Error Intervals

What you'll learn

How a given answer can represent a range of possible original values.
How to find the smallest boundary and largest boundary for that range.
How to write the range using inequality signs such as ≤ and <, which compare sizes.
How to handle digits after the decimal point (decimal places), counted important digits (significant figures), and truncation, which means cutting digits off.

The basic idea

Rounding means changing a number to a nearby simpler value. For example, 6.83 rounded to 1 decimal place is 6.8.

But if you are only told the rounded answer is 6.8, the original number might have been 6.76, 6.82, 6.849, and so on. That is why we use an error interval.

Definition

Key words

A variable such as $x$ or $y$ is a letter that stands for an unknown number.
An error interval is the range of values that could have given the rounded or truncated result.
The lower bound is the smallest boundary value.
The upper bound is the largest boundary value.

For ordinary rounding, the lower bound is included, but the upper bound is not included. This is because the upper boundary would round up to the next answer.

A number line shows that rounding to 6.8 includes values from 6.75 up to but not including 6.85.

Key Idea

Rounding uses halfway points

For ordinary rounding, go halfway down and halfway up from the rounded value. Write the interval as $\le x < b$ .

Correct to the nearest whole unit

If a measurement is correct to the nearest kg, cm, or m, it has been rounded to the nearest whole unit.

The step size is the gap between one rounded answer and the next. For nearest kg, nearest cm, or nearest metre, the step size is 1 whole unit. Half of 1 is 0.5.

Example

Measurement correct to the nearest kg

A sack has a mass of 18 kg, correct to the nearest kg. Find the lower and upper bounds.

The mass 18 kg correct to the nearest kg represents all masses from 17.5 kg up to but not including 18.5 kg.

The rounded value is 18 kg.
The step size is 1 kg, because the mass was rounded to the nearest kg.
Half of 1 kg is 0.5 kg.
Go 0.5 kg below and 0.5 kg above 18 kg.
The lower bound is 17.5 kg and the upper bound is 18.5 kg.
If $m$ is the mass in kg, the error interval is $\le m < 18.5$ .

Tip

Least and greatest wording

If a question asks for the least and greatest boundary values, give the lower bound and upper bound. In a full error interval, the upper bound still uses <, not ≤.

Rounding to decimal places

A decimal place is a digit after the decimal point. For example, 4.7 has 1 decimal place, and 4.72 has 2 decimal places.

When rounding to decimal places, look at the value of the last rounded digit.

Tip

Useful half-steps

Rounded to 1 decimal place: step size 0.1, so half-step 0.05.
Rounded to 2 decimal places: step size 0.01, so half-step 0.005.
Rounded to 3 decimal places: step size 0.001, so half-step 0.0005.

Example

Rounded to 1 decimal place

A number $y$ is rounded to 1 decimal place. The result is 9.6. Write down the error interval for $y$ .

Rounding to 1 decimal place uses a half-step of 0.05 on either side of 9.6.

Rounding to 1 decimal place means the step size is 0.1.
Half of 0.1 is 0.05.
Subtract 0.05 from 9.6 to get the lower bound: $9.6 - 0.05 = 9.55$ .
Add 0.05 to 9.6 to get the upper bound: $9.6 + 0.05 = 9.65$ .
The error interval is $\le y < 9.65$ .

Example

Rounded to 2 decimal places

A number $x$ is rounded to 2 decimal places. The result is 1.34. Write down the error interval for $x$ .

Rounding to 2 decimal places uses a half-step of 0.005 on either side of 1.34.

Rounding to 2 decimal places means the step size is 0.01.
Half of 0.01 is 0.005.
The lower bound is $1.34 - 0.005 = 1.335$ .
The upper bound is $1.34 + 0.005 = 1.345$ .
The error interval is $\le x < 1.345$ .

Common Mistake

Using ≤ on both sides

Do not write $\le x \le 1.345$ for normal rounding. The upper boundary is not included, because 1.345 would round to 1.35, not 1.34.

Rounding to significant figures

A significant figure is an important digit in a number, counted from the first non-zero digit. Significant figures are often shortened to s.f.

For example, in 0.472, the significant figures are 4, 7, and 2. The zero before the decimal point does not count as significant.

To find an error interval for significant figures:

Count to the last significant figure.
Find its place value.
Use half of that place value.

Tip

Do not confuse s.f. with d.p.

For significant figures, count from the first non-zero digit. For decimal places, count digits after the decimal point.

Example

A small decimal rounded to 3 significant figures

A number $x$ is rounded to 3 significant figures. The result is 0.472. Write down the error interval for $x$ .

The third significant figure in 0.472 is in the thousandths column, so the interval is half a thousandth either side.

Count the significant figures: 4 is first, 7 is second, 2 is third.
The third significant figure is in the thousandths column.
The step size is 0.001, so the half-step is 0.0005.
The lower bound is $0.472 - 0.0005 = 0.4715$ .
The upper bound is $0.472 + 0.0005 = 0.4725$ .
The error interval is $\le x < 0.4725$ .

Example

A large number rounded to 1 significant figure

A number $x$ is rounded to 1 significant figure. The result is 7000. Write down the error interval for $x$ .

For 7000 to 1 significant figure, the last significant digit is in the thousands column, giving a half-step of 500.

The first significant figure is 7.
The 7 is in the thousands column.
The step size is 1000, so the half-step is 500.
The lower bound is 7000 minus 500, which is 6500.
The upper bound is 7000 plus 500, which is 7500.
The error interval is $\le x < 7500$ .

Truncation

Definition

Truncation

Truncation means cutting off extra digits without rounding up.

This is different from rounding. If a positive number is truncated to 1 decimal place and the result is 6.2, the original number starts with 6.2 but is less than 6.3.

Truncation starts at the displayed value and goes up to the next possible value, rather than going halfway down.

Example

Truncated to 2 decimal places

A number $x$ is truncated to 2 decimal places. The result is 4.73. Write down the error interval for $x$ .

A number truncated to 4.73 starts at 4.73 and can approach 4.74 but cannot include it.

Truncating to 2 decimal places means the number has been cut off after the hundredths digit.
The smallest possible value is exactly 4.73.
The next 2-decimal-place value is 4.74.
So $x$ must be less than 4.74.
The error interval is $\le x < 4.74$ .

Common Mistake

Treating truncation like rounding

For truncation, do not go halfway down and halfway up. A number truncated to 4.73 starts at 4.73 and goes up to, but not including, 4.74.

Exam technique

In the exam

Underline whether the question says rounded, correct to the nearest, or truncated.
Find the step size: nearest whole unit, 1 decimal place, 2 decimal places, or the place value of the last significant figure.
For rounding, go half a step down and half a step up.
For truncation, start at the given value and go up to the next possible value.
Write the final interval with ≤ on the lower bound and < on the upper bound.

Self review

Check yourself

If a length is 23 cm correct to the nearest cm, what are the two boundary values?
A number rounds to 5.8 to 1 decimal place. What half-step should you use?
How is truncating to 2 decimal places different from rounding to 2 decimal places?

Error Intervals

What you'll learn

How a given answer can represent a range of possible original values.
How to find the smallest boundary and largest boundary for that range.
How to write the range using inequality signs such as ≤ and <, which compare sizes.
How to handle digits after the decimal point (decimal places), counted important digits (significant figures), and truncation, which means cutting digits off.

The basic idea

Rounding means changing a number to a nearby simpler value. For example, 6.83 rounded to 1 decimal place is 6.8.

But if you are only told the rounded answer is 6.8, the original number might have been 6.76, 6.82, 6.849, and so on. That is why we use an error interval.

Definition

Key words

A variable such as $x$ or $y$ is a letter that stands for an unknown number.
An error interval is the range of values that could have given the rounded or truncated result.
The lower bound is the smallest boundary value.
The upper bound is the largest boundary value.

For ordinary rounding, the lower bound is included, but the upper bound is not included. This is because the upper boundary would round up to the next answer.

A number line shows that rounding to 6.8 includes values from 6.75 up to but not including 6.85.

Key Idea

Rounding uses halfway points

For ordinary rounding, go halfway down and halfway up from the rounded value. Write the interval as $\le x < b$ .

Correct to the nearest whole unit

If a measurement is correct to the nearest kg, cm, or m, it has been rounded to the nearest whole unit.

The step size is the gap between one rounded answer and the next. For nearest kg, nearest cm, or nearest metre, the step size is 1 whole unit. Half of 1 is 0.5.

Example

Measurement correct to the nearest kg

A sack has a mass of 18 kg, correct to the nearest kg. Find the lower and upper bounds.

The mass 18 kg correct to the nearest kg represents all masses from 17.5 kg up to but not including 18.5 kg.

The rounded value is 18 kg.
The step size is 1 kg, because the mass was rounded to the nearest kg.
Half of 1 kg is 0.5 kg.
Go 0.5 kg below and 0.5 kg above 18 kg.
The lower bound is 17.5 kg and the upper bound is 18.5 kg.
If $m$ is the mass in kg, the error interval is $\le m < 18.5$ .

Tip

Least and greatest wording

If a question asks for the least and greatest boundary values, give the lower bound and upper bound. In a full error interval, the upper bound still uses <, not ≤.

Rounding to decimal places

A decimal place is a digit after the decimal point. For example, 4.7 has 1 decimal place, and 4.72 has 2 decimal places.

When rounding to decimal places, look at the value of the last rounded digit.

Tip

Useful half-steps

Rounded to 1 decimal place: step size 0.1, so half-step 0.05.
Rounded to 2 decimal places: step size 0.01, so half-step 0.005.
Rounded to 3 decimal places: step size 0.001, so half-step 0.0005.

Example

Rounded to 1 decimal place

A number $y$ is rounded to 1 decimal place. The result is 9.6. Write down the error interval for $y$ .

Rounding to 1 decimal place uses a half-step of 0.05 on either side of 9.6.

Rounding to 1 decimal place means the step size is 0.1.
Half of 0.1 is 0.05.
Subtract 0.05 from 9.6 to get the lower bound: $9.6 - 0.05 = 9.55$ .
Add 0.05 to 9.6 to get the upper bound: $9.6 + 0.05 = 9.65$ .
The error interval is $\le y < 9.65$ .

Example

Rounded to 2 decimal places

A number $x$ is rounded to 2 decimal places. The result is 1.34. Write down the error interval for $x$ .

Rounding to 2 decimal places uses a half-step of 0.005 on either side of 1.34.

Rounding to 2 decimal places means the step size is 0.01.
Half of 0.01 is 0.005.
The lower bound is $1.34 - 0.005 = 1.335$ .
The upper bound is $1.34 + 0.005 = 1.345$ .
The error interval is $\le x < 1.345$ .

Common Mistake

Using ≤ on both sides

Do not write $\le x \le 1.345$ for normal rounding. The upper boundary is not included, because 1.345 would round to 1.35, not 1.34.

Rounding to significant figures

A significant figure is an important digit in a number, counted from the first non-zero digit. Significant figures are often shortened to s.f.

For example, in 0.472, the significant figures are 4, 7, and 2. The zero before the decimal point does not count as significant.

To find an error interval for significant figures:

Count to the last significant figure.
Find its place value.
Use half of that place value.

Tip

Do not confuse s.f. with d.p.

For significant figures, count from the first non-zero digit. For decimal places, count digits after the decimal point.

Example

A small decimal rounded to 3 significant figures

A number $x$ is rounded to 3 significant figures. The result is 0.472. Write down the error interval for $x$ .

The third significant figure in 0.472 is in the thousandths column, so the interval is half a thousandth either side.

Count the significant figures: 4 is first, 7 is second, 2 is third.
The third significant figure is in the thousandths column.
The step size is 0.001, so the half-step is 0.0005.
The lower bound is $0.472 - 0.0005 = 0.4715$ .
The upper bound is $0.472 + 0.0005 = 0.4725$ .
The error interval is $\le x < 0.4725$ .

Example

A large number rounded to 1 significant figure

A number $x$ is rounded to 1 significant figure. The result is 7000. Write down the error interval for $x$ .

For 7000 to 1 significant figure, the last significant digit is in the thousands column, giving a half-step of 500.

The first significant figure is 7.
The 7 is in the thousands column.
The step size is 1000, so the half-step is 500.
The lower bound is 7000 minus 500, which is 6500.
The upper bound is 7000 plus 500, which is 7500.
The error interval is $\le x < 7500$ .

Truncation

Definition

Truncation

Truncation means cutting off extra digits without rounding up.

This is different from rounding. If a positive number is truncated to 1 decimal place and the result is 6.2, the original number starts with 6.2 but is less than 6.3.

Truncation starts at the displayed value and goes up to the next possible value, rather than going halfway down.

Example

Truncated to 2 decimal places

A number $x$ is truncated to 2 decimal places. The result is 4.73. Write down the error interval for $x$ .

A number truncated to 4.73 starts at 4.73 and can approach 4.74 but cannot include it.

Truncating to 2 decimal places means the number has been cut off after the hundredths digit.
The smallest possible value is exactly 4.73.
The next 2-decimal-place value is 4.74.
So $x$ must be less than 4.74.
The error interval is $\le x < 4.74$ .

Common Mistake

Treating truncation like rounding

For truncation, do not go halfway down and halfway up. A number truncated to 4.73 starts at 4.73 and goes up to, but not including, 4.74.

Exam technique

In the exam

Underline whether the question says rounded, correct to the nearest, or truncated.
Find the step size: nearest whole unit, 1 decimal place, 2 decimal places, or the place value of the last significant figure.
For rounding, go half a step down and half a step up.
For truncation, start at the given value and go up to the next possible value.
Write the final interval with ≤ on the lower bound and < on the upper bound.

Self review

Check yourself

If a length is 23 cm correct to the nearest cm, what are the two boundary values?
A number rounds to 5.8 to 1 decimal place. What half-step should you use?
How is truncating to 2 decimal places different from rounding to 2 decimal places?

Error Intervals

What you'll learn

The basic idea

Correct to the nearest whole unit

Rounding to decimal places

Rounding to significant figures

Truncation

Recap questions

You've reached the end

Error Intervals

What you'll learn

The basic idea

Correct to the nearest whole unit

Rounding to decimal places

Rounding to significant figures

Truncation

Recap questions

You've reached the end

Error Intervals

What you'll learn

The basic idea

Correct to the nearest whole unit

Rounding to decimal places

Rounding to significant figures

Truncation

Recap questions

You've reached the end

Error Intervals Revision Guide

Error Intervals

What you'll learn

The basic idea

Correct to the nearest whole unit

Rounding to decimal places

Rounding to significant figures

Truncation

Recap questions

You've reached the end

Error Intervals Revision Guide