procedure. When one applies a square root transformation, the square            root of every value is taken. However, as one cannot take the square            root of a negative number, if there are negative values for a                   variable, a constant must be added to move the minimum value of the             distribution above 0, preferably to 1.00 (the rationale for this                assertion is explained below). Another important point is that                  numbers of 1.00 and above behave differently than numbers between               0.00 and 0.99. The square root of numbers above 1.00 always become              smaller, 1.00 and 0.00 remain constant, and number between 0.00 and             1.00 become larger (the square root of 4 is 2, but the square root              of 0.40 is 0.63). Thus, if you apply a square root to a continuous              variable that contains values between 0 and 1 as well as above 1,               you are treating some numbers differently than others, which is                 probably not desirable in most cases. 
Log transformation(s).                  Logarithmic transformations are actually a class of transformations,            rather than a single transformation. In brief, a logarithm is the               power (exponent) to which a base number must be raised in order to              get the original number. Any given number can be expressed as y to              the x power in an infinite number of ways. For example, if we were              talking about base 10, 1 is 100, 100 is 102, 16 is 101.2, and so on.            Thus, log10(100)=2 and log10(16)=1.2. However, base 10 is not the               only option for log transformations. Another common option is the               Natural Logarithm, where the constant e (2.7182818) is the base. In             this case the natural log 100 is 4.605. As the logarithm of any                 negative number or number less than 1 is undefined, if a variable               contains values less than 1.0, a constant must be added to move the             minimum value of the distribution, preferably to 1.00. 
There are              good reasons to consider a range of bases. Cleveland (1984) argues              that base 10, 2, and e should always be considered at a minimum. For            example, in cases in which there are extremes of range, base 10 is              desirable, but when there are ranges that are less extreme, using               base 10 will result in a loss of resolution, and using a lower                  base (e or 2) will serve. (Higher bases tend to pull extreme values             in more drastically than lower bases). Readers are encouraged to                consult Cleveland (1984) for more details. 
Inverse transformation.            To take the inverse of a number (x) is to compute 1/x. What this                does is essentially make very small numbers very large, and very                large numbers very small. This transformation has the effect of                 reversing the order of your scores. Thus, one must be careful to                reflect, or reverse the distribution prior to applying an inverse               transformation. To reflect, one multiplies a variable by -1, and                then adds a constant to the distribution to bring the minimum value             back above 1.0. Then, once the inverse transformation is complete,              the ordering of the values will be identical to the original                    data. 
In general, these three transformations have been presented             in the relative order of power (from weakest to most powerful). A               good guideline is to use the minimum amount of transformation                   necessary to improve normality. 
Positive vs. Negative Skew. There             are, of course, two types of skew: positive and negative. All of the            above-mentioned transformations work by compressing the right side              of the distribution more than the left side. Thus, they are                     effective on positively skewed distributions. Should a researcher               have a negatively skewed distribution, the researcher must reflect              the distribution, add a constant to bring it to 1.0, apply the                  transformation, and then reflect again to restore the original order            of the variable. ISSUES SURROUNDING THE USE OF DATA                           TRANSFORMATIONSData transformations are valuable tools, offering            man benefits. However, they should be used appropriately, in an                 informed manner. Too many statistical texts gloss over this issue,              leaving researchers ill-prepared to utilize these tools                         appropriately. All of the transformations examined here reduce                  non-normality by reducing the relative spacing of scores on the                 right side of the distribution more than the scores on the left                 side. 
However, the very act of altering the relative distances                between data points, which is how these transformations improve                 normality, raises issues in the interpretation of the data. If done             correctly, all data points remain in the same relative order as                 prior to transformation. This allows researchers to continue to                 interpret results in terms of increasing scores. However, this might            be undesirable if the original variables were meant to be                       substantively interpretable (e.g., annual income, years of age,                 grade, GPA) because the variables become more complex to interpret              due to the curvilinear nature of the transformations. Researchers               must therefore be careful when interpreting results based on                    transformed data. CONCLUSIONS AND OTHER DIRECTIONSFour                    recommendations for researchers grow out of this discussion: 
1.               Always examine and understand data prior to performing analyses. To             do less is to increase the chance of drawing incorrect                          conclusions. 
2. Know the requirements of the data analysis                    technique to be used. As Zimmerman (1998) and others have pointed               out, even nonparametric analyses, which are generally thought to                be "assumption-free," can benefit from examination of the data. 
3.            Utilize data transformations with care--and never unless there is a             clear reason. Data transformations can alter the fundamental nature             of the data, such as changing the measurement scale from interval or            ratio to ordinal and creating curvilinear relationships, thereby                complicating interpretation. As discussed above, there are many                 valid reasons for utilizing data transformations, including                     improvement of normality, variance stabilization, and conversion of             scales to interval measurement 
4. Ensure that the variable is                 anchored at a place where the transformation will have the optimal              effect. In the case of the three transformations discussed, the                 anchor point should be 1.0. REFERENCESCleveland, W. S. (1984).            Graphical methods for data presentation: Full scale breaks, dot                 charts, and multibased logging. The American Statistician, 38 (4):              270-280. 
Hartwig, F. and Dearing, B.E. (1979). Exploratory Data               Analysis. Newberry Park, CA: Sage Publications, Inc. 
Judd, C. M.              and McClelland, G.H. (1989). Data Analysis: A Model-Comparison                  Approach. San Diego, CA: Harcourt Brace Jovanovich. 
Micceri,                  T. (1989). The unicorn, the normal curve, and other improbable                  creatures. Psychological Bulletin, 105: 156-166. 
Orr, J. M.,                  Sackett, P. R., & DuBois, C. L. Z. (1991). Outlier detection and                treatment in I/O psychology: A survey of  researcher beliefs and an             empirical illustration. Personnel Psychology, 44: 473-                          486. 
Zimmerman, D. W. (1998). Invalidation of parametric and                  nonparametric statistical tests by concurrent violation of two                  assumptions. Journal of Experimental Education, 67:                             55-68. 
----- 
This publication was prepared with funding from the            Office of Education Research and Improvement, U.S. Department of                Education, under contract ED-99-CO-0032. The opinions expressed in              this report do not necessarily reflect the positions or policies or             the U.S. Department of Education. Permission is granted to copy and             distribute this ERIC/AE Digest.                                              **Title:** Normalizing Data Transformations. ERIC Digest.                               **Document Type:** Information Analyses---ERIC Information Analysis Products (IAPs) (071); Information Analyses---ERIC Digests (Selected) in Full Text (073); **Available From:** ERIC Clearinghouse on Assessment and Evaluation, 1129 Shriver                   Laboratory, University of Maryland, College Park, MD 20742. Tel:                800-464-3742 (Toll Free); Web site: http://ericae.net.                       **Descriptors:**  Data Conversion, Research Methodology,  Statistical Analysis,                   Transformations (Mathematics)                                               **Identifiers:** ERIC Digests, Normalizing Transformation,  Variables                         
###
[Return to ERIC Digest Search Page]