9 A postscript: UK local elections and a coronation

Contents

9.0 A postscript

9.1 The coronation

King, royalty and monarchy: the distribution of opinion

The average opinion

The surface, one-sided aggregate percentage

The surface, net aggregate percentage (net score)

The underlying mean

Two measures, net score and mean, for different aspects of the monarchy

Members of the royal family

Spread

Social groups

9.2 The local elections in England, 2023

How the party votes have changed, 2022 to 2023

The size of the effect of the other two variables

The effect of the Leave vote

9.2.1 The aftermath of the local elections

The Lib Dem voters in May: their future flow to other parties

Conservative MPs: the distribution of opinion

9.3 Elections in general

Swing the same across the country – necessarily so?

Biggest losses in party’s heartlands – necessarily so?

Hypothesis: the loss proportion is constant

The best predictor of the current vote is the previous vote

Difference, proportion and swing

9.4

The proportional loss model (mathematical)

9.5 Change and trajectory

Change: local elections 2012

Change: descriptive

Change: the loss proportion model

Change: the mass and distance gravity model

Trajectory: time series

Trajectory: the mathematics of the damped pendulum

Overview

The previous chapters have charted the jubilee and the funeral of the queen and a succession of three prime ministers. At the end of it all, in September 2022, Charles became king and Rishi Sunak became prime minister. Eight months later in May 2023 there were local elections and Charles had his coronation. It is these events that we look at in this chapter - almost a year on from the tempestuous summer of 2022.

The coronation

The coronation was seen by some as providing a contrast between a unifying monarchy and a divisive party politics – a monarchy that was above party politics – a monarchy that the nation can be proud of. This is the Britannia view.

However an opinion poll carried out just before the coronation provided a better representation of the distribution of opinion, and its average and its spread. Britannia is just one part of the distribution, different from the average and on one side of the spread.

Headlines in newspapers tend to report the majority opinion but this measure is one-sided, aggregate and surface. A better measure is the net score and the best measure of all is the mean. Asked whether Charles will make a good or a bad job as king, the mean response is a low positive: +18 on a scale ranging from -100 to +100. Other aspects of the king, royalty and monarchy have means ranging from +5 to +22, in other words ranging between neutral and a quarter positive. In particular the net scores for members of the royal family range between -72 and +57. (Note that Charles is +26).

The spread of responses over the options is neither concentrated nor polarised. For each question both extremes attract at least some people. For each question with more than two options, the mode is a middle option, not an extreme option (in most cases). Moreover the distribution is single-peaked (unimodal) in almost all cases.

Different social groups have means for the king ranging from -27 to +60. The net scores show contrasts between social groups: Conservative v Labour; old v young; Leave v Remain; South Britain v London and Scotland. Groups more strongly associated with Britannia (the first-named in the pairs above) give somewhat higher means.

Local elections

Local elections were held in parts of England on 4th May 2023. Conservative poll ratings had plummeted in the summer of 2022 and had only partially recovered under Rishi Sunak’s premiership. So it was anticipated that the Conservatives would do badly, even wors than they had done four years earlier. And that was what happened: they lost 1000 seats. The gains were shared between the Labour, Liberal Democrat and Green parties.

An article by Robert Ford reported how the party votes had changed, looking at the “percentage point change in vote share across key wards from 2022 to 2023” for Conservative and Labour parties (and in one graph, the Green party); and the swing from Conservative to Labour. These voting changes were looked at for different Leave vote areas in the first graph; different graduate level areas in the second graph; and different Conservative 2022 vote areas in the third graph.

The variable with the greatest effect was the previous vote. In particular the largest decline was in areas where the Conservative vote had been largest. Straight-line equations gave approximations to the effect of previous voting on the Conservative vote, on the Labour vote and on the swing from Conservative to Labour. Straight-line equations also gave approximations to the effect of the Leave vote on the Conservative vote, on the Labour vote and on the swing from Con to Lab. The effect of the leave vote was seen as arising from the relationship between the Conservative vote and the Leave vote and an equation was identified to represent this relationship. A high graduate level was shown to give a high vote for the Greens.

These specific effects exemplified certain features of elections in general. The past vote was the best predictor of the present vote. The dominant feature of some elections is the sharp decline in the vote for one specific party. In such cases losses exhibit constant proportional loss. It is a secondary matter as to how the other parties gain from these losses. Proportional loss can be conceptualised in terms of the mean of a distribution of individual propensities to vote for a particular party.

Previous studies of change and trajectory are noted. The changes in voting between elections results are routinely analysed descriptively. Between 2007 and 2011 the votes in Scotland showed constant gains for the SNP across regions and constant proportional losses for the Liberal Democrats across regions. Between 2017 and 2019 the votes in the UK were analysed in terms of proportional flows between Remain and Leave parties. Between 2015 and 2017 the votes in the UK were analysed in terms of proportional flows between parties according to a mass-and-distance gravitational model. The trajectories in voting across elections are also routinely analysed descriptively – one study looked at elections in Ireland, 1901-2016. Another study presents a time series model for UK general elections, 1945-2015. A further study presents a damped pendulum model for US presidential elections, 1789-2020. Finally, Conservative voting in three consecutive elections is analysed in terms of a population binary process model.

9.0 A postscript

It was a postscript to last year. Last June, Boris Johnson was prime minister and Queen Elizabeth II was celebrating her Platinum Jubilee. Last September, Liz Truss replaced Boris Johnson; the queen died and Charles became king; and Liz Truss in her turn was replaced by Rishi Sunak. Now, this May, would the local elections be good or bad for Rishi Sunak? … would the coronation of King Charles III be good or bad for the British monarchy?

9.1 The coronation

“The coronation service provided the perfect illustration of why we can take pride in the survival of our particular type of monarchy, with its strange traditions and roots reaching back centuries. It is that rare thing in a noisy, divided world: a unifying institution above party politics. And as a product of our history, it is a distinctly British concoction.”

Iain Martin. The Times, May 11 2023: 23.

Iain Martin was not alone in seeing a contrast between a unifying monarchy and a divisive party politics – a monarchy that was above party politics – a monarchy that the nation can be proud of. However a recent opinion poll challenges all these notions. Certainly some people have the same opinions as Martin has but other people do not. The monarchy is not unifying – rather people have different opinions about it. Some are not so much proud of the monarchy – rather they are embarrassed by it. The monarchy is not above politics – rather it is strongly associated with the Conservative party and dissociated from the Labour party.

King, royalty and monarchy: the distribution of opinion

Just a couple of weeks before the coronation, YouGov carried out a survey of people’s opinions about the King, the royal family and the monarchy (April 25 to 26, 2023). The results were reported in The Times – see below.

Report: https://yougov.co.uk/topics/society/articles-reports/2023/05/03/where-does-public-opinion-stand-monarchy-ahead-cor;

Tables: https://docs.cdn.yougov.com/72bpekq9nm/Internal_RoyalTrackers_230426_W.pdf

The average opinion

There are a number of possible measures of average opinion. Here we consider three. Perhaps the measure most commonly used in newspaper reports, especially in the headlines, is what I refer to as the surface, one-sided aggregate percentage. Also common but an improvement on the first is the surface, net aggregate percentage (net score). Better than both those measures is the underlying mean. We apply each of these three measures in turn.

The surface, one-sided aggregate percentage

“Majority think Charles will be a good king. … 59%.”

The Times, May 2nd 2023: 1-2.

The headline refers to the fact that 59% said that Charles will make a good job of being king. I refer to such a number as a surface, one-sided aggregate percentage. I think it is a poor measure because it aggregates the ‘very positive’ and ‘fairly positive’ options and it is one-sided in that it ignores information about ‘very negative’ and ‘fairly negative’ – and ‘don’t know’. Finally, it is surface in that it ignores the underlying dimension.

The surface, net aggregate percentage (net score)

The net score for Charles being a good king is ‘middle positive’, 45.

There is a simple measure which overcomes the one-sidedness problem. The net score is the difference between the aggregate positive and the aggregate negative percentages. The net score can range from perfectly positive (+100) to neutral (0) to perfectly negative (-100). A score of +50 can be thought of as ‘middle positive’ and a

score of -50 can be thought of as ‘middle negative’.

For example the net score for Charles making a good or bad job of being king is +45. This is close to ‘middle positive’. The net score is obtained by taking the difference between the aggregate positive percentage of 59% and the aggregate negative percentage of 14%. (There were 27% don’t knows.)

The underlying mean

However I believe that the underlying mean is the best measure of the average. A person’s opinion is thought of as having a score somewhere between -100 and +100.

Consider the question about King Charles.

Here we assign scores to the options:

‘very positive’ +75; ‘fairly positive’ +25; ‘don’t know’ 0; ‘fairly negative’ -25; ‘very negative’ -75;

The percentage responses were:

‘very positive’ 20%; ‘fairly positive’ 39%; ‘don’t know’ 26%; ‘fairly negative’ 8%; ‘very negative’ 6%;

Combining the scores and the percentages we obtain the underlying mean of 18.

So in this case the underlying mean is closer to neutral than the other two measures. And my conclusion is that this is a better reflection of the underlying distribution of opinion.

Two measures, net score and mean, for different aspects of the monarchy

As well as the question about Charles, the survey asked about eight other aspects of the king, the royal family and the monarchy. The table below gives the net scores and the means for these different aspects. It also gives the number of substantive options offered by the question.

Each net score is higher than (roughly half of) the corresponding mean. The net scores range from ‘neutral’ to ‘middle positive’. The means range from ‘neutral’ to halfway between ‘neutral’ and ‘middle positive’. Overall one might say that average opinion is positive – but only somewhat positive.

The two measures place the aspects in much the same order. The top three aspects are Charles being good at the job; the monarchy being good or bad for Britain; and a monarchy being preferred to an elected head of state. However the difference in scores between the aspects is not large; and the means are not much above neutral.

Table Net scores for different aspects of the monarchy

. net score options mean

Charles: good or bad job +45 four 18

Monarchy: good or bad for Britain +43 two 21.5

Monarchy - or elected head of state +37 two 18.5

Charles: positive or negative +26 four 10

Monarchy: proud or embarrassed +25 four 15

Royal family in general: pos or neg +24 four 7

Royal family: good or bad value for money +23 four 10

Monarchy in general: pos or neg +19 four 5

Monarchy in 100 years time + 8 four 5

Members of the royal family

The following are the net scores for the positive or negative opinion for royal family members. The scores range between -72 and +57. Note that Charles is +26. Anne, Catherine and William are high. Andrew, Megan and Harry are low.

Anne +57; Catherine +54; William +52; Edward +32; Charles +26; Camilla +9; Harry -34; Megan -41; Andrew -72.

Spread

The spread of responses over the options is neither concentrated nor polarised. For each question both extremes attract at least some people. For each question with more than two options, the mode is a middle option, not an extreme option in most cases (not Harry, Megan or Andrew). Moreover the distribution is single-peaked (unimodal) in each case (not strength of feeling about pride/embarrassment).

Illustrative of these general points is the response to the question about King Charles.

The percentage responses were:

‘very positive’ 20%;

‘fairly positive’ 39%;

‘don’t know’ 26%;

‘fairly negative’ 8%;

‘very negative’ 6%;

The distribution is single-peaked and the peak is one of the middle options.

Social groups

The net scores show contrasts between social groups: Conservative v Labour; old v young; Leave v Remain; South Britain v London and Scotland. These contrasts illustrate the relationship between certain social groups and the monarchy and the notion of Britannia.

The youngest age group is negative about almost all aspects … the exceptions being that they are positive about Anne, Catherine and William – the latter two being closer to their age group.

Table Net scores for different social groups; Charles, positive or negative; in social categories A, B, C, D, E, and F

Cons 60 A

Age 65+ 52 B

Age 50-64 44 B

Leave 43 C

LibDem 37 A

Midlands 38 D

Rest of South 35 D

female 34 E

England 30 D

remain 29 C

ABC1 27 F

CDE 27 F

North 26 D

All 26

Wales 22 D

Male 20 E

Age 25-49 14 B

London 13 D

Lab 10 A

Scotland - 4 D

Age 18-24 -27 B

9.2 The local elections in England, 2023

Local elections were held in parts of England on 4th May 2023. These parts of the country had last had local elections four years previously in May 2019. That was a time when Conservative prime minister Theresa May was struggling to gain parliamentary approval for her Brexit proposals. It was a bad time for the Conservatives and they had done badly in the local elections. However, three years later in 2022 it was an even worse time for the Conservatives as Johnson and then Truss made way for Sunak. Since then, the new prime minister Rishi Sunak has been able to turn things round but only gradually: during the premiership of Liz Truss, the polls registered a collapse for the Conservative party from 30% plus down to 20% plus, and Rishi Sunak presided over an immediate partial recovery to mid-20%, then slowly recovering to just under 30% at the beginning of May.

https://en.wikipedia.org/wiki/Opinion_polling_for_the_next_United_Kingdom_general_election

So, it was anticipated that the Conservatives would do even worse in the May 2023 local elections. In order to manage expectations, the Conservatives suggested that they would lose 1,000 seats. In the event:

“PM’s worst fears realised as Tories shed 1,000 seats.”

The Times, May 6th 2023: 16-17.

The table below shows how only the Conservatives (and Other) had losses whereas the three other significant parties shared the gains. In particular it was not the case that Labour monopolised the gains – indeed it had slightly fewer gains than the combined number of gains for Lib Dem and Green.

Table Number of councillors, net losses and net gains by the different parties

Losses Gains

Conservatives -1063 Labour + 537

. Lib Dem + 407

. Green + 241

Other - 127 Other +5

Total -1190 +1190

Other: -127 = -89 Independent, -25 UKIP, -13 Residents

Source: Times as above. Also:

https://en.wikipedia.org/wiki/2023_United_Kingdom_local_elections

The BBC’s projected national share for the Conservatives fell by 2% from 28% in 2019 to 26% last week. John Curtice said that Conservatives would have got 26%, Labour would have got 35%, and Lib Dems 20% of the vote if the local elections had been held nationwide (4 May).

The latest opinion poll has Conservatives 28%; Labour 47%; and Liberal Democrat 9% (Deltapoll, 5-9 May).

Leave? = 34 =28 Cons + 5 Reform + 1 UKIP;

Remain? = 65 = 47 Lab + 9 LD + 3 SNP + 5 Green + 1 PC.

Observer, as below. Times, as above.

https://en.wikipedia.org/wiki/Opinion_polling_for_the_next_United_Kingdom_general_election

How the party votes have changed, 2022 to 2023

“This was bad enough for Sunak – tactical voting could make it even worse. Labour is bridging the Brexit gap but is not yet poised to win a general election. One thing could change all that…. An electorate on the move.” Robert Ford. Observer, May 7th, 2023.

https://www.theguardian.com/politics/2023/may/06/this-election-bloodbath-was-bad-enough-for-the-tories-tactical-voting-labour-brexit-general-election

Ford looks at how the party votes have changed – he looks at the “percentage point change in vote share across key wards from 2022 to 2023” for Conservative and Labour parties (and in one graph, the Green party); and the swing from Conservative to Labour. These voting changes are looked at for different Leave vote areas in the first graph; different graduate level areas in the second graph; and different Conservative 2022 vote areas in the third graph.

Sources: BBC key wards database; graduate data from 2021 census.

(Note: the numbers I use below are my estimates based on visual inspection of the graphs.)

The biggest change is -6.25%. This was the fall in the Conservative vote in the areas where the Conservative vote had been over 45% in 2022. Indeed, there seems to be a straight-line relationship between the fall in the Conservative vote and the level of Conservative vote in 2022. In fact it is a proportional relationship, the proportion being 0.11:

decrease in C vote = 0.11 x previous C vote in 2022 … [1]*

The decrease in the Conservative vote corresponds to the net increase in the aggregate vote for all other parties. So:

increase in other vote = 0.11 x previous C vote in 2022 … [2]

The table in the previous section above suggests that the increase in vote is shared between Labour, Lib Dem and the Green parties. Ford’s third graph also shows how the change in the Labour vote depends on the previous Conservative vote in 2022. It is not a proportionate relationship but it is a straight-line relationship (only approximately so – see the following):

change in Labour vote = 0.11 x (previous C vote in 2022 – 25) … approx. … [3]*

Equation [3] indicates that:

the change in the Labour vote is zero where the Conservative vote in 2022 was 25%; the change in the Labour vote is negative where the Conservative vote in 2022 was below 25%;

and the change in the Labour vote is positive where the Conservative vote in 2022 was above 25%.

This is what we see in Ford’s third graph – but only approximately so. In the third graph the change in the Labour vote stays zero or low for middling levels of the Conservative vote – and only turns negative or positive for the lowest or highest levels of the Conservative vote, respectively.

From the two change equations [1] and [3] we can deduce an approximate swing equation and this corresponds roughly with what is shown in the third graph.

[Note: equations [1] and [3] are averaged.]

swing from Cons. to Lab. = 0.11 x (previous C vote in 2022 – 12.5) … approx [4]*

We can also deduce from [1] and [3] an approximate equation for the change in the combined Lib Dem and Green vote.

[Note: equations [1] and [3] are subtracted.]

change in combined Lib Dem and Green vote = 2.75 … approx … [5]

Equation [5] can be modified to take account of the fact that the Labour change stays flattish in the middle. To compensate for this the change in combined Lib Dem and Green vote needs to have a peak in the middle part of the range – the gains are greatest in the middle.

*Note: Equations [1], [2] and [4] correspond to what is displayed in Ford’s third graph and are broadly in line with what is displayed. Equations [3] and [5] arise from my reasoning and I am unsure as to whether I have been careful enough with my reasoning here. Sorry.

The size of the effect of the other two variables

Ford’s three graphs look at the effect of the Leave vote; the effect of the graduate level; and the effect of the Conservative vote in 2022. The size of the effect of the Conservative vote is roughly double the size of the effect of the other two variables. This can be seen even from the labelled ranges for the vertical scale: -4 to +2 for the other two variables, but -6 to +4 for the Conservative vote label.

The effect of graduate level is rather small except where there are over 35% graduates. There: “Labour is down in areas with the highest share of graduates, losing ground to the Greens”.

The effect of the Leave vote

We now turn to Ford’s first graph which looks at the effect of the Leave vote: “Labour made its largest gains in areas with the highest Leave vote”.

In what follows I deliberately keep the gradient at 0.11. I estimate:

(Labour change - 0.5) = 0.11 (Leave -55) … [6]

I also estimate the following equations from the graph:

(Swing from Cons to Lab – 2) = 0.11 (Leave – 55) … [7]

(Cons change + 3.5) = - 0.11 (Leave – 55) … [8]

Note that the swing equation [7] is indeed the difference between [6] and [8] divided by two.

My suggestion here is that the results in the first graph are due to the correlation between the Leave vote and the Conservative vote. First of all I note that the numbered range for the leave vote is 48 to 63 and the numbered range for the Conservative vote is 15 to 45. The difference is 23 or so. It so happens that equations and [8] can be used to deduce equation [9]:

Leave = Cons + 23 … [9]

9.2.1 The aftermath of the local elections

What do the results of the local elections imply for the future? More than half those voting Liberal Democrat in the local elections intend to vote for a different party at the next general election. Anticipating that they will lose the next election, Conservative MPs are positioning themselves to replace prime minister and party leader Rishi Sunak.

…

The Lib Dem voters in May: their future flow to other parties

“Labour poised to snatch quarter of Lib Dem support at election.” 15, 8.

The Liberal Democrats did well in this month’s local elections – much better than they were doing in the national opinion polls. This suggests that the people who voted Lib Dem in the local election will flow away to other parties at the next general election.

A survey by You Gov reported “how Lib Dem local election voters would vote in a general election” (May 10-11):

44% would stay with Lib Dem

23% would vote Labour

17% did not know

9% would vote Tory

3% would Green

3% would vote Reform UK

1% would vote other

YouGov, Labour Together

…

These figures can be related to the proportional loss model. The equation dL=0.56L says that the anticipated change dL in the LibDem vote is 56% of the LibDem vote at the local election.

…

Conservative MPs: the distribution of opinion

“Fractious Tory tribes who could make it a lot harder for Sunak.” 16, 7.

“The Conservative Party is becoming restless after this month’s local election result, in which the Tories lost more than a thousand seats. The internal politics of the party – and the potential threats they pose to Rishi Sunak’s leadership – will be critical in the run-up to the next election.”

The article identifies seven different tribes, each with their own views, and gives a trouble rating T:

…

The One Nationers … centrist, moderate, liberal conservatives

… Jeremy Hunt … T=0/5

The Resolute Sunakites … sound money, steady government … T=0/5

The Silent Majority … T=1/5

Diehard Johnson Supporters … Priti Patel … T=2/5

[Conservative Democratic Organisation:

https://www.conservativedems.co.uk/;

https://www.conservativedems.co.uk/conference]

The Quiet Revolutionaries … focus on red-wall working class - not on affluent, southeast managerial class …T=3/5

Disenchanted Trussites … small state, low tax, pro-growth …

… Jacob Rees-Mogg … T=4/5

The Right-Wing Brexiteers … Eurosceptic, socially conservative, low tax …

… Suella Braverman … T=3/5

…

Recall that the Remain Scores for the following candidates last summer were:

Braverman 3; Truss 10; Mordaunt 12; Sunak 34; Hunt 55

[Aside: it was Mordaunt who bore the sceptre at the coronation.]

5 Johnson to Truss in ten weeks; social choice

It so happens that I wrote a rather obscure mathematical account of “Tory rebelliousness” back in 2012. If I remember correctly, one of the rebels was Jacob Rees-Mogg. It was the time of the Cameron-Clegg coalition.

Tory rebelliousness, 2012;

https://www.thetimes.co.uk/article/new-model-tories-are-emboldened-to-risk-careers-by-lack-of-retribution-88r3xp00hwm ;

Also: https://www.thetimes.co.uk/article/coalition-creaks-as-rebels-flex-muscles-w5ddfjsxlt7 .

9.3 Elections in general

Swing the same across the country – necessarily so?

Biggest losses in party’s heartlands – necessarily so?

“Biggest losses in party’s heartlands” has always been a popular headline. It was so for the big losses for the Liberal Democrats in the general election of 2015. It was so for Labour in the general election of 2019. And now in last week’s local elections it was so for Rishi Sunak’s Conservatives.

Hypothesis: the loss proportion is constant

The two previous sections appear to be saying opposite things. However they can be reconciled as follows:

Places which have the same party composition have the same swing.

Places which have different party composition have different swings.

And these two statements can be reconciled by the hypothesis:

The loss proportion is the same for all subgroups.

The best predictor of the current vote is the previous vote

The best predictor of the current vote is the previous vote …

Difference, proportion and swing

The vote for one party

Consider the vote for one party. Looking at the current vote and the previous vote, there are two basic measures of change which we can use: the difference and the proportion.

difference = current – previous

proportion = current / previous

These two measures are related as follows. The graph of difference against previous is a straight line with a slope of gradient m equal to (proportion - 1). The gradient m may be positive or negative and we can refer to the proportion gain or the proportion loss.

[An example of such a graph is Robert Ford’s third graph – see above.]

difference = (proportion - 1) previous = m previous

9.4 The proportional loss model (mathematical)

Although the real situation is more complex the broad features of Ford’s third graph can be represented by a simple model which I refer to as the proportional loss model. It seems plausible that this model is suited to situations where the dominant feature of an election is the loss of vote by one specific party.

If each former party voter has a certain probability of voting for another party, then any group of such voters will have a mean probability and this will be expressed as a proportion voting elsewhere. Similar groups of voters will have a similar proportion. In general, the proportional loss will be constant across groups - across wards or constituencies. Wards with large previous party voters will have the same constant proportional loss but a large loss in terms of percentage difference.

Cons

Consider a party ‘Cons’. Let its previous percentage vote be C and its current percentage vote be C’. Let dC denote the difference:

dC = C’ – C … [1]

Let p be the proportion:

p = C’/C … [2]

Then:

dC = p C – C = (p-1) C … [3]

Let us denote the ‘difference proportion’ (p-1) by m. So m=(p-1). So:

dC = m C … [4]

Note that if p is less than 1, then (p-1) is negative and m is negative.

Rest

We now turn our attention to ‘Rest’, the rest of the parties. Let its previous percentage vote be R and its current percentage vote be R’. Let dR denote the difference:

dR = R’ – R … [5]

Of course together Cons and Rest constitute everybody, 100% of the vote. So

C + R = 100 … [6]

C’ + R’ = 100 … [7]

dC + dR = 0 … [8]

So the change in R is the opposite of the change in C:

dR = - dC … [9]

So:

dR = - m C … [10]

Note for later that the swing from Cons to Rest is the difference of the differences divided by 2:

(dR – dC)/2 = (dR + dR)/2 = dR … [11]

The two parties in Rest

We now consider Rest to comprise of two parties Lab and Third. We denote changes in their vote by dL and dT. Together the changes constitute dR.

dR = dL + dT … [12]

Suppose that the change in Third is a constant k. Then the changes are:

dT = k … [13]

dL = dR – dT = - m C - k … [14]

If C=0 then dL=-k. If C=-k/m then dL=0.

Ford’s third graph displays equations [4] and [14]. It also displays the swing SCL from Cons to Lab:

SCL = (dL - dC)/2 = (- m C – k – m C)/2 = - m C – k/2 … [15]

If C=-k/2m then SCL=0. Thus the swing SCL and the difference dL both turn negative but do so at different levels of C.

Note that back in equation [4] I have deliberately referred to the difference proportion m, without saying whether m is positive or negative. The above argument runs irrespective of that matter. Of course in order to apply to Ford’s third graph, m is negative.

9.5 Change and trajectory

Change: local elections 2012

Change: descriptive

Change: the proportional loss model

Change: the mass and distance gravity model

Trajectory: time series

Trajectory: the mathematics of the damped pendulum

Trajectory: population binary processes

Change: local elections 2012

“The forthcoming local elections promise to exhibit large changes in the percentage

vote for Conservatives, Labour, Liberal Democrats and (in Scotland) SNP. Rallings

and Thrasher (2012) note that the three main parties gained 43%, 24% and 23%

respectively in the local elections of 2008 whereas the parties’ current standing in

national opinion polls is 29%, 40% and 11% (with UKIP and others having 10%

apiece).

Rallings and Thrasher (2012) Local voters ready to punish Tories. Sunday Times, April 29. p. 17.

Sunday Times (2012) Tories at lowest ebb for 8 years. April 29. p. 1.

How are these changes to be measured? The ‘swing’ is a commonly used method

based on the change in the percentages between the two leading parties. As well as

looking at the change in percentages, there is merit in looking at the proportional

change and also at the change in z-scores. Two examples are presented the victory of

George Galloway in the recent Bradford West bye-election; and the SNP gains and

the Liberal Democrat losses in last year’s Scottish Parliament elections.

Curtice, J. (2012) Was this the biggest shock in by-election history? Not quite ... The Independent. Saturday 31 March …”

2 How the vote changes: percentage change, proportional change orchange in z-score?

Change: descriptive

UK general elections, 2010-2015:

12: elections 1945-2015, pages 1-9

Change: the proportional model

See above discussion of UK local elections, 2023.