C1.03: Rounding

Section 2. Rounding.

Suppose we want to make a graph to summarize the heights of a class of 50 people. And we have their heights measured to the nearest tenth of an inch. The purpose of making this graph is to get a feeling for the variability of their heights. So the numbers as measured are more accurate than we really need – heights to the nearest inch would be adequate and easier to handle. Here is a portion of the dataset.

Original 61.3 68.5 71.4 65.8 64.3 63.4 67.2 72.3 69.5 70.1 62.8 63.7 65.2
Rounded 61 69 71 66 64 63 67 72 70 70 63 64 65

The usual rule for rounding is that, when the part you will drop is less than half, you “go down” and when the part you will drop is equal to or more than half, you “go up.”

In a later section in this Topic, we will also learn to “think backwards” to see the interval of actual values that are consistent with a particular rounded value.

Example 1. Round to the nearest hundredth.

  1. (a) 3.14738 goes to 3.15
  2. (b) 0.73372 goes to 0.73
  3. (c) 0.0032 goes to 0.00

Example 2. Round to the nearest ten.

  1. (a) 817 goes to 820
  2. (b) –1123 goes to –1120
  3. (c) 74.567 goes to 70

Example 3. Important idea: Sequential rounding does not always give correct results. You must do all the necessary rounding in one step in order to obtain correct results.

Problem: Round 64.7 to the nearest ten.

Correct Solution: The nearest ten means that the answer must either be 60 or 70. Since 4.7 is less than half of ten, then when we round to the nearest ten, we have 60.

Incorrect solution: If we round 64.7 to the nearest one first, it is between 64 and 65, and 0.7 is more than half, so it rounds to 65.   Then if we round that result to the nearest ten, it is between 60 and 70 and 5 is half of ten, so we round this to 70.

Notice that we did not obtain the same answer by these two solution methods. Yet, clearly, 64.7 is closer to 60 than to 70, so the first solution method must be correct.   This illustrates the incorrect answer that sometimes arises if we do our rounding in sequential steps rather than all in one step. 

Discussion. What about one-half?

Bookkeepers have noticed that, if you systematically round all numbers to the nearest dollar, rounding half-dollars up, and then take sums of those rounded numbers to estimate the sums of the original values, those estimates are a bit too high to be accurate. The problem is that there is some non-symmetry in the rounding rule. All those less than half “go down” and all those more than half “go up.” So far that’s symmetric. The problem is that the ones that are exactly half all go the same direction, which is “up.” So that’s not symmetric. So the rounded values are, on the average, overall, just a little bit higher than the original numbers.

Going deeper. In situations where dealing with one-half in a non-symmetric manner might be a problem, a more sophisticated rounding rule is adopted. See the course web pages for additional discussion of more sophisticated rounding rules.