Key points
- Measurements (for example, time, length, capacityThe volume of a material (typically liquid or air) held in a vessel or container. and temperature) are not exact. The accuracyAccuracy is how close a measured value is to the actual (true) value. that they are measured to depends on the equipment that is being used and how important it is for those measurements to be precise.
- Measurements can be slightly smaller or larger than recorded. The difference between the actual measurement and the recorded value is a rounding errorThe difference between a rounded off numerical value and the actual value..
- Rounding errors can be expressed using two inequality symbols (< >)The symbol > is used when a value is greater than another, eg 23 > 5. The symbol < is used when a value is less than another, eg 11< 40 symbols to represent the lowest and highest possible values.
- When rounding to the nearest one, divide the degree of accuracy by two. This provides the error interval of 0.5 either side of the whole number.
- Understanding rounding decimals and also significant figures will help with rounding errors.
Why rounding errors can vary
Rounding errors can vary in size depending on the accuracy of the equipment used to measure the value.
Events can be measured to a different degree of accuracy, depending on their importance. For example, an Olympic Games 100 m race would be measured to one hundredth of a second using specialist equipment. Whereas a school sports day 100 m race might only be measured to the nearest second using a simple stopwatch.
Whilst a different type of timer is used to measure both events, the time taken can only be recorded to a set degree of accuracy. This means the actual time could be slightly different, and the error intervalAn error interval gives the range of possible values that a number could have been before it was rounded. would vary.
Understanding rounding decimals will help with rounding errors.
Examples
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Question
A junior swimmer completes a 50m backstroke race in 34鈭6 seconds to the nearest one tenth of a second (nearest 0鈭1 seconds). What would the rounding error be?
When rounding to the nearest one tenth of a second (0鈭1), divide the degree of accuracy by two.
This provides the rounding error interval of 0鈭05 seconds either side of the time recorded.
The rounding error for 34鈭6 seconds is 0鈭05 seconds.
Meaning the time taken is between 34鈭55 and 34鈭65 seconds.
Using inequality symbols to express rounding errors
Rather than writing in full sentences, mathematicians often use symbols. The > symbol can be used instead of writing a value is greater than (or larger than).
Using two inequality symbols (< >)The symbol > is used when a value is greater than another, eg 23 > 5. The symbol < is used when a value is less than another, eg 11< 40 symbols can describe a value that is between two numbers eg, greater than one amount but smaller than the other.
Measured values can be slightly bigger or smaller than the actual amount. This means they can result in rounding errors. Rounding errors can be expressed using two inequality symbols to represent the lowest and highest possible values.
Examples
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Question
If 鈥榮鈥 stands for seconds, what does 120 鈮 s < 125 mean?
A The number of seconds is greater than 120 but less than 125
B The number of seconds is greater than 120 but less than or equal to 125
C The number of seconds is greater than or equal to 120 but less than or equal to 125
D The number of seconds is greater than or equal to 120 but less than 125
D The number of seconds is greater than or equal to 120 but less than 125
Practise rounding errors
Practise rounding errors in this quiz.
Quiz
Real-world maths
Rounding errors are important to show the precision of a measurement. This is important in manufacturing because no measuring device can be 100% accurate.
Using error intervals as part of a quality assurance process allow a manufacturer to ensure their business is efficient.
A manufacturer will have a degree of tolerance (error interval) that they decide is acceptable to them.
If the weight of their product is below their lowest error interval, customers are likely to complain and potentially ask for a refund.
If the weight of the product is higher than their highest error interval, their profit margins decrease as the cost of materials are higher than necessary.
Game - Divided Islands
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