Las Positas College
Physics 8A
Laboratory 1
“Designing a Parking Lot Space”
February 3, 2009
Marc Costantino
Lab Partner 1
Lab Partner 2
Abstract
We design a parking lot space by calculating the averages of the measured lengths and widths of 21
personal vehicles and adding space for opening doors. The average length and width of the vehicles
are 450  45 cm and 174  15 cm, respectively, where the uncertainty is one standard deviation. The
calculated parking space is 610 cm (20’0”) long and 260 cm (8’6”) wide. This space optimizes the use
of the available parking lot area under the constraint of only one size space.
Introduction
This laboratory is an introduction to the basics of measurement and data reduction. The objective is to
determine the dimensions of a parking space so that it is large enough to accommodate most personal
vehicles, but not so large that it wastes parking lot space. We do this by measuring the “footprint”
width and length of a sampling of cars in a Las Positas College parking lot, then calculate the average
and standard deviation of the sample. We then add approximately one foot all around to allow for
opening doors and for cars that are larger than two standard deviations from the average. Our result is
a parking space that is 610 cm (20’0”) long and 260 cm (8’6”) wide.
Theory
A parking lot contains n spaces in a fixed area ATotal. The objective of designing the parking lot is to
maximize the profit of the businesses that share the lot. Each space does not have to have the same
size (width and length), but the number of different sizes should be small to decrease the cost of laying
out and painting the lines. Increasing n permits more customers to park, but decreases the average size
of the spaces, making it more difficult to park. In addition, there must be adequate driving and turning
room in the regions between the spaces. Some important quantities are:
<l> =
<w> =
l =
Average length of a vehicle (a personal car/truck)
Average width of a vehicle
An amount added to <l> to allow for room at the ends of the vehicle
w =
ATotal =
An amount added to <w> to allow for room to open doors
The total area available for the parking lot
ARoads =
The total area of the roads in the parking lot
rturn =
The turning radius of a vehicle
wl = The width of the paint line defining a parking space
The total number of spaces, n, is the sum of number of spaces of each size:
n   ni ,
Eqn 1
i
where the index i is over the number of differently sized spaces. The total area of the parking lot is the
sum of the parking space area and the road area:
ATotal   ni Ai  ARoads
Eqn 2
i
where Ai is the area of one space. To find the parking space area, we assume there are i = 1, 2,
…different sized spaces. We calculate the area Ai using
w 

Ai   li   li  l    wi   wi  wl ,
2 

2
Eqn 3
where the average length <li> and width <wi> are found from the population making up ni. We then
adjust ni so that Equation 2 is satisfied. For the purposes of this design, we will constrain the number
of different sizes of parking spaces to one: i = 1.
We find <li> and <wi> by taking the averages of the lengths and widths, respectively, of at least 20
vehicles:
l  
 w 
1
jmax
Eqn 4
j max
l
j 1
1
j max
jmax
j 1
j
and
w ,
j
where jmax = number of vehicles. The “best” value for a measured quantity such as this is the
average value. We find the length and width of a parking space using
Eqn 5
wL
, and
2
W   w  2 w  w  wL ,
L  l   2 l  l 
where wL is the width of the line marking the space. The quantities l and w address the need to
make the parking space larger than the vehicle to make room for opening doors, and avoiding
contacting the vehicle in the next space. The standard deviation, , is found using:
N
x 
 ( x   x )
i 1
2
Eqn 6
i
N 1
,
where N is the number of values in a population {xi} having an average value <x>.
We leave for further investigation the layout of the parking lot, which requires information about the
turning radius of the vehicles, ingress and egress from city streets, and emergency access.
Experiment
Apparatus
The experimental apparatus is a common measuring tape, at least 7.6 m (25’) long, with a least count
of 3 mm (0.12”). The Instrumental Limit of Error (ILE) is 3 mm. We use the measuring tape asreceived from the manufacturer, without further calibration and take its accuracy to be the ILE.
Procedure
We select a random sample of 21 cars parked between 4:30 and 7:30 PM on Tuesday, January 27,
2009 in the parking lot north of Building 1800 at Las Positas College. Two people measure the width
and length of each car by having each person hold the tape at the estimated extent of each dimension.
We defined the length of the vehicle to be the distance between the maximum extent of the bumpers,
trailer hitches, etc.. We defined the width to be the maximum extent of the outside mirrors. We
repeated the measurement of the dimensions of one vehicle ten times to permit estimation of the
measurement error using a standard deviation. We then measured the dimensions of 20 other vehicles
and calculated their average length and width.
To minimize the error owing to “tape sag,” we pulled the tape so that the sag is no more than 3 cm
over its length. We estimate that the increase in tape length owing to pulling on it is less than 5 mm.
We take care to insure the tape is parallel to the ground and to the vehicle dimension within 8 cm over
3
its length. Since we intend to calculate the final dimension to the nearest centimeter, we measure to the
nearest 0.5 cm (approximately 0.2”) and measure each dimension once.
Results
A summary of the results for the length and width data for 21 cars is in Table 1. The raw data are in
the Appendix in Table 2. The average length is 450  45 cm and the average width is 174  15 cm,
where the variation is one standard deviation. The primary source of error is the random observational
error caused by the difficulty in determining the outermost extent of the vehicle footprint. We estimate
that this error may be as large as 5 cm at each end, resulting in a random error of about 10 cm. We
included the outside mirrors in the width. We included length extensions, such as brush guards and
winches on the front of trucks and trailer hitches.
Table 1. Summary of results
Length (cm)
Width (cm)
Mean:
450
174
Average Deviation:
36
11
Standard Deviation:
45
15
Mode:
430
198
Median:
432.5
175.0
Errors
Generally, to calculate a meaningful standard deviation for data having a normal distribution of
random variations, we need at least 7 data points. We could test the effect of changing the sample size
by doubling the number of measurements and calculating a new average and standard deviation. Just
getting the dimensions of all the cars made wouldn’t work, since that wouldn’t weight the data
properly, since there are many more cars on the road with “compact” or “standard” sizes than there are
“luxury” sizes.
Systematic errors include a possible mis-calibration of the tape, the effect of temperature on tape
length, a relatively constant amount of sag in the tape, and a consistent error by the measurers in
estimating the extent of the vehicle dimensions. We estimate the systematic errors to be negligible
compared to the random errors.
Random errors include small variations in estimating the extent of the vehicle dimension, rounding off
the measurement to the nearest reading on the tape, small, variable changes in the amount of sag of the
tape or of its alignment, and errors in recording data. We should emphasize that the size of the
standard deviation owes to the differences in sizes of the vehicles, and not in the random errors
associated with measuring the lengths and widths.
Some observational errors are:

Error in estimating the extent of the vehicle (i.e., lining up the end of the tape measure with the
end of the vehicle). We estimate this error to be 10 cm

Mis-reading the number on the tape. We estimate this error to be 0.5 cm

The error in the length measurement owing to 3 cm tape sag over a nominal length of 450 cm is
less than 1 mm and therefore is not significant. The error owing to a non-parallelism of 8 cm over
450 cm also is less than 1 mm.
The measuring instrument’s errors are:

The absolute calibration of the tape. This is unknown, but probably less than twice the least count,
or about 6 mm. The length of the metal tape is affected by the temperature, since L = L0(1 + T),
4
where  is the coefficient of linear expansion and T is the difference in temperatures from the
calibration temperature and the temperature at which the tape is being used. To estimate the effect
of temperature on the accuracy of the steel measuring tape, we use a nominal value for thermal
expansion of steel of 3(10-6)C-1, the largest possible error for thermal expansion of the tape is
L/L0 = (3(10-6)C-1)(40C) = 1.2(10-4),
which is not significant. We note that the dimension of the vehicle, since it is made of steel,
changes at about the same rate as the steel tape, so the systematic error owing to temperature is
small.

The least count (the Instrumental Limit of Error) of the tape is 3 mm.
The primary environmental error is the temperature. Some minor contribution to the random error
might be caused by the wind blowing the tape. Gravity also contributes by causing the tape to sag.
We conclude that instrumental and environmental errors are insignificant, with the major source of
error being observational. This error could be minimized by using special jigs to define better the
“footprint” of the vehicle, or by making a statistically significant number of readings for each
dimension for each vehicle. Both those are outside the scope of this work.
We choose the number of significant figures based on the estimated limit of error of the measuring
tape. In this case, we believe there may be an error of as much as 10 cm in estimating the extent of the
vehicle dimensions. We believe that, by taking a large number of measurements, we can justify an
answer to within 5 cm. So we carry the next significant digit to 0.5 cm. Another way to proceed would
be to acknowledge that we are going to round the final result to the nearest 6”. Therefore, we would
measure to the nearest 1”.
Adding two standard deviations and 60 cm (2 feet) to each average dimension gives the calculated size
for a parking space:
Length (cm)
=
450 + 2(45) + 60 = 600 cm  19’8”
Width (cm)
=
174 + 2(15) + 60 = 264 cm  8’8”
Since our data provide only a low level of accuracy, we round off the length and width dimensions to
the nearest 6”. The recommended length of the parking space is 20’ (610 cm) and the width is 8’6”
(260 cm).
Discussion
The length data appear to fall into two ranges. In Figure 1, thirteen of the vehicles had lengths of
450 cm or less, while 8 had lengths longer than 450 cm. There was no similar pattern in the width
measurements (Figure 2), with the data distributed more or less evenly from widths of 160 to 200 cm.
A Gaussian distribution does not describe the data. The reason for this is that the differences in the
lengths of the cars are not distributed normally, but are weighted by similarities between car types and
manufacturers.
The data may suggest use of a “compact” length of 450 cm and a “standard” length of 550 cm, but we
believe the sample is too small to support that conclusion. Further, we note that our sample could be
skewed since we measured cars in the parking lot north of Building 1800. This lot has about 50 spaces
and is used primarily by students. Thus a random sample would weight the sizes of the types of cars
students might drive, which could be different than for the overall population of the Las Positas
College parking space user. This hypothesis can be tested by taking a larger sample.
5
Conclusions
Based on measurement of the length and widths of 21 vehicles, we recommend a parking space of
20’0” in length and 8’6” in width. This space provides for the “footprint” of the average sized vehicle
plus two standard deviations, and allows a space of about 1’ all around to accommodate opening doors
and movement of people. Our data suggest use of two, different sized, spaces. However, we believe
we must increase the sample size of our measurements to test adequately that hypothesis.
6
Appendix
Table 2. Data for lengths and widths of personal vehicles.
N Length (cm) Width (cm) L - Lave |L - Lave| |L - Lave|2 W - Wave |W - Wave| |W - Wave|2
1
425.5
166.5
-24.3
24.3
588.6
-7.9
7.9
62.5
2
430.0
160.0
-19.8
19.8
390.5
-14.4
14.4
207.5
3
485.0
178.5
35.2
35.2
1241.7
4.1
4.1
16.8
4
449.0
164.5
-0.8
0.8
0.6
-9.9
9.9
98.1
5
400.0
155.0
-49.8
49.8
2476.2
-19.4
19.4
376.5
6
390.0
180.5
-59.8
59.8
3571.5
6.1
6.1
37.2
7
457.0
179.0
7.2
7.2
52.4
4.6
4.6
21.1
8
424.0
175.0
-25.8
25.8
663.7
0.6
0.6
0.4
9
399.0
165.0
-50.8
50.8
2576.8
-9.4
9.4
88.4
10
430.0
198.0
-19.8
19.8
390.5
23.6
23.6
556.7
11
448.0
198.0
-1.8
1.8
3.1
23.6
23.6
556.7
12
543.5
195.5
93.7
93.7
8786.8
21.1
21.1
445.0
13
503.0
182.5
53.2
53.2
2834.3
8.1
8.1
65.5
14
521.0
177.0
71.2
71.2
5074.9
2.6
2.6
6.7
15
424.5
173.5
-25.3
25.3
638.2
-0.9
0.9
0.8
16
468.0
172.0
18.2
18.2
332.6
-2.4
2.4
5.8
17
432.5
168.0
-17.3
17.3
298.0
-6.4
6.4
41.0
18
432.0
137.0
-17.8
17.8
315.5
-37.4
37.4
1399.1
19
383.5
190.5
-66.3
66.3
4390.6
16.1
16.1
259.1
20
500.5
162.5
50.7
50.7
2574.4
-11.9
11.9
141.7
21
499.0
184.0
49.2
49.2
2424.4
9.6
9.6
92.1
7
Frequency
Vehicle Length Distrbution
8
6
4
2
0
375
400
425
450
475
500
525
550
575
More
Length (cm)
Figure 1 Distribution of the lengths of 21 vehicles
Frequency
Vehicle Width Distrbution
8
6
4
2
0
120
130
140
150
160
170
Width (cm)
Figure 2 Distribution of the widths of 21 vehicles.
8
180
190
200
210
More