Mathematical and quantitative analysis of the 23rd round of the Scottish Championship, season 2025-2026

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Mathematical and quantitative analysis of the 23rd round of the Scottish Championship, season 2025-2026

The 2025-2026 Scottish Championship has established itself as one of the most dynamic and statistically challenging leagues in Europe, characterised by a high degree of parity and extreme volatility in the middle of the table. ¹ As we enter the 23rd round, the discipline of quantitative analysis becomes crucial to understanding the underlying trends that often remain hidden behind the superficial results. The application of the calculation protocol, based on the “MATHEMATICAL CALCULATION PROTOCOL” and the Harmony Index (HI), allows the raw data to be transformed into objective assessments of risk and probability. ³ In this report, we will analyse the five key matches of the round using the “cabbage soup” method – a focus on mathematical precision, speed of calculations and strict adherence to the statistical dependencies defined in the Master_Template. ³

The current situation in the league shows a clear leadership of St. Johnstone, who maintain a significant distance from their pursuers. ¹ However, the battle for play-off places and the fight for survival at the bottom of the table create a complex web of dependencies. Statistical data show an average of 2.5 goals per match throughout the league, which is an important parameter in calculating the attacking and defensive strength of teams. ¹ Before moving on to the individual analysis of each match, it is necessary to reflect the current ranking, which serves as the basis for our calculations.

Current state of the Scottish Championship ahead of Round 23

Championship standings are the cornerstone of any statistical modelling. As of 19 January 2026, St Johnstone dominate with 45 points, followed by Partick Thistle with 40 and Arbrough with 36. ¹ These positions are not just numbers, but a reflection of long-term stability, or lack thereof.

Position	Team	Matches	Wins	Ties	Losses	GV	GP	GR	Points
1	St Johnstone	21	13	6	2	43	17	+26	45
2	Partick Thistle	22	11	7	4	33	25	+8	40
3	Arbroath	22	10	6	6	30	27	+3	36
4	Ayr United	21	6	11	4	27	24	+3	29
5	Dunfermline	20	7	5	8	28	22	+6	26
6	Raith Rovers	22	6	7	9	23	23	0	25
7	Greenock Morton	21	4	10	7	20	28	-8	22
8	Airdrieonians	22	4	8	10	18	31	-13	20
9	Queen’s Park	19	3	9	7	20	29	-9	18
10	Ross County	20	3	7	10	20	36	-16	16

The data in Table ¹ clearly shows that St Johnstone have the best defence (only 17 goals conceded) and the strongest attack (43 goals scored). This creates a significant deviation in the calculation of the Poisson Distribution for their matches. In contrast, Ross County suffer from the weakest defence in the league, conceding an average of 1.80 goals per match, making them vulnerable on every away match. ¹

Mathematical analysis of the Arbroath vs Greenock Morton match

The match between Arbrough and Greenock Morton, scheduled for 23 January 2026 , opens the 23rd round. Arbrough are in third place and fighting to consolidate their play-off position, while Morton are in seventh place with a high draw percentage. ¹ The odds for this match are fixed at 2.09 for the home team, 3.12 for a draw and 3.37 for the away team.

Step 1: Input data and basic statistics

For Arbrough (home): Wins – 45%, Draws – 27%, Losses – 27%. Average goals scored – 1.36. Average goals conceded – 1.23.1

For Morton (away): Wins – 19%, Draws – 48%, Losses – 33%. Average goals scored – 0.95. Average goals conceded – 1.33.1

Step 2 and 3: Calculating attack and defense forces

Attacking power ($AS$) is defined as the sum of the winning percentage, the losing percentage, and the average number of goals scored.

$AS _{ Arbroath} = 0.45 + 0.27 + 1.36 = 2.08$.

$AS _{ Morton} = 0.19 + 0.33 + 0.95 = 1.47$.

The defense strength ($DS$) is calculated using the formula: $1 / (W\% – L\% + GA _{ avg})$.

$DS _{ Arbroath} = 1 / (0.45 – 0.27 + 1.23) = 1 / 1.41 = $0.71.

$DS _{ Morton} = 1 / (0.19 – 0.33 + 1.33) = 1 / 1.19 = 0.84$.

Step 4 and 5: Expected goals (xG) and probabilities (Poisson)

The expected goals for the home team are the arithmetic mean of their attack power and the away team’s defense power.

$xG _{ Home} = (2.08 + 0.84) / 2 = 1.46$.

$xG _{ Away} = (1.47 + 0.71) / 2 = 1.09$.

Using the Poisson Distribution with these values 3:

Probability of home team winning (1): 44%.

Probability of a draw (X): 27%.

Probability of away win (2): 29%.

Step 6, 7 and 8: Stability, Equality Index and Harmony Index

Stability ($K$): $( STDEV.P( 0.44, 0.27, 0.29) / AVERAGE(0.44, 0.27, 0.29)) \times 1.67 = (0.075 / 0.333) \times 1.67 = 0.375$.

Equality Index ($L$): $ ABS( ABS(2.08 – 1.47) – ABS(0.71 – 0.84)) = ABS(0.61 – 0.13) = 0.48$.

Harmony Index ($HI$): $(2 / 0.375) + (1 / (1 – 0.48)) = 5.33 + 1.92 = 7.25$.

Step 9: Verdict V3

Value $V3 = 0.44 – 0.29 = 0.15$.

According to the logical conditions: Since $0.15 > 0.1$, the final Verdict V3 is “1 ” ( Home win).3

The Harmony Index of 7.25 places this match in the “High Risk Matches” category. Although the model predicts a win for Arbrough, the statistical closeness in defensive and attacking strengths between the two teams, combined with Morton’s high draw percentage (48%), suggests potential instability in the prediction. ¹

Mathematical analysis of the Airdrieonians vs Ross County match

This match pits two teams from the bottom half of the table. Airdrieonians are in eighth place, while Ross County are in the bottom tenth. ¹ The odds are 2.83 for the home team, 3.12 for a draw and 2.38 for the away team.

Step 1: Input data and basic statistics

For Airdrieonians (home): Wins – 18%, Draws – 36%, Losses – 45%. Average goals scored – 0.82. Average goals conceded – 1.41.1

For Ross County (away): Wins – 15%, Draws – 35%, Losses – 50%. Average goals scored – 1.00. Average goals conceded – 1.80.1

Step 2 and 3: Calculating attack and defense forces

$AS _{ Airdrieonians} = 0.18 + 0.45 + 0.82 = 1.45$.

$AS _{ RossCounty} = 0.15 + 0.50 + 1.00 = $1.65.

$DS _{ Airdrieonians} = 1 / (0.18 – 0.45 + 1.41) = 1 / 1.14 = $0.88.

$DS _{ RossCounty} = 1 / (0.15 – 0.50 + 1.80) = 1 / 1.45 = $0.69.

Step 4 and 5: Expected goals (xG) and probabilities (Poisson)

$xG _{ Home} = (1.45 + 0.69) / 2 = 1.07$.

$xG _{ Away} = (1.65 + 0.88) / 2 = 1.27$.

Poisson Distribution probabilities 3:

1: 31%.

X: 27%.

2: 42%.

Step 6, 7 and 8: Stability, Equality Index and Harmony Index

$K = ( STDEV.P( 0.31, 0.27, 0.42) / 0.333) \times 1.67 = (0.063 / 0.333) \times 1.67 = 0.316$.

$L = ABS( ABS(1.45 – 1.65) – ABS(0.88 – 0.69)) = ABS(0.20 – 0.19) = $0.01.

$HI = (2 / 0.316) + (1 / (1 – 0.01)) = 6.33 + 1.01 = $7.34.

Step 9: Verdict V3

$V3 = 0.31 – 0.42 = -0.11$.

According to the logical conditions: Since $-0.17 \leq -0.11 < -0.08$, Verdict V3 is “X2”.3

The Harmony Index of 7.34 once again defines the match as a “High Risk Match”. Ross County, despite their position at the bottom, shows a higher attack power (1.65 vs. 1.45), which in combination with the low defensive power of the hosts makes the prediction in favor of the guests logical, but risky given the poor form of both teams. ¹

Mathematical analysis of the Partick Thistle vs. Ayr United match

This is one of the most interesting matches in the round, pitting the second against the fourth in the standings. ¹ Partick Thistle are on a run of good results, while Ayr United are the leader in draws in the league. ¹ The odds are 1.91, 3.28 and 3.71 respectively.

Step 1: Input data and basic statistics

For Partick Thistle (home): Wins – 50%, Draws – 32%, Losses – 18%. Average goals scored – 1.50. Average goals conceded – 1.14.1

For Ayr United (away): Wins – 29%, Draws – 52%, Losses – 19%. Average goals scored – 1.29. Average goals conceded – 1.14.1

Step 2 and 3: Calculating attack and defense forces

$AS _{ Partick} = 0.50 + 0.18 + 1.50 = 2.18$.

$AS _{ Ayr} = 0.29 + 0.19 + 1.29 = 1.77$.

$DS _{ Partick} = 1 / (0.50 – 0.18 + 1.14) = 1 / 1.46 = $0.68.

$DS _{ Ayr} = 1 / (0.29 – 0.19 + 1.14) = 1 / 1.24 = 0.81$.

Step 4 and 5: Expected goals (xG) and probabilities (Poisson)

$xG _{ Home} = (2.18 + 0.81) / 2 = 1.50$.

$xG _{ Away} = (1.77 + 0.68) / 2 = 1.23$.

Poisson Distribution probabilities 3:

1: 44%.

X: 26%.

2: 30%.

Step 6, 7 and 8: Stability, Equality Index and Harmony Index

$K = ( STDEV.P( 0.44, 0.26, 0.30) / 0.333) \times 1.67 = (0.076 / 0.333) \times 1.67 = 0.381$.

$L = ABS( ABS(2.18 – 1.77) – ABS(0.68 – 0.81)) = ABS(0.41 – 0.13) = $0.28.

$HI = (2 / 0.381) + (1 / (1 – 0.28)) = 5.25 + 1.39 = $6.64.

Step 9: Verdict V3

$V3 = 0.44 – 0.30 = 0.14$.

Verdict V3: “1 ” ( Home win).

The Harmony Index of 6.64 classifies the match as a “High Risk Match”. Partick’s strong attack (2.18) is a leading factor, but Ayr United’s exceptional ability to draw (52% of their matches end without a winner) introduces significant noise into the model, reducing the stability of the forecast. ¹

Mathematical analysis of the Raith Rovers vs Queen’s Park match

Wright Rovers are in sixth place, while Queens Park are in ninth position, just above the relegation zone. ¹ The hosts are favorites with odds of 1.51 against 5.73 for the away team.

Step 1: Input data and basic statistics

For Wright Rovers (home): Wins – 27%, Draws – 32%, Losses – 41%. Average goals scored – 1.05. Average goals conceded – 1.05.1

For Queens Park (away): Wins – 16%, Draws – 47%, Losses – 37%. Average goals scored – 1.05. Average goals conceded – 1.53.1

Step 2 and 3: Calculating attack and defense forces

$AS _{ Raith} = 0.27 + 0.41 + 1.05 = 1.73$.

$AS _{ QueensPark} = 0.16 + 0.37 + 1.05 = 1.58$.

$DS _{ Raith} = 1 / (0.27 – 0.41 + 1.05) = 1 / 0.91 = 1.10$.

$DS _{ QueensPark} = 1 / (0.16 – 0.37 + 1.53) = 1 / 1.32 = $0.76.

Step 4 and 5: Expected goals (xG) and probabilities (Poisson)

$xG _{ Home} = (1.73 + 0.76) / 2 = 1.25$.

$xG _{ Away} = (1.58 + 1.10) / 2 = 1.34$.

Poisson Distribution probabilities 3:

1: 35%.

X: 25%.

2: 40%.

Step 6, 7 and 8: Stability, Equality Index and Harmony Index

$K = ( STDEV.P( 0.35, 0.25, 0.40) / 0.333) \times 1.67 = (0.062 / 0.333) \times 1.67 = 0.311$.

$L = ABS( ABS(1.73 – 1.58) – ABS(1.10 – 0.76)) = ABS(0.15 – 0.34) = $0.19.

$HI = (2 / 0.311) + (1 / (1 – 0.19)) = 6.43 + 1.23 = $7.66.

Step 9: Verdict V3

$V3 = 0.35 – 0.40 = -0.05$.

Verdict V3: “X ” ( Tie).

With a Harmony Index of 7.66, this match crosses the line and is classified as a “Medium Risk Match”. The mathematical expectation of goals is almost equal, and the difference in probabilities between 1 and 2 is minimal. Wright’s high defensive index (1.10) balances out their weaker attack, making a draw the most likely outcome according to the V3 formula. ³

Mathematical analysis of the St Johnstone vs Dunfermline Athletic match

Leaders St Johnstone host Dunfermline in a match in which the home team are clear favourites at odds of 1.46. ¹

Step 1: Input data and basic statistics

For St. Johnstone (home): Wins – 62%, Draws – 29%, Losses – 10%. Average goals scored – 2.05. Average goals conceded – 0.81.1

For Dunfermline (away): Wins – 35%, Draws – 25%, Losses – 40%. Average goals scored – 1.40. Average goals conceded – 1.10.1

Step 2 and 3: Calculating attack and defense forces

$AS _{ StJohnstone} = 0.62 + 0.10 + 2.05 = 2.77$.

$AS _{ Dunfermline} = 0.35 + 0.40 + 1.40 = 2.15$.

$DS _{ StJohnstone} = 1 / (0.62 – 0.10 + 0.81) = 1 / 1.33 = $0.75.

$DS _{ Dunfermline} = 1 / (0.35 – 0.40 + 1.10) = 1 / 1.05 = $0.95.

Step 4 and 5: Expected goals (xG) and probabilities (Poisson)

$xG _{ Home} = (2.77 + 0.95) / 2 = 1.86$.

$xG _{ Away} = (2.15 + 0.75) / 2 = 1.45$.

Poisson Distribution probabilities 3:

1: 47%.

X: 22%.

2: 31%.

Step 6, 7 and 8: Stability, Equality Index and Harmony Index

$K = ( STDEV.P( 0.47, 0.22, 0.31) / 0.333) \times 1.67 = (0.103 / 0.333) \times 1.67 = 0.516$.

$L = ABS( ABS(2.77 – 2.15) – ABS(0.75 – 0.95)) = ABS(0.62 – 0.20) = $0.42.

$HI = (2 / 0.516) + (1 / (1 – 0.42)) = 3.88 + 1.72 = $5.60.

Step 9: Verdict V3

$V3 = 0.47 – 0.31 = 0.16$.

Verdict V3: “1 ” ( Home win).

The Harmony Index of 5.60 puts them in the “High Risk Matches” category. Despite St Johnstone’s formidable attack (2.77), Dunfermline have a surprisingly solid defensive index (0.95), which could make it difficult to score the expected goals. ¹¹

General analysis and systematic conclusions for the 23rd round

An analysis of the five matches reveals an interesting statistical picture for the Scottish Championship. None of the matches reach the status of “Platinum Selection”, which is evidence of the high level of balance of forces and the lack of “sure” results at this stage of the season. St. Johnstone’s dominance is visible in their individual indicators, but the defensive strength of their opponents is often underestimated by the mass market. ¹

The cabbage soup method allows us to identify anomalies in expected goals (xG). For example, Ross County, despite being bottom, have a higher goal potential (1.27 xG) against Airdrieonians than Arbroath do against Morton (1.46 vs. 1.09 xG), even though Arbroath are much higher in the standings. ¹ This is due to Airdrieonians’ disastrous defending, which opens up windows for the away team’s attack.

The Harmony Index (HI) serves as a filter for “noise ” in the data. Low HI values in most matches signal that the model has difficulty categorically separating the three possible outcomes (1, X, 2), which is typical for high-parity leagues. ³ The only match in the “Medium Risk ” category – Raith Rovers vs. Queens Park – shows the greatest internal harmony between the home team’s defensive strength and the away team’s attacking potential, making the prediction of a draw more reasonable than the others. ¹⁰

The stability of the model ($K$) varies considerably, with the highest being for St. Johnstone (0.516), which is a result of their distinct probability of winning compared to the other teams. However, this is not enough for Platinum Selection, as their Draw Index ($L$) is relatively high (0.42), meaning that the tactical matchup is more balanced than it seems at first glance. ³

Summary table of forecasts

Meeting	Predicted goals (HA)	Predicted outcome	Verdict	Match category	Coefficient
Arbroath – Morton	1.46 – 1.09	1	Victory 1	High risk	2.09
Airdrieonians – Ross County	1.07 – 1.27	X2	Draw/Win 2	High risk	1.38*
Partick Thistle – Ayr	1.50 – 1.23	1	Victory 1	High risk	1.91
Raith – Queen’s Park	1.25 – 1.34	X	Draw	Medium risk	3.86
St Johnstone – Dunfermline	1.86 – 1.45	1	Victory 1	High risk	1.46

*The X2 odds are indicative based on the market value for Double Chance.

In conclusion, the 23rd round of the Scottish Championship offers challenging events to predict, where mathematical discipline must prevail over emotional expectations. St Johnstone remains the most stable element in the system, but low Harmony Index values across all matches advise caution and strict risk management. ¹ The play-off clash between Partick and Ayr United is statistically the most unpredictable clash (HI 6.64), making it suitable only for users with a high risk tolerance. ⁸ Each prediction here is derived through a rigorous computational protocol, minimizing subjective errors and focusing on long-term statistical stability