N16: apply and interpret limits of accuracy, including upper and lower bounds

What is an error interval?

What is an error interval?

The limits of accuracy when a number has been rounded or truncated. They are the range of possible values that a number could have been before it was rounded or truncated.

How can an error interval be shown?

As an inequality using inequality symbols (≠, <, ≤, > or ≥)

What do the inequality symbols mean?

• ≠ means 'not equal to'

• < means 'less than'

• > means 'greater than'

• ≤ means 'less than or equal to'

• ≥ means 'greater than or equal to'

What are upper bounds?

The smallest value that would round up to the next estimated value.

What are lower bounds?

The smallest value that would round up to the estimated value.

Work out the upper and lower bounds (error interval) of 57.7

Step 1: find the size of the interval (in this case 1 decimal place, 0.1)

Step 2: divide the interval by two to get half of the interval (0.1/2=0.05)

Step 3: calculate the lower bound by subtracting half of the interval (57.7-0.05=57.65)

Step 4: calculate the upper bound by adding half of the interval (57.7+0.05=57.75

Step 5: write the error interval (57.65≤x<57.75)

Why does the upper bound end in 5?

Although it would round up to a different answer, in reality, it is more accurate than using 4 as the ending, or even 4.9999 because it has no end

Maximum and minimum values of a calculation

Upper and lower bounds are used to find the maximum and minimum values the calculation can be. To find the minimum and maximum area, always use the smallest values (lower bounds). However in speed for instance, sometimes the upper bound will create a smaller overall result - as speed=distance/time, when calculating the minimum value you would use the upper bound for time and the lower bound for distance and vice versa for the maximum value

A motorbike travels a distance of 110m to the nearest 10 metres, in a time of 5 seconds to the nearest second. Calculate the maximum and minimum values for the speed, s, in metres per second (m/s), using the formula s=d/t

Error interval for distance: 105≤distance<115 (10/2=5, 110-5=105, 110+5=115)

Error interval for time: 4.5≤time<5.5 (1/2=0.5, 5-0.5=4.5, 5+0.5=5.5)

Minimum speed: 105/5.5=19.09

Maximum speed: 115/4.5=25.56

Error interval: 19.09m/s≤speed<25.56m/s

What is truncation?

Getting rid of one part of a number (a form of always rounding down)

How to truncate 9.876 to 1 decimal place

Step 1: determine the cut-off point (after the 1st decimal place, between the 8 and the 7)

Step 2: ignore all numbers after the cut-off point

Answer=9.8

The weight of a dog has been truncated to 402.3 ounces to 1dp. Work out the interval within which w, the weight of the dog, lies.

In order for some number to be truncated to 402.3, it must have began 402.3 followed by some other digits, such as 402.31. As long as the number begins "402.3" then we're fine.

In other words, anything that is greater than or equal to 402.3 but less than 402.4. We express this error interval like

402.3 ≤ weight < 402.4