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

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.
2

• 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.
3

B : Base x
 q B x
 r x
x
: Quotient r x
: Remainder
( )
 q B x

( x
)
Types of rounding:
• Unbiased rounding: E[R(r)|r] = r
• Sum-unbiased rounding: E[R(r)] = E(r)
4

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

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.
6

4. Unbiased
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.
7


B

1
2

B
2 
1
12
In general,
( )


( | )
 
( | )

8

V x
 2
( )

B
2 
1
12
III. 2 Conventional rounding when B is even
[
 
]

B
2

B

1
2 for unrounded number.
9

[
 
]

B
2
4

B
2 
1
12 for unrounded number .

( )
 
( )

B
2
,
4 and

( )


( )

B
2
4

1
.
10

III.3 Conventional rounding when B is odd
 
]

B
2

1
,

B
2
4

1
.

B
2 
1
12 for unrounded number
[ ( )]

3 V r
11

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

IV.
U

[ ( )
 x ]
2
 x
R r x
 r x
]
2 x
Define
 x


1,
0, if r is rounded up x
, otherwise .
13

Reexpressing the numerator of U, we have
(
 x
 x
)
2
With conventional rounding with B=10,
  x
1 with r x

5, 6, . . , 9
Then we have
E
 x
( | )
 x

1 

0


(
 x x
 x
)
2
|
 x P (
 x
).
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Expected value of U

 q r




   q x r x
 x
[
(
 x x
 x
)
2 x P
 x
P r P q x x
We define
U
1
 r


( | ) |
    r x
 x
1

0
[
(
 x
 x q B r x
 x
)
2 x
 x x
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IV.1 Conventional rounding with even B
U
1

[
B
 r x

1 r x
2 q B
 r x

B

1
 r x

B / 2
( x

 x
) x
2
]
1 q B r B which can be reexpressed as
U
1

[
B
 r x

1 r x

B

1
 r x

B / 2
(
 x r x
2
) 1
]
B

[
B
 r x

1 r x
2 q B
 r x

B

1
 r x

B / 2
( B x
 r
 x
) x
2
]
1 q B r B
16

U
1


[
B
 r x

1
[
B r x


1 r x
 r x
B



1
B / 2
(
 x r x
2
) 1
]
B r x
2

B

1
 r B x

/ 2
(
 x
2
) 1 x
] q B B
The upper and lower bounds for harmonic series
H n
1 2

1 
3

1 
. .
 n

1 are ln( n 1) H n
1 ln( )
The upper bound for the first term of U
1
B ln[
2( B

1)
]
B

2

B

1
2
17

B
2 
2
12 B
Note the second term of E(U) is,
E q x
1
[ ][ q
B x r x


1 r x
2 
B

1
 r x

B / 2
(
 x
1
B
2
IV.2 Modified conventional rounding with even B
Has the same E(U) as the conventional rounding.
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IV.3 50/50 rounding
The first term of U
1
B ln ( B 1)
2
1
2
The second term
2 B
2 
3 B

1
6 B
IV.4 Unbiased rounding
The first term of U
1 B

1
2
19

The second term:
B
2 
1
6 B
IV.5 Comparisons of three rounding rules
Conv 50/50 Unbiased
1 st
Term
B ln[
2( B

1)
B

2
]

B

1
2
B ln ( B 1)
2
1
2
B

1
2
2 nd
Term
B
2 
12 B
2
E

1

 
 q x

2 B 2 
6
3
B
B

1
E


 q
1 x



2 B
2 
6
3
B
B

1
E


1 q x


20

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

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 term 83.33
E

1

 q x

322.83
E

1

  q x
166.67
E

1

  q x
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IV.6 E(1/q) for log-normal distribution y N
 
2
)
Let y = ln x
Then, x has a lognormal distribution, i.e., f x
  2   
2
 x )

1 e

1
2
( ln x

 
 
) 2
, 0
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Let c
 
1
 f x
  2
) dx
Then
E
1
( | q q

1,
  2
)

1 c e
1
2
 
2   
[1
 
(
  

 

)]
IV.6 E(1/q) for Pareto distribution of 2nd kind
The Pareto distribution of the second kind is,
 ak a q a

1
, a

0, 0
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E
1
( )

1 q

( a k

1) 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
( ,
1 2
, . .
q k
| ,
1 2
, . .
p k
)
 n !
i k


1 q i
!
i k


1 p i q i q i
= 0,1,2,
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Note,
E (
1
)

E ( q i q i
1

1
)

E [
3
( q i

1)( q i

2)
] for all i.
Let be the size of the category i and i n
  n
 i j

1


1 n i
1
E ( q q
2
. .
q k
)
 i k


1
[
5( n

1)( n

2) p r i

1  j

1 n j
2( n

1)( n

2) p i
2

2( n

2) p i

6
(1
 r i i

1  j

1 n j
)
]
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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
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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
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