Lecture 2

AccuracyAccuracyandandPrecision,Precision,DerivedDerivedVariablesVariables

AccuracyAccuracyandandPrecisionPrecision

Definition 1

Accuracy is the closeness of a measured or computed value to its true value. Precision is the closeness of repeated measurements to each other.

Example 1

A biased but sensitive thermometer may yield inaccurate but precise temperature measurements. For example if the temperature outside is $-10.7^{\circ} \mathrm{C}$, a precise but inaccurate thermometer might give the readings $-9.8^{\circ} \mathrm{C}$,$-9.7^{\circ} \mathrm{C}$, $-10.0^{\circ} \mathrm{C}$, which are all close to each other, but far from the true value.

Example 2

An insensitive scale might result in accurate readings which are imprecise; other measurements of the same object would give a different values, but the average of these values would be close to the true value. For example if the blood pressure of a patient is $117\,\mathrm{mmHg}$, an imprecise but accurate apparatus might give the readings $128\,\mathrm{mmHg}$,$99\,\mathrm{mmHg}$, $123\,\mathrm{mmHg}$, which are far from each other, but their average is close to the true value.

Remark

Discrete variables are usually recorded accurately and precisely (4 eggs in a nest). Continuous variables (e.g. weight) are measured with finite precision, so the values are approximate. The exact value of a continuous variable is unknowable.

Example 3

Consider a meter with measurement precision of $0.1 \,\mathrm{mm}$. A length measurement of $12.3 \,\mathrm{mm}$ implies (based on the precision of the meter) that the true length is between $12.25$ and $12.35 \,\mathrm{mm}$.
If however we find a better instrument with precision $0.01 \,\mathrm{mm}$ and obtain the value $12.28 \,\mathrm{mm}$, the implied limits now are $ 12.275$ to $12.285 \, \mathrm{mm}$.

Example 4

The highest possible precision of a measurement depends on the measurement device rather than on the variable. However we might choose to record measurements with less precision than the instrument allows. Regardless, a recorded measurement implies a range of possible values.$$\begin{array}{c|c}\text { Measurement } & \text { Implied Limits } \\ \hline 193 & 192.5-193.5 \\ 192.8 & 192.75-192.85 \\ 192.76 & 192.755-192.765\end{array} $$

Example 5

Discrete variables are usually recorded accurately and precisely (with infinite precision). However when the numbers are large the values could be recorded approximately as well: a count of $36\,000$ insects implies that the true number lies between $35\,500$ and $36\,500\,$insects.

Remark

A question which arises in practice is this: Given that we have a precise enough measurement device, with what should we record measurements?

Rule of Thumb

The number of unit steps from the smallest value to largest value obtainable in the measurement should be between $30$ and $300$.

Remark

Let the range from the smallest value to the largest value obtainable in the measurement correspond to $100\%$. If we subdivide this range in $300$ steps, each step (the measurement precision) corresponds to $\sim 0.3\%$. If we subdivide this range in $30$ steps, each step corresponds to $\sim 3\%$ of the range.

Example 6

Up to the nearest mm shells are between $4 \,\mathrm{mm}$ and $8 \,\mathrm{mm}$ wide. If we measure to the nearest $\mathrm{mm}$, how many unit steps are there:$$ \frac{8-4}{1}=4 \text{ unit steps}$$Not good enough - we need to measure more precisely.
We went back to the station and got a more precise instrument, which measures up $0.1 \,\mathrm{mm}$. Now assume that shells are between $4.1 \,\mathrm{mm}$ and $8.2 \,\mathrm{mm}$ wide. $$\frac{8.2-4.1}{0.1}=41 \,\text{ unit steps} $$ With this meter we have more than 30 unit steps, but less than 300, which is good.

Example 7

Heights of certain plants are between $26.6$ and $173.2\,\mathrm{cm}$. If we measure this height with precision $0.01 \,\mathrm{cm}$, we will have:$$\frac{173.2-26.6}{0.01}=14660 \, \text{ unit steps }$$This is unnecessarily too many. Heights could be recorded to the nearest centimeter. $$\frac{173-27}{1}=146 \, \text{ unit steps } $$Good! We have between 30 and 300 unit steps.

Example 8

pH readings are known to have a lowest value of $7.434$ and a highest value$7.456$. How many decimal places should we keep?$$\frac{7.45-7.43}{0.001}=20 \, \text{ unit steps }$$Not enough!We need to measure with precision 4 digits.$$ \frac{7.4560-7.4340}{ 0.0001 }= 220 \text{ unit steps }$$Good! We have between 30 and 300 unit steps.

RoundingRoundingNumbersNumbers

Rule of Thumb

A digit to be rounded will not be changed if the following digit is $<5$. A digit to be rounded will be increased by 1 if the following digit is $\geq 5$.
See for an exception below.

Example 9

$$ \begin{array}{l|c|c} \text { Value } & \text { Significant digits required } & \text { Rounded value } \\ \hline 26.58 & 2 & 27 \\ 133.7137 & 5 & 133.71 \\ 0.03725 & 3 & 0.0372 \\ 0.03715 & 3 & 0.0372 \\ 18316 & 2 & 18000 \\ 14.3476 & 3 & 14.3 \end{array}$$

Note

Notice that if the following digit is 5 and there are no other digits to the right, the digit to be rounded is not changed if it is even and increased by $1$ if it is odd. In the table above, both $0.03715$ and $0.03725$ are rounded to $0.0372$.

DerivedDerivedVariablesVariables

Definition 2

Aderived variable is a variable not directly measured, but rather computed from measured values.

Example 10

Ratio

A study has $72$ females and $48$ males. The ratio of females to males is $$\frac{72}{48}=\frac{3}{2}=1.5$$ As a second example: the percentage of the females in the study is $$\frac{72}{72+48}=\frac{72}{120}=\frac{3}{5}=0.6=60 \% $$

Example 11

Index

An animal is given six behavioural tests where the score ranges from $0$ to $5$. Say the scores are $3, 4, 1, 4, 5, 4.$An index of its behaviour could be the average score on the six tests:$$\frac{3+4+1+4+5+4}{6} = 3.5$$The index is a derived variable.

Example 12

Rates

Weight gain per unit time, reproductive rates per unit population size, death rates per unit population are all derived variables. (Notice that these are ratios.)

Remark

Derived (computed) variables have reduced precision, when compared to the precision of the measured variables. This could lead to misleading results if the measured data is not precise enough. We will illustrate this loss of precision with two examples.

The decrease of precision in derived variables is illustrated in terms of relative error. The relative error is defined as the error (lack of precision) divided by the true value.

Example 13

Consider the product $2.4\times 3.5$ of two measured values. The measurement of $2.4$ implies the range $[2.35,2.45]$ for the true value. The relative error is $\frac{2.45-2.4}{2.4}=0.020=2.0\%$. Similarly, the measurement of $3.5$ indicates true range of $[3.45,3.55]$. The relative error is $\frac{3.55-3.5}{3.5}=0.014=1.4\%$
The value of the product is $2.4 \times 3.5=8.4.$ The minimum value for the product allowed by the implied precision is $2.35 \times 3.45=8.108$, where we multiplied the smallest values within the implied bounds for the measurements. The maximum value is $2.45\times 3.55=8.698$. The relative error is $\frac{8.698-8.4}{8.4}=0.034=3.4\%$, and it is higher than the relative error of the individual measurements.

Example 14

Consider the ratio $\frac{1.2}{1.8}$. The measurement of $1.2$ indicates true range $[1.15,1.25]$. The relative error is $\frac{1.25-1.2}{1.2}=0.042=4.2\%$. Similarly, the measurement of $1.8$ indicates true range of $[1.75,1.85]$. The relative error is $\frac{1.85-1.8}{1.8}=0.028=2.8\%$
The minimum value for the ratio is $\frac{1.15}{1.85}=0.622$. Here we took the smallest within the implied bounds value in the numerator and the largest in the denominator. The maximum value is$\frac{1.25}{1.75}=0.714$. The ratio of empirically measured values $\frac{1.2}{1.8}=0.667$ will be used as the `anchor` for evaluating the relative error. This relative error is $\frac{0.714-0.667}{0.667}=7.0\%$, higher than the relative error of the individual measurements.
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