sum(abs((refrence_signal-measured_signal).^2))/length(refrence_signal); use this code to detect the average (over all) error; this term called "Mean Square Error" and it gives a better indecation. 

For tracking the amplitude changes use this code, 

abs((abs(refrence_signal)-abs(measured_signal)).^2), you will get on the amplitude changing at all domain. 

For tracking the phase changes use this code, 

abs((angle(refrence_signal)-angle(measured_signal)).^2), you will get on the phase changing at all domain. 

if you need to compute without MSE just remove the (.^2) from each code, good luck.

Are you expecting there to be a time shift?

Regardless, if you are just expecting the signals (x & y) to be scaled versions of each other, I.e. y=b*x,  you could regress one signal against the other, usingy std linear regression with 1 degree of freedom (b), to minimise the sum of squared errors. You could use an error weighting function to emphasise certain parts.

First of all you have to go from what you have to probability distributions. For this you should normalize your data such that it sums up to one.

Then use the KL divergence (KL) to measure the distance between the two probability distributions. KL is sensitive to any difference between the histograms.

One disadvantage of the KL divergence is that when the two histograms are completely non-overlapping, the KL divergence approaches infinity.

By figures interpretation and reading your problem statement ......

The figures looks like same by visual inspection but you can quantify it by MSE or by correlation coefficient that will vary between -1 to 1 (A value of 0 indicates that there is no association between the two variables. A value greater than 0 indicates a positive association; that is, as the value of one variable increases, so does the value of the other variable) or by statistical criteria.

But all these depends on application and criteria used by domain researcher. 

As you mention that it is for automotive or car then that system has some model or characteristic frequency for that case you analyse you signal in frequency domain. Find the spectrum of both the signal and compare the signals with respect to frequency and its amplitude.  

Dear Nasir,

The root mean square error  RMSE is the most expressive quantity for the difference between two sets of similar samples x(t), xr(t), where x(t) is the measured signal and xr(t) is the reference signal.

RMSE= SQROOT of (Integral (x-xr)^2 ) over the interested time interval,

For discrete time signal x[n] and xr[n],

the RMSE= SQROOT of (SUM (x[n]- xr[n])^2) over all samples N.

One can use this scheme to compare one dimensional as yours and higher dimensional similar signals.

Best wishes

Cross entropy is popular choice.

Thanks a lot to all for ur time and expert opinions . It was a great help for me. I have Tried some techniques and I am having some issues or errors so instead of updating this question I am positing a new qurrey .

Thanks a lot once again.