Introduction

No measurement is exact. Every time we make ameasurement 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 graduationson a measuring instrument make it possible to get more precise measurements. The greaterthe number of graduations on theinstrument, the more precise the measurement.

For example,

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

Brian's measurement 21.5 cm implies that themeasurement was carried out to the nearest tenth of a centimeter.

Amy's measurement 21.46 cm implies that themeasurement 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 thetwo.

Let's look at one more example.

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

Scale-1 hadmarking in kg.

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

The weight of theobject on Scale-1 was 67 kg. The measurement of the same weight on Scale-2 was 67.3kg. 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 mustfind out the smallest unit on the instrument.

By definition,

?The Greatest Possible Error in a measurement is half the smallest unitused. ?

Some examples on ?GreatestPossible Error? should help make things clearer.

Example 1

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

Solution:

We have no clue how your friend measured the height.

Let's just trust your friend and go forward.

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

Nothing has been stated about how the measurement isrounded. Let's assume your friend rounded the measurement to the nearest inch. Inthat case, what could be the EXACT height of the table?

If your friend's measurement is right, then any number that would roundoff 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 greatestpossible error is one half of the smallest measuring unit used.? 

The height of the table is 28 inches and thismeasurement is made to the nearest inch. So, the Greatest Possible Error will behalf of one inch which is 0.5 inch.

Example 2

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

Solution:

?The greatest possible error is one half of thesmallest measuring unit used.? 

The length of a stick is stated as 5.4 cm. Thismeasurement is made to the nearest tenth of a centimeter. So, the GreatestPossible 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 thegreatest possible error of the measurement?

Solution:

?The greatest possible error is one half of thesmallest measuring unit used.? 

The length of a window is 16 feet and thismeasurement is made to the nearest foot. So, the Greatest Possible Error willbe 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 themeasurement.

Solution:

?The greatest possible error is one half of thesmallest measuring unit used. ?

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

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

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