In the last post we discussed how measurements are subject to errors and the different types of errors that one might come across.In this post we shall discuss how to measure these errors and how to present our data with specifying errors in it.
We begin our discussion with the concept of ‘True value – T‘ .We talked about the true value in our earlier post too ,but never thought about it in greater detail.So then , what is the true value of a measurement? The true value of a quantity cannot be known prior to experimentation or measurement.If it was known, why would one conduct the experiment to measure it again in the first place? True value is generally obtained by finding out the central tendency of a set of data.So basically , we measure some quantity a couple of times, and then find a single value that attempts to describe that set of data.Thus, we try to find out the central tendency of the data value set.Mean, Median and Mode are the measures of central tendency. All the three mean,Median and Mode are valid measures of central tendency but depending on the data set values , one chooses the most appropriate of the three and uses that to find the true value ,T.
Point to remember –
True value ⇒ A single value that describes the set of data ⇒ obtained by finding out central tendency i.e mean/median/mode(generally mean is always preferred).
A]Arithmetic Mean – This is the most popular measure of central tendency and we use it many times in our daily life.The mean is equal to the sum of all the values in the data set divided by the number of values in the data set.
Mean = x̄ = (x_{1}+ x_{2} +x_{3}+….. + x_{n }) / n
where,
x_{1,} x_{2} ,x_{3,} x_{n}. ⇒ values of the set .
and n ⇒ Total no.of values.
It can also be represented as ,
x̄ = ∑x /n.
‘∑’ sign means summation i.e addition of all values.
e.g. Suppose we want to find out the average height of students in a classroom.The height (in cm ) of all students in the class are as follows –
132, 135,131,137,133,132,135,136,131,135. What is the average height ?
Mean = (132+135+131+137+133+132+135+136+131+135)/10 = 1337/10 = 133.7 cm.
Thus, the mean value here is 133.7cm.
In the above example the mean value i.e 133.7 is kind of representative of all other data values in that set(as all other values are close to 133.7). It gives us a rough idea of the height of each student in that classroom.But now let us try to find average height of employees in an office .The heights(in cm) are as follows –
142,165,175,188,170,155,178,149,152,151.
Mean = 142+165+175+188+170+155+178+149+152+151/10 = 1625/10= 162.5cm.
Now the value ‘162.5 cm’ is not giving us idea of the height of all employees correctly as some values are far too greater than 162.5 e.g. 178 and some are very low e.g. 142 ! So, here finding the mean is not helpful. In such situations we opt for median or mode.
B] Median – Median means ‘situated in the middle’.Here, we arrange all the values in ascending order and then find the middle value .
e.g. In the above example , we arrange values starting from lowest to the highest.
142,149,151,152,155,165,170,175,178,188 .Here there are 10 values(even number) so we have two middle values – 155 and 165. So, we use the mean of these two values.
Median = (155+165)/2 = 160.
C] Mode – The value which has the highest frequency of occurrence in the data set is the mode.
e.g. – If the values in a data set are 6,8,9,3,7,9,2,1,2,9. Here number ‘9’ has the highest frequency of occurrence. So,the mode is 9 .
Mode as a form of central tendency is not used widely because most of the times it is impractical to find it.As seen in the earlier example where the data set consists of heights of students, we don’t have a value that repeats itself. In such a case, one cannot find the mode.
So, now that we know how to find the true value(T) for a data set , we can go ahead and determine the errors i.e study the departure of measured value from the true value(T).
CALCULATION OF ERRORS –
We calculate errors in the following ways –
- Absolute Error – The difference between the measured value and the true value gives us the absolute error.
A.E = x_{i }– T
where,
A.E ⇒ Absolute error.
x_{i }⇒ Measured Value.
T ⇒ True/Accepted Value.
Absolute error could be positive or negative. If the measured value is more than the true value we get a positive error and if it is less than T then we get a negative error.
e.g. Suppose, the true value of length of a stick is 12cm. After measurement,
person A finds it to be 12.2cm then(A.E)_{1} = 12.2 – 12 = +o.2cm (positive error).
person B finds it to be 11.9cm then(A.E)_{2} = 11.9 – 12 = -o.1 cm (negative error).
person C finds it to be 11.8cm then(A.E)_{3} = 11.8-12 = -0.2 cm (negative error).
2.Mean Absolute Error – This is found by taking the average of the magnitudes(i.e + or -ve sign is not taken into account) of all absolute errors.
Mean absolute error =(A.E)_{m}= {⎮(A.E)_{1}⎮+⎮(A.E)_{2}⎮+…..+⎮(A.E)n⎮}/n
Mean absolute error =(A.E)_{m}= {⎮(A.E)_{1}⎮+⎮(A.E)_{2}⎮+⎮(A.E)_{3}⎮}/3 =(0.2+0.1+0.2)/3 = 0.5/3 ≈0.167.
‘⎮A.E⎮’ ⇒ Magnitude of Absolute error (+/- sign of the error is not considered).
The mean absolute error is always positive.
3.Relative Error – To understand this parameter let us consider the following example
Suppose a shopkeeper is asked to weigh a bag full of rice and another bag full of wheat.He weighs both and finds out that,
Weight of the rice bag = 5kg ± 0.5kg .Here absolute error is ±0.5kg i.e the weight could be 4.5,5 or 5.5 kg.
Weight of the wheat bag = 20kg ± 1kg. Here the absolute error is ±2kg i.e the weight could be 19,20,21 kg.
In which of the above measurement has the shopkeeper incurred more error ?To find this out we need to find out the relative error in both cases.
Relative Error (R.E) = Absolute error/True value = (x_{i }– T)/T.
Relative error in ppt = [(x_{i }– T)/T]× 1000 (ppt = parts per thousand).
Relative error in pph = [(x_{i }– T)/T]× 100 (pph = parts per hundred).
So,for the above example ,
For Rice, R.E in pph = (0.5/5)×100 = 10% relative error.
For wheat,R.E in pph = (1/20)×100 =5% relative error.
So, though A.E (wheat) > A.E(rice) , R.E(wheat)< R.E(Rice).
Thus, relative error tells us how much is the error in relation to the true value.
4.Constant / Zero error – Before starting any experiment, we always set our instrument to zero.Although sometimes, the device may read slightly higher or lower than zero.In such cases ,where the instrument cannot be set to zero or calibrated, we later add or subtract the difference from each observation.Thus, here the A.E remains constant but R.E changes with the change in sample size.
e.g.Suppose during a titration, the burette adds 0.1cm^{3 }extra everytime.So, the absolute error for any measurement will be +0.1cm^{3} always. Although, the relative will decrease with increase in sample size as follows –
Absolute Error |
Titre Value |
Relative Error(pph) |
0.1cm^{3} |
50cm^{3} |
0.2% |
0.1cm^{3} |
10cm^{3} |
1% |
0.1cm^{3} |
1cm^{3} |
10% |
Note →
Relative error(pph) = (Absolute error / True value)×100.
In the above example,
R.E = (0.1/50) × 100 = 0.2% for the first case.The titre value(volume of the titrant delivered by the burette) is considered as the true value as that is the exact quantity which is required to reach the endpoint.The titre values for three different titrations are mentioned above.
Thus, Sample size ↑ R.E ↓ for Zero Error.
5.Proportionate error – Here, the amount of error will always be a consistent percentage.The magnitude of relative error i.e ⎮R.E⎮ remains constant.Absolute error changes with sample size.
e.g.Determination of lead in a sample with barium impurity.Here, along with the lead sulphate PbSO_{4},Barium sulphate BaSO_{4} also precipitates.
Sample Size |
Amount of PbSO_{4 }+ BaSO_{4} |
Absolute error i.e amount of BaSO_{4} |
Relative error. |
0.5g =500mg |
100mg+2mg |
2mg |
(2/500)× 100 = 4% |
1g = 1000mg |
200mg+4mg |
4mg |
(4/1000)× 100=4% |
0.25g=250mg |
50mg+1mg |
1mg |
(1/250) ×100 =4% |
Note →
In the above example the ratio of amt of BaSO_{4} / amt of PbSO_{4} , remains constant(2mg/100mg=4mg/200mg=1mg/50mg).
In the next post , we shall discuss two very important parameters related to this topic.Till then,
Be a perpetual student of life and keep learning …
Good day !
References and Further Reading –
1) https://statistics.laerd.com/statistical-guides/measures-central-tendency-mean-mode-median.php
2)http://www.ehow.com/info_12216420_constant-error.html
3)http://www.ehow.com/info_12185218_difference-between-constant-proportional-error.html