Statistics on Commute Times around Aix-en-Provence / Marseille

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 school-term journey time" as 100%, and show the individual values as percentages of this.

Someone moving into the Aix-Marseille region, or starting at a new place of work, for example, could measure the autoroute-dominated 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 rush-hour. 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 school-term journey time (irrespective of day of week, or whether it is school term or vacation).

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 term-time, 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 mid-day on the Wednesday and Saturday). Primary schools do not have the Saturday half-day, and usually do not have the Wednesday half-day 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 part-time 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 rush-hour 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 / zⁿ

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.

Week-by-week 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 5-week 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 work-force 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.

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(w₁) + fn(w₂) + ... + fn(w_k) ) / k )

The geometric mean uses

fn(x)=ln(x)

fn^-1(x)=exp(x)

but all the others use

fn(x)=xⁿ

fn^-1(x)=x^1/n

So that n=2 for a root-mean-square, 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 off-cut 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 Bouches-du-Rhône 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 (north-west 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 south-east 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.

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.