1. 1 WP.31 Effects of Rounding on Data Quality Jay J. Kim, Lawrence H. Cox, Myron Katzoff, Joe Fred Gonzalez, Jr. U.S. National Center for Health Statistics

2. 2 I.Introduction Reasons for rounding Rounding noninteger values to integer values for statistical purposes; To enhance readability of the data; To protect confidentiality of records in the file; To keep the important digits only.

3. 3 Purpose: Evaluate the effects of four rounding methods on data quality and utility in two ways : (1) bias and variance; (2) effects on the underlying distribution of the data determined by a distance measure.

4. 4 B : Base : Quotient : Remainder Types of rounding: Unbiased rounding: E[R(r)|r] = r Sum-unbiased rounding: E[R(r)] = E(r)

5. 5 II. Four rounding rules 1. Conventional rounding Suppose r = 0, 1, 2,...,9 (=B-1). If r (B/2), round r up to 10 (=B) else round down r to zero (0). If B is odd, round r up when r

6. 6 r is assumed to follow a discrete uniform distribution 2. Modified Conventional rounding Same as conventional rounding, except rounding 5 (B/2) up or down with probability ½. 3. Zero-restricted 50/50 rounding Except zero (0), round r up or down with probability ½.

7. 7 4.Unbiased rounding rule Round r up with probability r/B Round r down with probability 1 - r/B III. Mean and variance III.1 Mean and variance of unrounded number r = 0, 1, 2, 3,... B-1.

8. 8 In general,

9. 9 III. 2 Conventional rounding when B is even for unrounded number.

10. 10

11. 11 III.3 Conventional rounding when B is odd for unrounded number

12. 12 Modified conventional rounding, 50/50 rounding and unbiased rounding have the same mean, variance and MSE as the conventional rounding with odd B.

13. 13 IV. Distance measure Define

14. 14 Reexpressing the numerator of U, we have With conventional rounding with B=10, Then we have

15. 15 Expected value of U We define

16. 16 IV.1 Conventional rounding with even B which can be reexpressed as

17. 17 The upper and lower bounds for harmonic series are The upper bound for the first term of

18. 18 The second term of Note the second term of E(U) is, IV.2 Modified conventional rounding with even B Has the same E(U) as the conventional rounding.

19. 19 IV.350/50 rounding The first term of The second term IV.4 Unbiased rounding The first term of

20. 20 The second term: IV.5Comparisons of three rounding rules Conv 50/50 Unbiased 1 st Term 2 nd Term

21. 21 Comparisons of three rounding rules B=10 Conv 50/50 Unbiased 1 st term 2.61 11.49 (4.4) 4.5 (1.7) 2 nd term.85 2.85 1.65

22. 22 Comparisons of three rounding rules B=1,000 Conv 50/50 Unbiased 1 st term 193.65 3,453.88 (18) 499.5 (2.6) 2 nd term83.33 322.83 166.67

23. 23 IV.6E(1/q) for log-normal distribution Let y = ln x Then, x has a lognormal distribution, i.e.,

24. 24 Let Then IV.6 E(1/q) for Pareto distribution of 2nd kind The Pareto distribution of the second kind is,

25. 25 where k is the minimum value of q and c is the cumulative probability from 1. IV.7 Upper limit for E(1/q) for multinomial distribution The multinomial distribution has the form = 0,1,2,

26. 26 Note, for all i. Let be the size of the category i and

27. 27 V.Concluding comments Various methods of rounding and in some applications various choices for rounding base B are available. The question becomes: which method and/or base is expected to perform best in terms of data quality and preserving distributional properties of original data and, quantitatively, what is the expected distortion due to rounding? This paper provides a preliminary analysis toward answering these questions

28. 28 References Grab, E.L & Savage, I.R. (1954), Tables of the Expected Value of 1/X for Positive Bernoulli and Poisson Variables, Journal of the American Statistical Association 49, 169-177. N.L. Johnson & S. Kotz (1969). Distributions in Statistics, Discrete Distributions, Boston: Houghton Mifflin Company. N.L. Johnson & S. Kotz (1970). Distributions in Statistics, Continuous Univariate Distributions-1, New York: John Wiley and Sons, Inc. Kim, Jay J., Cox, L.H., Gonzalez, J.F. & Katzoff, M.J. (2004), Effects of Rounding Continuous Data Using Rounding Rules, Proceedings of the American Statistical Association, Survey Research Methods Section, Alexandria, VA, 3803-3807 (available on CD). Vasek Chvatal. Harmonic Numbers, Natural Logarithm and the Euler-Mascheroni Constant. See www.cs.rutgers.edu/~chvatal/notes/harmonic.html www.cs.rutgers.edu/~chvatal/notes/harmonic.html