# Introduction

Although Canada has maintained an official and legal English-French bilingual status since 1969, only a few cities showcase significant proportion of both native English and French speakers. This post tries to determine if there exists regions where both English and French coexist. It specifically focuses on Montreal and Ottawa, which are the only large metropolitan areas with a significant share of both official languages.

Empirical evidence shows that English, French and non-official languages can indeed live side-by-side. However, data also reveals clear spatial clustering patterns, which suggests that even within a given metro, there exists important linguistic segregation.

# Montreal

As the choropleths below highlight, there appears to be considerable linguistic clustering 1 in the Montreal census metropolitan area (CMA).

The periphery of the metropolis is almost exclusively French. The only exceptions are Brossard and Longueuil on the South Shore, which have a sizable proportion of speakers of non-official languages as well as areas like Hudson with a large proportion of anglophones. French speakers are also well represented in the East of the island as well as in Laval.

As can be expected given the historical East-West linguistic divide of the island, English speakers cluster in the West-Island an in areas near Hudson and Vaudreuil. Perhaps surprisingly, this is the only area in the region where they are present in significant numbers. On the other hand, non-official languages are relatively well spread out on the island as well as Laval and Brossard.

Nonetheless, the following tricolor map also suggests that there are areas where different groups coexist. Laval, Brossard, Rivière des Prairies and Montréal Nord namely showcase numerous dissemination areas (DAs) 2 with both high proportion of French and non-official languages. Parts of the West-Island also have a mix of English and non-official languages. Some of the most mixed areas in terms of all 3 three groups, represented in dark grey, can be found in Pierrefonds, Vaudreuil, Verdun and downtown.

Another observation emerges from the figure below: when francophones form a plurality in the dissemination area, they are much likelier to coexist with non-official speakers than anglophones. Anglophones are also slightly likelier to live with non-official speakers than francophones when they form a plurality. Finally, allophones are likelier to pair with francophones in DAs with a plurality of non-official speakers.

This is interesting and might suggest it is easier for anglophones or francophones to speak their mother tongue on a daily basis if their neighbours are predominantly allophones rather than the other official language group. Indeed, allophones likely speak multiple languages and might have a greater tendency to switch language easily. Immigrants might namely be more willing to speak French or English to facilitate their integration.

Although interesting, the figure above only shows the number of prevalent language pairs per DA. The 2D contour plot below gives further details on the exact linguistic composition of the area by presenting the plurality gap for each DA, which represents the difference in the proportion for the 2 most prevalent languages.

We immediately notice that non-official and English speakers both have a greater tendency to live with other language groups than francophones. Indeed, the red and green distributions have a large density of points close to the bottom left corner which reflects the fact that even when forming a plurality, these groups rarely form more than 50% of the DA’s population. It is interesting to note that while a few DAs are formed by a large plurality of non-official speakers (80% or more), this is never the case for anglophones.

# Ottawa

The Ottawa CMA, which straddles Ontario and Quebec, also shows some very interesting patterns. We can immediately make out the Ottawa/Outaouais river forming the boundary between Gatineau (Quebec) and Ottawa (Ontario). Francophones cluster north of the river while English speakers concentrate in the Ontario portion of the CMA. On the other hand, non-official languages are more prevalent closer to downtown.

The tricolor map below confirms these findings, with English speakers making up the important majority of rural regions on the south Ontario side and French speakers also very prevalent in rural areas on the Quebec side, albeit to a lesser degree. This is also reflected in the clusters of points in the triangle.

As can be expected, rural areas are much more homogeneous linguistically than the central urban parts of the CMA. The most mixed areas seem to be close to Vanier, just over the Rideau Canal and to some extent further East towards Orleans. Both sectors were historically French and have witnessed in increase in anglophone and allophone speakers in recent years.

On the Ontario side, non-official languages mainly coexist with the predominant official language, in this case English. When official language groups form a plurality, the second most prevalent language is also less likely to be the other offical language group.

Conversely, French plurality regions are likelier to have English as the second most prevalent language on the Quebec side. This can namely be explained by the presence of important English communities in cities like Chelsea, West of the Gatineau river as well as the fact that there is a smaller proportion of non-official language speakers in the north of the CMA.

The figure below shows that English and French dominated DAs are the most prevalent. This is not the case of Montreal where French and non-official speakers were likelier to form pluralities.

Contrary to Montreal, the figure below shows that both English and French speakers are likely to cluster in very homogeneous DAs. Indeed, both official language groups, particularly francophones, have high density of points closer to the top right corner in the figure below.

Non-official languages rarely form pluralities, and even when they do, they never form more than 70% of the DA’s population. This could suggest non-official speakers have a greater likelihood of mixing in with official speakers in Ottawa than in Montreal.

# A quick summary

The analysis above reveals important differences between both cities, namely that allophones and francophones have a greater likelihood of living side-by-side in Montreal while anglophones and francophones tend to be closer in Ottawa.

However, results also call for caution in drawing definitive conclusions. Many observations might simply be due to the fact that more prevalent languages naturally have a higher probability of being found together. Indeed, as shown in first bar plot in the previous post, non-official speakers are more prevalent than anglophones in Montreal, while this is the opposite in Gatineau. Similarly, there are more allophones than francophones in the Ontario part of Ottawa.

Linguistic coexistence could therefore simply be attributable to the demographic weight of sub-populations rather than some sort of mutual repulsion or attraction. The greater number of non-official speakers in Quebec’s largest city might also explain why non-official languages seem to display greater clustering there.

# Which city is more linguistically mixed?

As a final comparison between both cities, we define the following simple homogeneity index per DA:

$h(x)= \sqrt{\sum_{i=1}^3 (x_i-\frac{1}{3})^{2} } \qquad, \forall x \in \mathbb{R}^3$

which measures the degree of language mixing. This index is given by a norm, hence its minimal value is $$0$$ when $$x_i=\frac{1}{3}, \forall i=1,2,3$$, which means that all language groups are present in the same proportion in that DA.

We can also show that it is maximized when $$x_i=1, x_j =0, \forall j \neq i, \forall i=1,2,3$$. In other words, when exactly one of the languages is present and the others are completely absent. In that case, the index achieves its maximal value of $$\sqrt{2/3} \approx 0.816$$. 3

We compute this index for all DAs in each CMA and weight the results by population. This results in the following cumulative homogeneity distribution below. Results indicate it is not possible to assert that one city is definitely more diverse than the other.

The curves indicate that a larger proportion of Montreal’s population lives in linguistically diverse DAs with an index of 0.32 or less than Ottawa. However, the 75% most homogeneous DAs in Montreal are more homogeneous on average than those in Ottawa. This difference might jointly be explained by the greater urban (hence mixed) character of Montreal combined with the fact that the rural parts of Ottawa are more diverse than those in Montreal.

# Conclusion

The analysis above provides compelling insights on the linguistic composition of both multilingual cities. Although both present very clear spatial clustering patterns that suggest speakers of a given language tend to group together, it is also clear that many areas are extremely linguistically mixed.

In Montreal, non-official language speakers and francophone have the greatest likelihood of living side-by-side while in Ottawa, English and French tend to be closer. This observation might simply be due to the fact that more prevalent languages naturally have a higher probability of being found together.

We also observe that urban areas are likelier to be linguistically mixed than rural zones. Indeed, linguistic composition can change quickly as cities undergo rapid demographic shifts. It therefore remains to be seen whether areas like Orleans and Vanier in Ottawa can maintain diversity after a recent combination of increased density and migratory influx.

1. Using the local and global Moran’s I statistics confirm the existence of clustering for all three groups.

2. Dissemination areas represent the smallest geographical unit for which linguistic data is collected by Statistics Canada for the census.

3. For any $$y \in \mathbb{R}^n$$ the following inequality holds $$||y||_2 \leq ||y||_{1}$$. Since $$||y-\frac{1}{3}||_2 = \sqrt{||y||_2^2 - \frac{2}{3} y^{\top}1 + ||\frac{1}{3}||_2^2}$$ is also true, it follows that :\begin{align} \max_{y \in \mathbb{R}^n} & \sqrt{||y||_2^2 - \frac{2}{3} y^{\top}1 + ||\frac{1}{3}||_2^2 } & \leq & \quad \max_{y \in \mathbb{R}^n} & \sqrt{||y||_1^2 - \frac{2}{3} y^{\top}1 + ||\frac{1}{3}||_1^2 } \\ \text{s.t.} & y^{\top}1 =1 & & \quad \text{s.t.} & y^{\top}1 =1\\ & y \geq 0 && &y \geq 0 \end{align}Since $$y^{\top}1 = ||y||_1, \forall y \geq 0$$ it follows that both optimization problems have maximal value $$\sqrt{\frac{2}{3}}\approx 0.816$$.