Rounding Methods

There are many ways to round numbers ...

Firstly, what is "Rounding" ?

Rounding means reducing the digits in a number while trying to keep its value similar. The result is less accurate, but easier to use.

Example: 7.3 rounds to 7

Because 7.3 is closer to 7 than to 8

(Note: we round to whole numbers here, but you can round to tens, tenths, etc)

But what about 7.5? Is it closer to 7 or closer to 8? It is half-way in between, so what should we do?

Half Round Up (the common method of rounding)

The common method of rounding is to make 0.5 go up, so 7.5 rounds up to 8 (see Rounding Numbers).

7.5 usually rounds up to 8

But this is not a law or anything, it is just what people normally agree to do, and we get this:

  • 7.6 rounds up to 8
  • 7.5 rounds up to 8
  • 7.4 rounds down to 7

Half Round Down

But 5 can go down if we want. In that case 7.5 rounds down to 7, and we get this:

  • 7.6 rounds up to 8
  • 7.5 rounds down to 7
  • 7.4 rounds down to 7

But we should always let people know we are using "Half Round Down".

Why make 0.5 go down? Maybe there are lots of 0.5's in our numbers and we want to see what rounding down does to our results.

Have a Play ... try different rounding methods on the Rounding Tool.

Negative Numbers

But what about -7.5?

  • Does it round to -8 (and is that going "up" or "down" ?),
  • Or does it round to -7

Help! I am confused!

In fact the whole world is confused about rounding negative numbers ... some computer programs round -7.5 to -8, others to -7

But we can agree here that "up" means heading in a positive direction, like on this number line:

Half Round Up (including negative numbers)

So we get this:

  • 7.6 rounds up to 8
  • 7.5 rounds up to 8
  • 7.4 rounds down to 7
  • -7.4 rounds up to -7
  • -7.5 rounds up to -7
  • -7.6 rounds down to -8

Half Round Down (including negative numbers)

When we round 0.5 down we get this:

  • 7.6 rounds up to 8
  • 7.5 rounds down to 7
  • 7.4 rounds down to 7
  • -7.4 rounds up to -7
  • -7.5 rounds down to -8
  • -7.6 rounds down to -8

 

"Symmetric" Rounding

But maybe you think "7.5 rounds up to 8, so -7.5 should go to -8", which is nice and symmetrical.

Well you are in luck because that is rounding towards or away from zero:

Round Half Away From 0

For this method, 0.5 rounds the number so it is further away from zero, like this:

  • 7.6 rounds away to 8
  • 7.5 rounds away to 8
  • 7.4 rounds to 7
  • -7.4 rounds to -7
  • -7.5 rounds away to -8
  • -7.6 rounds away to -8

Round Half Towards 0

Or we can have 0.5 round the number closer to zero, like this:

  • 7.6 rounds away to 8
  • 7.5 rounds to 7
  • 7.4 rounds to 7
  • -7.4 rounds to -7
  • -7.5 rounds to -7
  • -7.6 rounds away to -8

But Being Consistent Can Be Bad

Choosing any of those methods can be bad, though!

Especially if we have lots of 0.5s, like 5.5, 7.5, 6.5, 9.5, etc. All the rounding goes in the same direction and we get a bias.

How can we stop the rounding being all one direction?

We can decide to round towards even (or odd) numbers, or we can just choose randomly.

Round to Even

We round 0.5 to the nearest even digit

Example:

7.5 rounds up to 8 (because 8 is an even number)

but 6.5 rounds down to 6 (because 6 is an even number)

Other numbers (not ending in 0.5) round to nearest as usual, so:

  • 7.6 rounds up to 8
  • 7.5 rounds up to 8 (because 8 is an even number)
  • 7.4 rounds down to 7
  • 6.6 rounds up to 7
  • 6.5 rounds down to 6 (because 6 is an even number)
  • 6.4 rounds down to 6
  • etc

Round to Odd

Just like "Round To Even", but 0.5 heads towards odd numbers

Example:

7.5 rounds down to 7 (because 7 is an odd number)

but 6.5 rounds up to 7 (because 7 is an odd number)

Round Randomly

We could also choose to round 0.5 up or down randomly, but how? By tossing a coin? Or a computer function?

With a large list of numbers this can give good results, but also gives a different answer each time (unless we use a fixed list of random choices).

Floor and Ceiling

There are two other methods that don't even consider 0.5. They are called Floor and Ceiling.

Floor gives us the nearest integer down (and ceiling goes up).

Example: What is the floor and ceiling of 2.31?

Floor and Ceiling function

The Floor of 2.31 is 2
The Ceiling of 2.31 is 3

Floor

Using "floor", all digits go down, no matter what the dropped digit is:

Example: 7.8 goes down to 7

so does 7.2, 7.5, 7.9, etc.

And 7 goes to 7, too.

Ceiling

And "ceiling" goes up:

Example: 7.1 goes up to 8

so does 7.2, 7.5, 7.8, etc.

But 7 stays at 7.

 

Summary

Number Half
Up
Half
Down
Half
Away 0
Half
To 0
Half
Even
Half
Odd
Floor Ceiling
 8 8 8 8 8 8 8 8 8
 7.6 8 8 8 8 8 8 7 8
 7.5 8 7 8 7 8 7 7 8
 7.4 7 7 7 7 7 7 7 8
 7 7 7 7 7 7 7 7 7
-7 -7 -7 -7 -7 -7 -7 -7 -7
-7.4 -7 -7 -7 -7 -7 -7 -8 -7
-7.5 -7 -8 -8 -7 -8 -7 -8 -7
-7.6 -8 -8 -8 -8 -8 -8 -8 -7
-8 -8 -8 -8 -8 -8 -8 -8 -8

 

Rounding to Tens, Tenths, Whatever ...

In our examples we rounded to a whole number, but you can round to tens, or tenths, etc:

Example: "Half Round Up" to tens:

15 rounds up to 20

14.97 rounds down to 10

Example: "Half Round Up" to hundredths:

0.5168 rounds up to 0.52

1.41119 rounds down to 1.41