Wikijunior:The Book of Estimation/Accuracy and precision

What is precision? What is accuracy? How do we determine the precision and accuracy of something? Read this chapter to find out!

Significant figures

Consider the number 987,654,321.000000001. Let's round it off to a neater number. Say, for example, to the nearest hundred. That's 987,654,300. How accurate is that?

Excluding the last two zeros, there are seven digits. That means the number was rounded to seven significant figures (sig. figs.). A significant figure is a figure that is accurate and important in an estimated number. In the example, we know that the actual last two digits are 2 and 1, so we can be sure that the figure was rounded up to seven sig. figs.

However, sometimes we are asked to find out the number of significant figures without the actual value. If this is the case, we do not know whether the zeros are exact or not. So, we can only write '7, 8 or 9'.

Let's round it off to the nearest hundredth instead. That's 987,654,321.00. This time, there seems to be nine significant figures. Or are there? Let's take a closer look. The last two zeros are exact. That's because the tens and hundreds places are 0 at first. This still holds true when we are asked to name the number of sig. figs. without the exact value, because we assume that unnecessary zeros are always removed.

Here are a couple more examples to illustrate this:

Example 1
Question	What is the number of significant figures in a) 123,456,700 b)123.5001 c) 0.0001000
Solution	a)7, 8 or 9 b) 7 c)1000

Note that the three zeros after the decimal point of c) are not significant figures.

Example 2
Question	Determine whether the following underlined digits are significant figures: a) 9,876,543,200.234 b) 543.2001 c) 0.0004000
Solution	a) Cannot be determined. b) Yes. c) First zero: No. Second zero: Yes.

Sometimes we are asked to round up, down or off a number to a number of sig. figs. This is done the same way as rounding to a certain number of place or a certain digit.

Example 3
Question	Round up, down and off the following numbers to three significant figures: a) 999.99 b) 2103.5 c) 0.000000320
Solution	a) Round up: 1000; Round down: 999; Round off: 1000 b) Round up: 2110; Round down: 2100; Round off: 2100 c) Round up: 0.000000320; Round down: 0.000000320; Round off: 0.000000320;

Notice that the last number is already in three significant figures. There is no need to change it.

Accuracy and precision

Accuracy is how close an estimate is to the true value. Precision is the number of significant figures you give in your estimate. These are two different things. Suppose the true population of a city is 2,432,543. If you estimate the population at 2.4 million, that is reasonably accurate. This will be accurate enough for most purposes. However, it is not very precise, being only to the nearest 100,000. If you estimate the population at 7,754,632, that is very precise. Since the number of people in a city must be a whole number, it could not be more precise. However, it is hugely inaccurate, being over three times too high.

We have just learnt how to measure precision (with significant figures). Accuracy is related to error, which we will discuss later in the book.

Example 4
Question	John won $12,000 in the lottery. Mary reported that John had won $10,000, while Jane reported that John had won $31,242. Who is more accurate? Who is more precise?
Solution	Jane is more precise and Mary is more accurate.

Vocabulary table

• Accuracy
• Precision
• Significant figures
• Most significant figure
• Least significant figure
• Placeholder

Exercises

1. In each of the following numbers, find the number of significant figures and determine if the underlined digits are significant figures.
   1. 123,456,789.1000
   2. 0.00000001234012340
   3. -230.567
2. Round up, down and off the following numbers to the three significant figures.
   1. 1,234,456
   2. -0.00012393546
   3. 345

Answers:

1. This question tests the ability to identify significant figures.
   1. 123,456,789.1000
   2. 0.00000001234012340
   3. -230.567
2. This question tests the ability to round numbers to a certain number of significant figures, rather than to a number of places or a certain digit.
   1. Round up: 1,240,000; Round down: 1,230,000; Round off: 1,230,000
   2. Round up: 0.000123; Round down: 0.000122; Round off: 0.000124
   3. Round up, down and off: 345