1.3 Working with errors

In this module, we shall introduce some statistical analysis techniques to improve our understanding about error and enable reporting of error in the measurement of a quantity. There are basically three related approaches, which involves measurement of :

• Absolute error
• Relative error
• Percentage error

Absolute error

The absolute error is the magnitude of error as determined from the difference of measured value from the mean value of the quantity. The important thing to note here is that absolute error is concerned with the magnitude of error – not the direction of error. For a particular $n^{th}$ measurement,

$$\left|\Delta x_n\right|=|x_m-x_n|$$

where " $x_m$ " is the mean or average value of measurements and " $x_n$ " is the $n^{th}$ instant of measurement.

In order to calculate few absolute values, we consider a set of measured data for the length of a given rod. Note that we are reporting measurements in centimeter.

$$x_1=47.7cm,x_2=47.5cm,x_3=47.8cm,x_4=47.4cm,x_5=47.7cm$$

The means value of length is :

$$\Rightarrow x_m=\frac{47.7+47.5+47.8+47.4+47.7}{5}=\frac{238.1}{5}=47.62$$

It is evident from the individual values that the least count of the scale (smallest division) is 0.001 m = 0.1 cm. For this reason, we limit mean value to the first decimal place. Hence, we round off the last but one digit as :

$$x_m=47.6cm$$

This is the mean or true value of the length of the rod. Now, absolute error of each of the five measurements are :

$$\left|\Delta x_1\right|=|x_m-x_1|=|47.6-47.7|=|-0.1|=0.1$$

$$\left|\Delta x_2\right|=|x_m-x_2|=|47.6-47.5|=\left|0.1\right|=0.1$$

$$\left|\Delta x_3\right|=|x_m-x_3|=|47.6-47.8|=|-0.2|=0.2$$

$$\left|\Delta x_4\right|=|x_m-x_4|=|47.6-47.4|=\left|0.2\right|=0.2$$

$$\left|\Delta x_5\right|=|x_m-x_5|=|47.6-47.7|=|-0.1|=0.1$$

Mean absolute error

Earlier, it was stated that a quantity is measured with a range of error specified by half the least count. This is a generally accepted range of error. Here, we shall work to calculate the range of the error, based on the actual measurements and not go by any predefined range of error as that of generally accepted range of error. This means that we want to determine the range of error, which is based on the deviations in the reading from the mean value.

Absolute error associated with each measurement tells us how far the measurement can be off the mean value. The absolute errors so calculated, however, may be different. Now the question is : which of the absolute error be taken for our consideration? We take the average of the absolute error :

$$\Delta x_m=\frac{\Delta x_1+\Delta x_2+\dots \dots \dots .+\Delta x_n}{n}$$

$$\Rightarrow \Delta x_m=\underset{0}{\overset{n}{\Sigma }}\frac{\Delta xi}{n}$$

The value of measurement, now, will be reported with the range of error as :

$$x=x_m\pm \Delta x_m$$

Extending this concept of defining range to the earlier example, we have :

$$\Rightarrow \Delta x_m=\frac{0.1+0.1+0.2+0.2+0.1}{5}=\frac{0.7}{5}=0.14=0.1cm$$

We should note here that we have rounded the result to reflect that the error value has same precision as that of measured value. The value of measurement with the range of error, then, is :

$$\Rightarrow x=47.6\pm 0.1cm$$

What we convey by writing in terms of the range of possible error. A plain reading of above expression is “the length of rod lies in between 47.5 cm and 47.7 cm”. For all practical purpose, we shall use the value of x = 47.6 cm with the caution in mind that this quantity involves an error of the magnitude of “0.1 cm” in either direction.