Significant Digits

This tutorial covers significant digits.You will learn about:

1. Significant Digits
2. False Precision

1. Significant Digits

“Significant digits” refers to the level of exactitude or precision that we can talk about with our measurements.

Significant Digits

The number of digits to which a value can reasonably be measured in a real-life scenario, or the number of digits we are able to convey without implying false precision.

IN CONTEXT

Suppose a CEO of a company makes exactly $351,815.84. Oftentimes, the exact number is not required and this salary will be reported as $350,000.

Basically, this is rounded to the nearest $10,000.

Rounding to the nearest 10,000 in a salary is an example of going to two significant digits: using the 300,000's and the 50,000's as the digits and rounding off on the rest, as shown in the table below.

• You could go to three significant digits and round off on the third number. But instead of calling it $351,000, you’re going to say that it's closer to $352,000 than it is to $351,000. That's three significant digits, the 100,000's, the 10,000's, and the 1,000's.
• You could take it to four significant digits or five as well, as the table above shows. Notice that, with 5 significant digits, the 15 is rounded to 20 because 351,815.84 is closer to 351,820 than it is to 351,810.
• You could even take it to six significant digits, all the way down to the dollar amount.

Suppose that a weatherman says there's a 40% chance of rain tomorrow. That's something that you hear quite a bit on the news. But what if you heard the weatherman say, there's a 42.156% chance of rain.

Does that sound believable?

If you heard that estimate, you might look at the TV funny because it would sound like that weatherman knows the probability to a highly precise level. And he probably doesn't. When he conveys the chance of rain, 40% is a reasonable number because he can only be sure to within the nearest 10%.

So you use one significant digit if you can only reasonably report to within 10% of the actual quantity.

If you were able to report to within 1% of the actual amount, then maybe you could have changed this 40% chance of rain to a 42% chance of rain. If you can report to more accurate levels, you can give more significant digits.

When you calculate a statistic, you want to use precise values, but precise values that are reasonable. You can round when you need to.

2. False Precision

There are many calculations that you might do that will require intermediate steps. For example, calculating a standard deviation has many steps. During calculations, use very exact values all the way through and then round only at the end, not during the middle steps.

Here is a list of the heights of each member of the Chicago Bulls basketball team, for which there is a calculation of the standard deviation. The last step demonstrated here is taking the square root of 14⅔.

Doing that calculation results in a value with many digits. Keeping all these digits is a good thing. But once you’re at the end of the problem, you’re going to turn this value back into inches.

Can you honestly say that the standard deviation is 3.82970843102535 inches? Is it fair to believe that you can measure reasonably down to the hundred-trillionth of an inch?

Including this degree of precision may be an example of false precision.

False Precision

Including too many figures when reporting a value, implying that the value can, in fact, be measured that precisely.

It is not reasonable to imply that you can measure exactly to the hundred-trillionth of an inch, or even that you can measure accurately to the thousandth of an inch. Even a hundredth of an inch may be too small. It's probably best to go with 3.8 as our standard deviation for our answer.

We can reasonably measure down to tenths of an inch, but maybe not even to hundredths of an inch. So two significant digits is probably all we're going to need in this particular problem. It will be up to you to make such a judgment call about what you think you can measure precisely.

Significant digits are tools that we use to determine the precision. This is a way of relaying the precision with which you're making your measurement to a reader. The more digits that you can put in, the more exact you claim the measurement to be.

If you don't think that you can claim a measurement to be highly exact, you're not going to include many digits. Indeed, it's dangerous to report digits beyond what you can reasonably measure, because it results in false precision, where you're implying that you can be incredibly precise when you just can't based on the way that you can measure things.

Thank you and good luck!