Greatest Possible Error

Introduction

No measurement is exact. Every time we make a measurement a certain degree of error creeps in, however careful we are.

Measurement errors depend on the measuring instrument used and how we use the instrument. The graduations on a measuring instrument make it possible to get more precise measurements. The greater the number of graduations on the instrument, the more precise the measurement.

For example,

Brian and Amy measured the length of the same notebook with two different rulers. Brian measured the length of the notebook to be 21.5 cm and Amys measurement was 21.46 cm. Choose the more precise measurement.

Brians measurement 21.5 cm implies that the measurement was carried out to the nearest tenth of a centimeter.

Amys measurement 21.46 cm implies that the measurement was carried out to the nearest hundredth of a centimeter.

Smaller measurements are more precise.

So 21.46 cm would be a better approximation of the two.

Lets look at one more example.

An object was weighed using two different scales, Scale-1 and Scale-2.

Scale-1 had marking in kg.

Scale-2 was graduated. It had 10 divisions between any two whole numbers on the scale which means each small graduation is equal to 0.1 kg.

The weight of the object on Scale-1 was 67 kg. The measurement of the same weight on Scale-2 was 67.3 kg. Which scale is more precise?

Clearly, Scale-2 is more precise.

Whenever we use a graduated measuring instrument, we must first determine the unit that the instrument is using, and then we must find out the smallest unit on the instrument.

By definition,

The Greatest Possible Error in a measurement is half the smallest unit used.


Some examples on Greatest Possible Error should help make things clearer.

Example 1

Suppose your friend says that the height of her study table is ABOUT 28 inches. Find the greatest possible error of the measurement.

Solution:

We have no clue how your friend measured the height.

Lets just trust your friend and go forward.

Your friend says that the height of the table is ABOUT 28 inches. So, this measure is just an ESTIMATE not the actual measure.

Nothing has been stated about how the measurement is rounded. Lets assume your friend rounded the measurement to the nearest inch. In that case, what could be the EXACT height of the table?

If your friends measurement is right, then any number that would round off to 28 inches would be reasonable.

Any number between 27.5 and 28.5 feet is reasonable for the exact height. Because, any number greater than 27.5 would round up to 28 and any number less than 28.5 would round down to 28.

The exact height could be 28 inches + or 0.5 inch.

Therefore the Greatest Possible Error of the measurement is 0.5 inch.

In other words, by definition,

The greatest possible error is one half of the smallest measuring unit used.

The height of the table is 28 inches and this measurement is made to the nearest inch. So, the Greatest Possible Error will be half of one inch which is 0.5 inch.

Example 2

The length of a stick is stated as 5.4 cm. Find the greatest possible error of the measurement.

Solution:

The greatest possible error is one half of the smallest measuring unit used.

The length of a stick is stated as 5.4 cm. This measurement is made to the nearest tenth of a centimeter. So, the Greatest Possible Error will be one half of 0.1 cm which is 0.05 cm.


Example 3

The length of a window is 16 feet to the nearest foot. What is the greatest possible error of the measurement?

Solution:

The greatest possible error is one half of the smallest measuring unit used.

The length of a window is 16 feet and this measurement is made to the nearest foot. So, the Greatest Possible Error will be half of one foot which is 6 inches. (1 feet = 12 inches)

Example 4

A finger ring weighs 3.75 grams. Find the greatest possible error of the measurement.

Solution:

The greatest possible error is one half of the smallest measuring unit used.

The weight of a finger ring is 3.75 grams and this measurement is made to the nearest hundredth of a gram.

So, the greatest possible error is one half of 0.01 grams which is 0.005 grams.

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