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# Uncertainty

Posted by on May. 11, 2013 at 7:38 AM
• 2 Replies

If you roll 1 die, it is totally uncertain which number you'll get between 1 and 6.

If you roll 2 dice and add them together, it is still somewhat uncertain which number you'll get between 2 and 12, but you know you have better odds of getting a 7 than a 2:

If you roll 100 dice and add them all together, you could theoretically get any number between 100 and 600, but your odds of getting 100 exactly are vanishing small (even if all 7 billion people on Earth tried it a trillion times a second, for 20 billion years, the odds of even one of them getting 100 exactly are worse than you simultaneously getting struck by lighting AND getting elected president of the USA).

In fact (being a maths geek, I calculated the above, specifically for this post), you've got a much better than a 95% chance that your total on rolling 100 dice will be somewhere between 300 and 400.

This is what scientists mean by "uncertainty".   To a scientist, it doesn't mean "I don't know what the heck I'm talking about".   To a scientist, uncertainty is something you can quantify and put bounds on.   You can calculate how uncertain you are about something.

You can also add two number together and, if you know how uncertain you are about each number, you can use that to calculate how uncertain you ought to be about the combined total.

by on May. 11, 2013 at 7:38 AM
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by Ruby Member on May. 11, 2013 at 7:39 AM

Here's a basic example:

(source)

#### Combining uncertainties in several quantities: adding or subtracting

When one adds or subtracts several measurements together, one simply adds together the uncertainties to find the uncertainty in the sum.

Dick and Jane are acrobats. Dick is 186 2 cm tall, and Jane is 147 3 cm tall. If Jane stands on top of Dick's head, how far is her head above the ground?

```                        combined height  =  186 cm  +  147 cm

=  333 cm

uncertainty in combined height       =    2 cm  +    3 cm

=    5 cm

combined height  =  333 cm  plus-or-minus  5 cm```

Now, if all the quantities have roughly the same magnitude and uncertainty -- as in the example above -- the result makes perfect sense. But if one tries to add together very different quantities, one ends up with a funny-looking uncertainty. For example, suppose that Dick balances on his head a flea (ick!) instead of Jane. Using a pair of calipers, Dick measures the flea to have a height of 0.020 cm 0.003 cm. If we follow the rules, we find

```                        combined height  =  186 cm  +  0.020 cm

=  186.020 cm

uncertainty in combined height       =    2 cm  +    0.003 cm

=    2.003 cm

???   combined height  =  186.020 cm  plus-or-minus  2.003 cm  ???```

But wait a minute! This doesn't make any sense! If we can't tell exactly where the top of Dick's head is to within a couple of cm, what difference does it make if the flea is 0.020 cm or 0.021 cm tall? In technical terms, the number of significant figures required to express the sum of the two heights is far more than either measurement justifies. In plain English, the uncertainty in Dick's height swamps the uncertainty in the flea's height; in fact, it swamps the flea's own height completely. A good scientist would say

`                        combined height  =  186 cm   plus-or-minus 2 cm`

because anything else is unjustified.

by Ruby Member on May. 11, 2013 at 7:45 AM

When you look at it in more detail, you find that you sometimes don't just add the uncertainty from each variable being added together, because if the errors are negatively correlated they will tend to cancel out.   This makes more sense when you'd adding more than two factors together.   Some of those uncertainly are bound to cancel out, if they are independent of each other.

For further info on the maths of how to do it properly: LINK

Another interesting point is what happens when you subtract two quantities.

If you have

Dick is 186 plus-or-minus 2 cm   (that's an error of just 1%)
Jane is 147 plus-or-minus 3 cm  (that's an error of about 2%)

and want to guess the difference between their two heights, you get:

39 plus-or-minus 5 cm  (which is a whopping 13% error)