Measurement

Lesson

The aim of a measurement is to obtain the true value of a quantity, be it the height of a tree, the temperature of a room, the mass of a rock and so on. We can carefully design a measurement procedure so that the measured value is as close as possible to the true value, but there will always be some difference between the two.

The difference between the measured value and the true value is called the error:

$\text{Error }=\text{Measured value }-\text{True value }$Error =Measured value −True value

Notice that the error may be positive or negative depending on whether the measured value overshoots or undershoots the true value.

In general the true value and the error are both unknown to us. Instead we can use the measured value, together with the uncertainty of the measurement, to produce an interval within which the true value will likely lie. The lower value of the interval is called lower bound, and the higher value is the upper bound.

One useful value is the absolute error and is defined as half the distance between the lower bound and upper bound. Usually the measured value is centred between the lower bound and upper bound, so we can also find the absolute error by subtracting the measured value from the upper bound.

Measurements taken from an measuring device have an accuracy of $\pm\frac{1}{2}$±12 of the smallest unit measured. Rounded measurements also have an accuracy of $\pm\frac{1}{2}$±12 of the unit the measurement is being rounded to. So someone who has a height of $160$160 cm using a ruler, will have a lower bound of $159.5$159.5 cm and an upper bound of $160.5$160.5 cm. Conventionally when we use a measuring device, or are rounding, we naturally round up in the case that we have a tie. In other words, the lower bound is always **included** in the uncertainty range while the upper bound is always **excluded** from the uncertainty range.

Remember!

- The uncertainty range obtained from a measurement is an interval that contains the true value.
- The lower value of the interval is the lower bound.
- The upper value of the interval is the upper bound.
- The absolute error is half the distance between the lower bound and upper bound.

The lower bound is always **included** in the uncertainty range. The upper bound is always **excluded** in the uncertainty range.

A person's height is measured to be $1.68$1.68 metres rounded to the nearest centimetre.

(a) What is the lower bound of their height?

**Think:** What heights would be rounded up to $1.68$1.68 m? What is the lower bound of these values?

**Do:** The lower bound is $1.675$1.675 m, since any height $h$`h` in the range $1.675\le h<1.68$1.675≤`h`<1.68 will round up to $1.68$1.68.

**Reflect:** The difference between the measured height and the lower bound is $0.005$0.005 m, or $0.5$0.5 cm, which is half of the smallest unit of the measurement. This is the **absolute error**.

(b) What is the upper bound of their height?

**Think:** What heights would be rounded down to $1.68$1.68 m? What is the upper bound of these values?

**Do:** The upper bound is $1.685$1.685 m, since any height $h$`h` in the range $1.68`h`<1.685 will round down to $1.68$1.68.

**Reflect:** Even though $1.685$1.685 is the upper bound of the person's height, it does not belong to the uncertainty range since $1.685$1.685 rounds up to $1.69$1.69.

The population of species is $95000$95000 to the nearest $1000$1000 organisms.

(a) What is the minimum possible population?

**Think:** What possible values round up to $95000$95000 to the nearest $1000$1000 organisms?

**Do:** The lower bound is $94500$94500, since any population $p$`p` in the range $94500\le p<95000$94500≤`p`<95000 will round up to $95000$95000. Since the lower bound is always in the uncertainty region, the minimum possible population is $95000$95000 as well.

(b) What is the maximum possible population?

**Think:** What possible values round down to $95000$95000 to the nearest $1000$1000 organism?

**Do:** The upper bound is $95500$95500, since any population $p$`p` in the range $9500095000<`p`<95500 will round down to $95000$95000. A calculation of $95500$95500 will round up to $96000$96000. However, since population is discrete, $95499$95499 is included in the uncertainty range and is the largest possible population. So the maximum possible population is $95499$95499.

**Reflect:** In this case, the upper bound is not necessarily the maximum possible value being measured.

Careful!

The upper bound is not necessarily the maximum possible value being measured.

Between what limits does the cost of a CD lie if it is known to be $£50$£50 correct to the nearest $£5$£5?

Upper bound = $£$£$\editable{}$

Lower bound = $£$£$\editable{}$

State the limits of accuracy for a distance measured to be $13.45$13.45 km.

Upper bound = $\editable{}$ km

Lower bound = $\editable{}$ km

The length of a piece of rope is measured to be $19.99$19.99 m using a ruler. What is the upper bound of the largest possible length of this rope?