Uncertainties
Introduction
The purpose of this experiment is to investigate random errors in measurements. To do this we will
analyze our reaction time. We will determine our average reaction time (with uncertainty) by catching a
dropped meter stick. Finally, we will histogram our data to see if it approximates a Gaussian probability
distribution.
Equipment
•
Meter stick
Theory
All measurements are subject to uncertainty. Measurement uncertainties fall into one of two categories:
systematic & random errors. As the purpose of this experiment is to investigate random errors, systematic error
will not be discussed.
Random errors are those which result from unpredictable variations in the experiment. For example, an
experiment may be subject to random temperature fluctuations that cannot be completely controlled. Random
errors are described by a Gaussian probability distribution. Gaussian distributions are characterized by their
bell shape and adherence to the following rule:
i. At least 68% of the data fall within one standard deviation of the mean.
ii. At least 95% of the data fall within two standard deviations of the mean.
iii. Almost all of the data fall within 3 standard deviations of the mean.
N
If we calculate the average value (x ) of a set of data using the equation: x =
(xi ) then it can be shown that the standard deviation (σ x )
1
σx =
N
∑x
i =1
i
for a set of N data points
N
of the set of data is given by:
2
σ
 N 
N ∑ x −  ∑ xi  and that the uncertainty in x is given by: δx =
.
N
i =1
 i =1 
N
2
i
To generate our data set, we will calculate our reaction times from the equation Y =
the distance a meter stick falls before we can catch it and t is time.
1
9.8t 2 where Y is
2
Data
For this experiment a meter stick was suspended between the thumb and forefinger of my lab partner. My lab
partner then released the meter stick and I recorded the distance the meter stick fell before I caught it.
Y (m)
0.13
0.17
0.1
0.11
0.16
0.25
0.19
0.12
0.2
0.23
0.13
0.17
0.15
0.2
0.18
0.05
0.15
0.18
0.16
0.09
0.23
0.15
0.11
0.12
0.22
t(s)
0.163
0.186
0.143
0.150
0.181
0.226
0.197
0.156
0.202
0.217
0.163
0.186
0.175
0.202
0.192
0.101
0.175
0.192
0.181
0.136
0.217
0.175
0.150
0.156
0.212
t^2 (s^2)
0.027
0.035
0.020
0.022
0.033
0.051
0.039
0.024
0.041
0.047
0.027
0.035
0.031
0.041
0.037
0.010
0.031
0.037
0.033
0.018
0.047
0.031
0.022
0.024
0.045
Y (m)
0.2
0.03
0.19
0.08
0.25
0.15
0.16
0.14
0.18
0.25
0.26
0.21
0.16
0.2
0.24
0.3
0.15
0.18
0.25
0.16
0.17
0.19
0.17
0.18
0.42
Sums
t(s)
0.202
0.078
0.197
0.128
0.226
0.175
0.181
0.169
0.192
0.226
0.230
0.207
0.181
0.202
0.221
0.247
0.175
0.192
0.226
0.181
0.186
0.197
0.186
0.192
0.293
9.321
t^2 (s^2)
0.041
0.006
0.039
0.016
0.051
0.031
0.033
0.029
0.037
0.051
0.053
0.043
0.033
0.041
0.049
0.061
0.031
0.037
0.051
0.033
0.035
0.039
0.035
0.037
0.086
1.800
If we histogram this data we get:
Figure 1: Reaction time histogram.
Calculations
To calculate our reaction time:
Solve Y =
1
9.8t 2 for t to get: t =
2
Y
. For example, t =
4 .9
Then, the average reaction time is:
N
t=
∑t
i =1
N
i
=
0.163 + 0.186 + 0.143 + L
= 0.186s .
50
To find the standard deviation in t:
0.13
= 0.163s .
4 .9
σt =
1
N
2
N
1
 N 
2
N ∑ t i2 −  ∑ t i  =
50 0.163 2 + 0.186 2 + 0.143 2 + L − (0.163 + 0.186 + 0.143 + L) = 0.035s .
50
i =1
 i =1 
Finally, the uncertainty in t is: δt =
(
)
σ
N
=
0.035
50
= 0.0049s
Conclusions
From this experiment I was able to determine my average reaction to be t = (0.186 ± 0.005)s.
This seems a reasonable value especially in light of the bell shape of the histogram in Figure 1. As seen in our
histogram, 76% of our data fall within 1σ of the average value of t, ~94% is within 2 σ, & ~100% are within
3 σ, in agreement with theoretical expectations. One source of error in this experiment is that it is difficult not
to anticipate my lab partner dropping the meter stick. I would expect then that the average reaction time as well
as the standard deviation reported may be slightly lower than it should be. This problem could be alleviated by
using a computer controlled release mechanism that would truly release the meter stick at random time
intervals.