Lecture notes #1, Physical Measurements 201 Lab:
Dr. Erkal
Gaussian Distribution:
If we make a measurement x_{1}, of a quantity x, we expect that the measurement to approximate the quantity, but do not expect our measurement to be exactly equal to the quantity x. If we continue to make measurements to obtain a set of measurements (x1,x2,x3,x4,x5,…..), we will observe discrepancies between the measurements because of random errors. However, after making many measurements, a pattern will emerge from the data. Some measurements will be too large, some will be too small. On the average, the measurements will distribute around the average. We assume that we correct for the systematic error. Here is an example (see below) of a set of measurements I made in our lab. The measurements is made using PASCO 750 workshop using a motion sensor to measure the muzzle velocity of the launchers you used in the lab.
v(m/s) 




1.824 

Bin 
Frequency 

1.726 

1 
0 

1.653 

1.2 
2 

1.624 

1.4 
4 

1.608 

1.6 
9 

1.568 

1.8 
4 

1.538 

2 
1 

1.53 

More 
0 

1.505 




1.488 

Column1 


1.477 




1.458 

Mean 
1.4774 

1.441 

Standard
Error 
0.03830717 

1.425 

Median 
1.4825 

1.4 

Mode 
#N/A 

1.362 

Standard
Deviation 
0.17131485 

1.328 

Sample
Variance 
0.02934878 

1.27 

Kurtosis 
0.13412346 

1.194 

Skewness 
0.1230388 

1.129 

Range 
0.695 



Minimum 
1.129 



Maximum 
1.824 



Sum 
29.548 



Count 
20 






First column is the data; graph is the histogram (a visual representation of the data using bins or intervals). A histogram is a frequency graph, which displays the number of times the data falls within a specified bin (interval). The horizontal axis represents the bins chosen.
Data analysis also provides descriptive statistics including average (or mean), standard deviation, median and many others that we are not interested in. Just ignore other statistics that you do not recognize.
Note that the distribution is symmetric and we can almost draw a bell curve if we connect the middle points of each bar smoothly If we make many more measurements, sat infinitely many, we could describe exactly the distribution of the muzzle velocities. Since this is not possible, we hypothesize the existence of such distribution called PARENT DISTRIBUTION (OR POPULATION DISTRIBUTION). We can assume that the measurements we have made are samples from the parent distribution and they form the SAMPLE DISTRIBUTION. In the limit of infinite number of measurements, the sample distribution becomes the parent distribution. We assume that a bell curve exists, called Gaussian function that first smoothly to the histogram. We can use this function to compute various properties of the data using the Gaussian function, and the results would be very similar to those obtained using directly the data.
Parent distribution provides us with a tool to determine the probability of getting any particular observation in a single measurement. For example, the area under the Gaussian curve represents probability. To be exact, the area under the Gaussian curve between two values of x (say x_{1} and x2), when divided by the total area (between ∞ and +∞) gives the probability that a given measurement would fall between x_{1} and x_{2}.
Here we see a probabilistic interpretation of the
Gaussian representation of the data.
Parameters of the Parent Distribution:
Mean:
_{}
Median:
The value of x such that divides the ordered data (from high
to low or from low to high) in half. Half the observations will be less than the median, and
half will be greater than the median.
Mode:
Mode is the most probable value of the observations. It is the value for which parent distribution
(or sample distribution) has greatest value.
Standard Deviation σ:
_{}
Standard deviation also represents the width or the spread of the data.
Standard Deviation of the Mean (σ_{m}):
If we obtain several samples (for example if 20 people in the class measured the muzzle velocity and obtained 20 samples), we can talk about the distribution of the mean value of the muzzle velocity. Off course, we shall never have a chance to repeat out experiment with so many people. Often we are required to report our measurement without the help of other groups. The best estimated error we can report is the standard deviation of the mean σ_{m, }and it represents a probabilistic interpretation. Thus, a particular sample mean has a 68% chance of falling between _{} + σ_{m } and _{}  σ_{m}., likewise a 95% chance of falling between _{} + 2σ_{m } and _{}  3σ_{m.}
_{ }
_{ }