PHSC1021
Measurements and Errors
• Objectives
–
–
–
–
Make simple measurements and estimate errors
Understand different errors, how to minimize them and how they affect results
Create and utilize a histogram.
Understand the meaning and use of average and standard deviation
Errors
• Systematic
– Caused by faulty or un-calibrated equipment
– Can be compensated for if known or determined
• Random
– Due to precision of measuring device and individual procedures
– Can be minimized by taking many many measurements
• Which GUN would you want?
Errors
• IF both guns were set on a tripod and fired in the same way…
• The gun on the left is more accurate. It has a systematic error,
probably due to the fact the site is not aligned with the barrel. When
that is done it would be the better gun because of its accuracy.
• The gun on the right is less accurate and the random errors in firing
are larger than with the gun on the left, perhaps due to manufacturing
flaws.
Error Estimation
A histogram of measurements helps determine the most probable
value and its error.
As you can see, there are random errors associated with measurement.
Using the average and standard deviation, we can make estimates of
the most probable value (average) and the size of our error (standard
deviation).
4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70
Calculating Average
N
A Set of Values (11, 10, 13, 11)
x

x
i 1
i
N
Average x = [11 + 10 + 13 + 11]  4
= 11.25
MS EXCEL formula*:
=AVERAGE(B1:B4)
*In the cell with the formula,
calculate the average for the values in column B, rows 1 through 4.
Calculating Standard Deviation
A Set of Values (11, 10, 13, 11)
Average x = 11.25
N
x

 (x  x )
i 1
i
N 1
Standard Deviation = { [ (11-11.25)2 + (10-11.25)2 +
(13-11.25)2 + (11-11.25)2 ]  (3) }1/2
= 1.26
MS EXCEL formula*:
=STDEV(B1:B4)
*In the cell with the formula, calculate the standard deviation
for the values in column B, rows 1 through 4.
2
Measurements and Errors
Pre-Lab Questions
? Taking numerous measurements helps to identify ________ errors and
helps to reduce _________ errors.
? The peak in a histogram is
a. the most probable value
b. the average value
c. all of the above
? Calculate the mean for the following measurements (1.2, 1.4, 1.3, 1.1)
? Calculate the standard deviation for the same set of measurements
? If we measure some characteristic of two groups (age, intellect,
height, etc.) and wish to determine which group is more diverse,
explain how you might use a histogram and the standard deviation to
determine those things.