17.Analysis of errors(2).

We have been studying all about errors in the last few posts.We continue our discussion further and talk in greater detail about ways of expressing accuracy and precision in this post.

We have already studied how to calculate absolute and relative error in post 15.We studied that –

A.E = xi – T
where,
A.E ⇒ Absolute error.
xi ⇒ Measured Value.
T ⇒ True/Accepted Value

and

Relative Error (R.E) = Absolute error/True value = (xi – T)/T.
Relative error in ppt =  [(xi – T)/T]× 1000  (ppt = parts per thousand).
Relative error in pph =  [(xi – T)/T]× 100    (pph = parts per hundred).

Let us begin to learn ways of expressing precision. We use deviation to study precision. Deviation tells us how much the data points are deviating  from the mean value i.e how widely spread the data is from the mean value. More the deviation less is the precision.In other words, if data values are precise i.e close to each other, deviation will be low.We study precision in the following ways –

GROUP I –

A]Deviation(D) – Deviation shows us how much a given value differs from the mean value.

D=X_i – X_mean ,where ,
D = Deviation.
X_i= Observed value.
X_mean=Mean value.

 

B]Relative Deviation(R.D) –  We saw the significance of relative error in an earlier post(kindly refer to post 15), The same explanation is applicable to relative deviation too.

R.D = D/X_mean .

C]Average Deviation (A.D) – Average or mean  of all deviation values gives us average deviation.

A.D = (D₁+D₂+D₃+……+D_n)/n = ∑D/n, where,
D₁,D_2,D₃⇒ Deviations.

 

D]Relative average deviation (R.A.D) –

Relative average deviation (R.A.D) = A.D / X_mean.
Relative average deviation in pph =A.D / X_mean  ×100.
Relative average deviation in ppt =  A.D / X_mean×1000.

e.g. Suppose we have five measurements – 50.7,51,54,55,52.4.

X_mean = (50+51+54+55+52)/5 = 262/5 = 52.4.

Deviation	Relative Deviation
D1   = 50-52.4 = -2.4	R.A.D1=2.4/52.4 = 0.046
D2 = 51-52.4 = -1.4	R.A.D2=1.4/52.4= 0.027
D3 = 54-52.4 =  1.6	R.A.D3=1.6/52.4 =0.03
D4 = 55-52.4 =  2.6	R.A.D4=2.6/52.4=0.05
D5 = 52-52.4 = -0.4	R.A.D5=0.4/52.4=0.008

Average deviation = (2.4+1.4+1.6+2.6+0.4)/5 = 8.4/5 = 1.68

R.A.D =A.D / X_mean= 1.68/ 52.4 = 3.2%

 

GROUP II –

A]Range –  This gives the difference between the highest and lowest value of the set.

Range= R = X_max – X_min.

B] Standard Deviation – This is the measure of spread.It provides the spread of the numerical data around the mean.It tells us how the data is clustered around the mean(X_mean ) value.So, it can be referred to as the ‘mean of the mean’.Thus standard deviation  gives us information about the distribution of dats around the mean value.

For small observations (less than 20 observations) –

For large observations/Population (more than 20 observations)-

Note that-

1. Only denominator is different in the two formulas . For less observations we divide by ‘(n-1)’ and for more observations we divide by ‘n’.
2. We  consider only the magnitude of the value (X_i – X_m).In the above example, D₁₌-2.4.This quantity is negative, but while calculating standard deviation we have only considered its magnitude i.e 2.4 .
3. Why do we square the value (X_i – X_m)? For understanding this, let us consider four measurements → 4,4,-4,-4.Here, the mean is zero. So,this is not a correct measure of the central tendency.
   

What if we just take the magnitudes i.e ignore the -ve sign on the values? Then,
Mean = (4+4+4+4)/4 = 16/4= 4. This gives us correct value of the central tendency !

Now let us consider another four set of measurements  → 7,1,-6,-2.

 

This also gives us mean as 4!! But the values are more spread out here.So, What quantity will tell us about the spread? It cant surely be the mean! Thus, we use standard Deviation here.

s for the first set of values      =      √(4² + 4² + 4² + 4²4) = √(644) = 4.
s for the second set of values =     √(7² + 1² + 6² + 2²4) = √(904) = 4.74.

Here, the standard deviation (s) is more when the results are more spread out.So,this parameter works well when we need to find the spread of data properly.

C] Variance – It is just the square of standard deviation.

Variance = s².

D] Coefficient of variance – This helps us to present the variance in percentages.

Coefficient of variance = (s/x)×100.

This post was all about formulas.In my next post, we shall learn the concept behind deviation in greater detail. Till then,

Be a perpetual student of life and keep learning…

Good Day !

 

References And Further Reading

1)http://www.robertniles.com/stats/stdev.shtml

2)http://www.mathsisfun.com/data/standard-deviation.html