Statistics on Commute Times around AixenProvence / Marseille
This page has been prepared
as a potential aid for anyone who might be thinking of moving home
to live in Provence, in the southeast of France.
It gives some idea of the time that is likely to be spent
in travelling, in the daily commute to and from work.
The default speed limits for cars on French roads are as follows:
130km/hr on autoroutes, 110km/hr on dualcarriageways,
90km/hr on main roads and 50km/hr in builtup areas.
In times of precipitation, this reduces to:
110km/hr on autoroutes, 100km/hr on dualcarriageways,
80km/hr on main roads and 50km/hr in builtup areas.
Typical journeys average lower speeds than this, of course.
A very useful rule of thumb is to suppose that longhaul journeys, dominated by autoroute driving,
will average 100km/hr (allowing for rest periods, and other delays);
and that shorthaul journeys, dominated by major roads,
will average 60km/hr.
Thus, "Marseille 325km" on an autoroute will take about 3.25 hours to cover,
and "Forcalquier 44km" on main roads will take about 44 minutes.
During rushhour periods, though, the speeds are much lower even than this.
This page reports the results of an experiment that was run over several years (19982006)
using my own car as a probe to measure traffic density.
Graph of mean speeds (as a percentage of 66km/hr)
plotted against hour of the day.
The calculated figure of 66km/hr for the normal termtime morning commute
is not too far from the published national statistics
showing that the average Frenchman commutes 31km and takes 26 minutes each morning.
Interpolated speed (km/hr, mornings, termtime) 
Year 
Overall 
Mondays 
Tuesdays 
Wednesdays 
Thursdays 
Fridays 
2006 (evenings)  74  80  73  73  70  
2006  66  58  59  73  64  77 
2001/2  70  68  68  73  68  72 
2000/1  71  69  68  74  70  73 
1998/9  50  48  47  52  47  53 
Interpolated speed (km/hr, mornings, school vacation) 
Year 
Overall 
Mondays 
Tuesdays 
Wednesdays 
Thursdays 
Fridays 
2006 (evenings)  77  84  72  69  80  
2006  86  88  85  79  90  88 
2001/2  80  80  78  81  79  81 
2000/1  76  75  77  77  77  74 
Mean Normalised Journey Time
The data are more conveniently expressed
as percentages of some nominal journey time,
as shown in the next two tables.
These tables take the "mean morning schoolterm journey time" as 100%,
and show the individual values as percentages of this.
Someone moving into the AixMarseille region, or starting at a new place of work,
for example, could measure the autoroutedominated part of the journey on a map,
and use the 66km/hr mean as a very initial estimate of how long this part of the journey will take
in the morning rushhour.
The morning commute during school vacation is anticipated to be about 82%
(irrespective of the day of week),
and returning home in the evening is anticipated to take about 91%
of the mean morning schoolterm journey time
(irrespective of day of week, or whether it is school term or vacation).
Mean journey time (%, mornings, termtime)
(percentage of overall morning mean value) 
Year 
Overall 
Mondays 
Tuesdays 
Wednesdays 
Thursdays 
Fridays 
2006 (evenings)  92  86  93  93  95  
2006  100  110  109  92  102  89 
2001/2  100  102  102  96  103  96 
2000/1  100  102  105  95  101  97 
1998/9  100  104  105  95  105  93 
Mean journey time (%, mornings, school vacation)
(percentage of overall morning mean termtime value) 
Year 
Overall 
Mondays 
Tuesdays 
Wednesdays 
Thursdays 
Fridays 
2006 (evenings)  89  84  93  96  86  
2006  82  81  83  87  80  81 
2001/2  87  87  89  86  88  86 
2000/1  93  94  93  92  93  95 
As already noted,
the above tables show that the figures for the evening commute,
and for the morning commute, during school vacations,
are all fairly constant, and hardly vary according to the day of the week.
The times for the morning commute during termtime, though,
are highly dependent on the day of the week
(and the variability is certainly noticeable, in the car, while making the journey).
The schools in the area start fairly early in the morning
(08:30 for primary schools, and 08:00 for secondary schools) on Monday to Saturday
(finishing at midday on the Wednesday and Saturday).
Primary schools do not have the Saturday halfday,
and usually do not have the Wednesday halfday either.
Thus, the school traffic on the road on Wednesday mornings
is less than that on the other weekdays.
Moreover, since someone needs to be at home on Wednesdays to look after those children,
the work traffic is less on Wednesday mornings, too,
with most parttime staff not working on those days.
(The same reasoning also explains the reduced traffic during school vacations.)
The reason for the drop in journey time on Friday morning is less obvious.
People are preparing for their weekend breaks,
so might take the Friday off
(strangely not the Monday, though, probably for psychological reasons);
or else go earlier to work on the Friday, so as to be able to leave earlier in the evening,
thereby spreading the rushhour traffic more thinly.

Analysing the Impact of Measurement Error
 Handling the absolute error of terms in a summation is easy,
but handling their relative error is less so.
Starting with:
 w = x + y – z
 this becomes:
 w±∆w = x±∆x + y±∆y – z±∆z
 and hence (allowing for the worst case):
 ∆w = ∆x + ∆y + ∆z
 With product terms, it is the other way round:
handling their relative error is easy,
but handling their absolute error is less so.
Taking an expression like:
 w = x . y^{m} / z^{n}
 the relative error of the final answer
is the sum of the relative errors of
each of the product and quotient terms.
This can be easily demonstrated by taking logs of the initial expression, giving:
 ln(w) = ln(x) + m.ln(y) – n.ln(z)
 and then noting that ln(r) is the integral of (1/r).dr,
so ln(r)≈∆r/r, thus giving:
 ∆w/w = ∆x/x + m.∆y/y – n.∆z/z
 Noting that in the worst case, errors accumulate, rather than cancel, this gives:
 ∆w/w = ∆x/x + m.∆y/y + n.∆z/z

Handling General Functions
 To handle the error ranges in general functions, such as:
 w = f(x)
 a simple differentiation can be used:
 ∆w = ∆x.f’(x)
 For the relative error:
 ∆w/w = ∆x . f’(x) / f(x)
 Thus for w = sin(x):
 ∆w/w = ∆x.cos(x)/sin(x) = ∆x.cot(x)
 Generalising even further:
 w = f( x, y, z )
 the overall error is the sum of errors obtained by
taking the partial differential with respect to each of the variables in turn:
 ∆w = ∆x.(∂/∂x)f(x,y,z) + ∆y.(∂/∂y)f(x,y,z) + ∆z.(∂/∂z)f(x,y,z)

Standard Deviation for the Normalised Journey Time
The standard deviations of the data,
which were used for computing the above mean values,
give an idea of the level of variability of the data within each sample.
These have been expressed, here, as percentages of their respective means.
Standard deviation/Mean journey time (%, mornings, termtime) 
Year 
Overall 
Mondays 
Tuesdays 
Wednesdays 
Thursdays 
Fridays 
2006 (evenings)  13  11  18  8  9  
2006  16  14  18  11  13  9 
2001/2  11  12  9  11  13  7 
2000/1  9  9  10  7  10  7 
1998/9  11  10  9  13  11  8 
Standard deviation/Mean journey time (%, mornings, school vacation) 
Year 
Overall 
Mondays 
Tuesdays 
Wednesdays 
Thursdays 
Fridays 
2006 (evenings)  7  6  1  2  4  
2006  7  7  8  8  7  5 
2001/2  4  5  5  3  4  3 
2000/1  6  6  5  4  5  8 

Weekbyweek Fluctuations in Normalised Journey Time
There is a noticeable fluctuation, over the weeks,
in the amount of traffic on the roads in the morning.
The following table attempts to quantify this.
The figures have been derived as a 5week moving average
(plus or minus 2 weeks either side of the central week)
with the figures for 2006 weighted more heavily than those for the other years.
These fluctuations are presumably an indication of weeks
where nearly 100% of the workforce has decided to work the complete week
(and to conserve their days off for later),
versus weeks where some people have taken one or more days off,
for one reason or another.
However, the trends are not consistent over the years,
and these figures are only presented here for interest.
Mean journey time (%, mornings, termtime)
(percentage of overall morning mean value) 
Week 
0109 
1019 
2029 
3039 
4049 
5053 
0   97  100   103  102 
1  104  96  101   104  102 
2  103  96  101   106  102 
3  104  95  102   106  
4  106  94  103  90  105  
5  105  93  102  101  102  
6  102  95  102  101  101  
7  101  97  101  100  101  
8  100  98  101  101  100  
9  97  98  99  103  101  

Taking Means from Repeated Measurements
 Suppose that an experiment is repeated k times,
each time producing a slightly different value for w,
and that we now take the mean of these values:
 v±∆v = mean( i=1..k, w_{i}±∆w )
 Many different types of mean can be used,
depending on the context.
In general, each of them takes the form:
 v = fn^{1}( ( fn(w_{1}) + fn(w_{2}) + ... + fn(w_{k}) ) / k )
 The geometric mean uses
 fn(x)=ln(x)
 fn^{1}(x)=exp(x)
 but all the others use
 fn(x)=x^{n}
 fn^{1}(x)=x^{1/n}
 So that n=2 for a rootmeansquare,
n=1 for an arithmetic mean,
n=0 for a geometric mean,
and n=1 for a harmonic mean.
 For the geometric mean,
the product of all the values of w_{i} would,
according to the earlier analysis for product terms, make no difference, since:
 ∆v/v = (1/k) . (k.∆w/w) = ∆w/w
 In the more usual case of the arithmetic mean, though,
we can note that ∆w should behave like 3σ of w
(provided that the sample and population conform to a normal distribution).
In that case:
 ∆v/v = (∆w/w) / √k

Summation Expressions
 Looking closer at the case of addition, in w=x+y,
in an attempt to find the relative error:
 ∆w/w = ( x.∆x/x + y.∆y/y ) / ( x + y )
 the closest that we can get is to note that, provided that x and y have the same sign:
 ∆w/w ≤ max( ∆x/x , ∆y/y ) ≤ ( ∆x/x + ∆y/y )
 This also works for subtraction, provided that x and y have opposite signs.
However, it most definitely does not work for subtraction
when the two parameters have the same sign
(or for addition, when x and y have opposite signs).
 If a 10 cm rod, ±1%, has a 6 cm section cut off, ±10%,
it leaves an offcut of length 4 cm, ±0.7 cm, i.e. ±17.5%.
Indeed, the sky's the limit,
since dividing by (x–y) can involve multiplying the numerator by huge values,
as y tends to x.

Details of the Source Data
The data actually collected consists of the times of departure and arrival,
between La Tour d'Aigues and the car park at the destination.
La Tour d'Aigues
(43.73°N, 5.55°E, 270m alt.) is 6km from Pertuis,
in the southeast corner of the Department of Vaucluse in France.
The border with the Department of the BouchesduRhône is reached immediately on
leaving Pertuis, via a suspension bridge over the River Durance.
The 1998/9 records span most of the academic year,
from Wednesday, 4th November, in week 45 of 1998,
continuing at Monday 4th January, in week 2 of 1999,
to Friday, 25th June, in week 26 of 1999.
These timings involve a distance of about 35km,
into the centre of Aix, rather than just round its outskirts.
At that time, the Rocade Ouest (Western Bypass) was undergoing major construction works:
there were traffic lights on the Célony side of Aix (northwest side),
and daily traffic jams that stretched back to Les Platanes.
All of this explains why the average speeds are so much slower than for the other years.
The 2000/1 records start at Tuesday, 1st February, in week 6 of 2000,
continuing at Wednesday, 3rd January, in week 1 of 2001,
and end on Thursday, 31st May, in week 22.
These timings involve a distance of about 51km,
to the car park at work in Rousset (just southeast of Aix).
Thanks to the adoption of flexitime at work,
the data are collected over a variable range of morning times.
The 2001/2 records start at Tuesday, 5th June, in week 23 of 2001,
continuing at Wednesday, 2nd January, in week 1 of 2002,
and end on Wednesday, 14th August, in week 33.
These timings involve a distance of about 50km,
to the car park at work in Rousset.
From the beginning of June (week 23),
the route to work was shortened by 1km
by the opening of a connecting road at Château Neuf le Rouge.
From the beginning of July (around week 28),
the route through Pertuis was improved
by the opening of a connecting road across the River Eze on the east side of the town.
The 2006 records start at Monday, 2nd January, in week 1 of 2006.
These timings involve a distance of about 58km to L'Estaque.
The road between La Tour d'Aigues and Pertuis had been widened and straightened in 2005,
which has marginally improved the journey time from La Tour d'Aigues
to the suspension bridge.

The following general comments can also be made on how the statistics were collected:
 It is important to note
that the original aim of the exercise was as an aid for planning my own journey times,
to help minimise the amount of my working day spent in the car.
All other applications of these statistics are byproducts
for which they were not initially designed.
 The data are independent of the route taken.
That is, the travel times represent the best available journey times to work,
including the use of skill and judgement to pick detours where appropriate.
 A detour, of course, adds extra kilometres to the route,
but this is not taken into account.
The use of a detour is like a gambit in chess:
adding kilometres in the slightly paradoxical hope of reducing the number of minutes.
 Delays caused by unusual events, such as traffic jams behind accidents,
were considered to be part of the statistics.
 If minor accidents were more likely to occur on one particular day of the week, for instance,
this was to be included in the data.
 However, not wanting abnormally serious incidents to cloud the statistics,
data were considered to be not part of the population,
and hence excluded, if they were outside 3 standard deviations of the mean value.
 Similarly, the statistics were lumped together regardless of the weather conditions.
In bad weather,
the traffic speed should adapt automatically to the bad driving conditions,
but French law also enforces lower speed limits on main roads and autoroutes
(both during periods of precipitation, and for pollution control during a heatwave).
However, this was considered just to be part of the statistical noise
(except, again, where freak conditions, such as snow,
took the figures outside the 3 standarddeviation limits).
 The raw data include the starting and finishing times to the nearest minute.
 Originally, the journey time was defined as the time between turning the ignition on,
and turning it off again at the other end.
The time to get to and from the car is not included.
 The averaged speeds,
for commuting in the morning and evening,
have been derived by interpolation:
the suspension bridge is 9.2km away from La Tour d'Aigues,
and on a normal day takes 14 minutes to cover in the morning and evening commuting periods.
These figures have been subtracted out when interpolating the average speed,
so as to give a rough estimate of the traffic conditions in the BouchesdeRhône.
 Termtime mean journey times (averaged across the days of the week)
were plotted against the journey starting time
(as shown at the top of the page).
 The percentages have been expressed with respect to the mean, termtime,
Wednesdaymorning, commute times.
They, therefore, need to be scaled twice,
using figures from the earlier tables, to put them back into normal units.
Wednesday mornings were chosen as the most stable reference, with respect to starting time,
for which values had been collected for all the past records.
 Beware, though.
Most of the measurements were taken between 07:1508:15
(and 17:4518:45 for the evening times),
and were definitely not collected as a random sample.
Indeed, the very purpose of the exercise was originally to help me
to find how I could bias my journey times,
to minimise the time wasted in traffic jams.
