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

I. IntroductionReasons 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

• 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

B : Base

: Quotient

: Remainder

Types of rounding:

• Unbiased rounding: E[R(r)|r] = r

• Sum-unbiased rounding: E[R(r)] = E(r)

x xx q B r

xq

xr( ) ( )x xR x q B R r

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

5

12

B

6

r is assumed to follow a discrete uniform distribution

2. Modified Conventional rounding

Same as conventional rounding, exceptrounding 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

4. Unbiased rounding rule

Round r up with probability r/B

Round r down with probability 1 - r/B

III. Mean and varianceIII.1 Mean and variance of unrounded number

r = 0, 1, 2, 3, . . . B-1.

8

In general,

1( )2

BE r

2 1( )12

BV r

( ) ( | ) ( | )V x V E x q E V x q

9

III. 2 Conventional rounding when B is even

for unrounded number.

1( )2

BE r

22 1

( ) ( )12

BV x B V q

[ ]2BE R r

10

2

[ ]4

BV R r

2 1( ) .12

BV r for unrounded number

2

2( ) ( ) ,4

BV R x B V q and

2

2 1( ) ( ) .4

BMSE R x B V q

11

III.3 Conventional rounding when B is odd

for unrounded number

1[ ] ,2

BE R r

2 1[ ( )] .4

BV R r

2 1( )12

BV r

[ ( )] 3 ( )V R r V r

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

IV. Distance measure

Define

2

2

[ ( ) ]

[ ( ) ]x x

R x xUx

R r rx

1, ,0, .

xx

if r is rounded upotherwise

14

Reexpressing the numerator of U, we have

With conventional rounding with B=10,

Then we have

2( )x xB r

21

0

( )( | ) | ( ).

x

x

x xx

B rE U x x P

x

1 5, 6, . . , 9x xwith r

15

Expected value of U

We define

2( )

( | ) [ | ] ( ) ( ) ( ).x x x

x xq r x x x

q r

B rE E E U x x P P r P q

x

21

10

( )( | ) | [ | ] ( ) ( ).x x

x xr x x x

r x x

B rU E E U x q q P P rq B r

16

IV.1 Conventional rounding with even B

which can be reexpressed as

2 2/ 2 1 1

1 / 21

( ) 1[ ]

x x

B Bx x

x x x xr r B

r B rU

q B r q B r B

2/ 2 1 1

11 / 2

( ) 1[ ]

x x

B Bx

xr r B x

B rU r

r B

2 2/ 2 1 1

1 / 2

( ) 1[ ]

x x

B Bx x

x x x xr r B

r B rq B r q B r B

17

The upper and lower bounds for harmonic series are

The upper bound for the first term of

2/ 2 1 1

11 / 2

( ) 1[ ]

x x

B Bx

xr r xB

B rU r

r B

2 2/ 2 1 1

1 / 2

( ) 1[ ]x x

B Bx x

x xr r B

r B rq B q B B

ln( 1) 1 ln( )nn H n

1U

1ln[

2

2( 1)]

2B

BB

B

1 1 11 2 3 . .nH n

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.

1U

/ 2 1 12 2

21 / 2

1 1[ ][ ( ) ]

x

x x

B B

q x xx r r B

E r B rq B

2 212

BB

19

IV.3 50/50 rounding

The first term of

The second term

IV.4 Unbiased rounding

The first term of

1U

22 3 16

B BB

1ln ( 1)2 2B B

1U 12

B

20

The second term:

IV.5 Comparisons of three rounding rules

Conv 50/50 Unbiased

1st

Term

2nd

Term

1ln[

2

2( 1)]

2B

BB

B

1ln ( 1)2 2B B

12

B

2 212

1

x

BB

Eq

22 3 1 16 x

B B EB q

22 3 1 16 x

B B EB q

2 16

BB

21

Comparisons of three rounding rules

B=10 Conv 50/50 Unbiased

1st term 2.61 11.49 (4.4) 4.5 (1.7)

2nd term .85 2.85 1.65 1

x

Eq

1

x

Eq

1

x

Eq

22

Comparisons of three rounding rules

B=1,000 Conv 50/50 Unbiased

1st term 193.65 3,453.88 (18) 499.5 (2.6)

2nd term 83.33 322.83 166.67

1

x

Eq

1

x

Eq

1

x

Eq

23

IV.6 E(1/q) for log-normal distribution

Lety = ln x

Then, x has a lognormal distribution, i.e.,

2( , )y N

212

ln( )2 1( | , ) ( 2 ) , 0x

f x x e x

24

Let

Then

IV.6 E(1/q) for Pareto distribution of 2nd kind

The Pareto distribution of the second kind is,

21

( | , )c f x dx

21221 1( | 1, , ) [1 ( )]E q e

q c

1( ) , 0, 0a

a

akf q a q kq

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,

11( )( 1)a kE

q c a

1 2 1 21

1

!( , , . . | , , . . )!

ik

qk k ik

ii

i

nf q q q p p p pq

iq

26

Note,

for all i.

Let be the size of the category i and

1 1 3( ) ( ) [ ]

1 ( 1)( 2)i i i i

E E Eq q q q

in

1

1

j

ii

n n n

1

1

1

1

2

11 22

5( 1)( 2) 2( 2) 61( ) [ ]. .

2( 1)( 2) (1 )

i

jj

i

jj

n

n

ki i i

ik

i i

n n p r n pEq q q

n n p r

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

ReferencesGrab, 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