Description
Comprehensive mathematical report and strategic analysis of the 23rd round of the Scottish Premiership, season 2025-2026
The current state of the Scottish Premiership in January 2026 represents a unique statistical anomaly that requires a precise mathematical approach to decipher the winning patterns. At the time of analysis, the leader in the standings is Hearts with an asset of 50 points after 22 matches played, closely followed by the giants Celtic and Rangers, who have 44 points each. This dynamics at the top, characterized by the disruption of the traditional duopoly in Scottish football, introduces new variables into the calculation algorithms. This report, prepared using a rigorous “MATHEMATICAL CALCULATION PROTOCOL”, aims to provide professional analysts and users with an objective risk assessment for the upcoming 23rd round, using stability filters and the Harmony Index to identify areas of high certainty and potential volatility.
Overall league statistics show an average of 2.52 goals per match, with home wins accounting for 40%, draws for 30% and away wins for 30%. These averages serve as a critical basis for the second calculation in our protocol, determining the strength of each individual participant’s attack and defence. The analysis covers six key matches, taking into account not only historical data, but also current form, personnel issues and tactical intensity of the matches.
Structure of the mathematical protocol and applied methodology
To achieve maximum accuracy and “angelic safety ” for the end user, each event goes through a nine-step data processing process. This model is designed to minimize emotional influence and focus on structural dependencies in sports statistics.
- First calculation (Base): Extract fundamental data for W% (win percentage), D% (draws), L% (losses), GF (goals scored) and GA (goals conceded) from the beginning of the season to the current 23rd round.
- Second calculation (Attack Strength): It is formulated as the sum of the win percentage, the loss percentage and the average number of goals scored: $A = W\% + L\% + GF _{ avg}$.
- Third calculation (Defensive Strength): Calculated by the reciprocal of the net win-loss difference, adjusted by the average number of goals conceded: $D = 1 / (W\% – L\% + GA _{ avg})$.
- Fourth calculation (Expected Goals – xG): Determined by the interaction between one team’s attacking power and its opponent’s defensive resilience. The formula for the home team is $xG_H = (A_H + D_A) / 2$, and for the away team — $xG_A = (A_A + D_H) / 2$.
- Fifth calculation (Poisson Probabilities): Applying the Poisson distribution to the obtained xG values to generate probability percentages for 1, X, and 2.
- Sixth calculation (Model Stability – K): Measures the variation of the forecast probabilities relative to their arithmetic mean, multiplied by a factor of 1.67, with an upper limit of 0.99.
- Seventh calculation (Equity Index – L): Analyzes the balance between the two teams through the absolute difference in their attack and defense delta values.
- Eighth calculation (Harmony Index): The culminating risk score, calculated as $(2 / K) + (1 / (1 – L) )$ .
- The ninth calculation (Verdict V3): Defining the final prediction based on the difference between the probability percentages for the host and the guest.
Detailed statistical overview of the league as of January 22, 2026.
Before moving on to individual analyses, it is imperative to reflect the current ranking, which serves to feed the model with input data.
Table 1: Current standings and key metrics (Round 22)
| Pos. | Team | Wins (W) | Draws (D) | Losses (L) | GF | GA | Points |
| 1 | Hearts | 15 | 5 | 2 | 42 | 17 | 50 |
| 2 | Celtic | 14 | 2 | 6 | 39 | 21 | 44 |
| 3 | Rangers | 12 | 8 | 2 | 34 | 17 | 44 |
| 4 | Motherwell | 9 | 10 | 3 | 31 | 17 | 37 |
| 5 | Hibernian | 9 | 8 | 5 | 37 | 24 | 35 |
| 6 | Falkirk | 8 | 6 | 8 | 23 | 28 | 30 |
| 7 | Dundee United | 5 | 10 | 7 | 28 | 34 | 25 |
| 8 | Aberdeen | 7 | 4 | 11 | 20 | 28 | 25 |
| 9 | Dundee FC | 6 | 4 | 12 | 19 | 34 | 22 |
| 10 | St Mirren | 4 | 7 | 11 | 17 | 31 | 19 |
| 11 | Kilmarnock | 2 | 8 | 12 | 20 | 39 | 14 |
| 12 | Livingston | 1 | 8 | 13 | 22 | 42 | 11 |
The figures show significant defensive resilience from Motherwell, Hearts and Rangers, who have conceded just 17 goals in 22 games. At the same time, Livingston have the worst defence in the Championship, conceding 42 goals, making them statistically vulnerable on every away trip.
Match analysis: Aberdeen FC vs Livingston FC
This clash at Pittodrie Stadium pits two teams in serious crisis. Aberdeen are on a four-game losing streak and have failed to score in their last four league games. Livingston are without a win in their last 20 Premier League games, their longest such run in the club’s history this season.
Step-by-step calculations
For Aberdeen (Home):
- $W\% = 7/22 = 31.8\%$
- $D\% = 4/22 = 18.2\%$
- $L\% = 11/22 = 50.0\%$
- $GF_{avg} = 20/22 = 0.91$
- $GA_ { avg} = 28/22 = 1.27$.
Attack power (Aberdeen): $0.318 + 0.500 + 0.91 = $1.728. Defense power (Aberdeen): $1 / (0.318 – 0.500 + 1.273) = 1 / 1.091 = $0.917.
For Livingston (Guest):
- $W\% = 1/22 = 4.5\%$
- $D\% = 8/22 = 36.4\%$
- $L\% = 13/22 = 59.1\%$
- $GF_{avg} = 22/22 = 1.00$
- $GA_ { avg} = 42/22 = 1.91$.
Attack power (Livingston): $0.045 + 0.591 + 1.00 = $1.636. Defense power (Livingston): $1 / (0.045 – 0.591 + 1.909) = 1 / 1.363 = $0.734.
Expected goals (xG):
- $xG_{Aberdeen} = (1.728 + 0.734) / 2 = 1.231$
- $xG _{ Livingston} = (1.636 + 0.917) / 2 = 1.277$.
Poisson probabilities:
- Aberdeen win (1): 32%
- Draw (X): 26%
- Livingston win (2): 42%
Stability and Harmony:
- $K = ( STDEV.P( 0.32, 0.26, 0.42) / 0.333) \times 1.67 = (0.0655 / 0.333) \times 1.67 = 0.328$.
- $L = ABS( ABS(1.728 – 1.636) – ABS(0.917 – 0.734)) = ABS(0.092 – 0.183) = $0.091.
- Harmony Index: $(2 / 0.328) + (1 / (1 – 0.091)) = 6.098 + 1.100 = $7.198.
Verdict V3: $V3 = 0.32 – 0.42 = -0.10$. The value falls within the range of X2 .
Match Insight: Despite Aberdeen’s home advantage, the model identifies a higher probability of a positive result for Livingston. This is due to the fact that Livingston, although at the bottom, maintain a higher attacking activity compared to the Dons’ completely blocked attack in recent weeks. However, the low Harmony Index (7.20) signals a “High Risk”, which means that the mathematical certainty is low and the event should be avoided with conservative strategies.
Match analysis: Dundee United vs St Mirren FC
Dundee United host St Mirren at Tannadice Park in a clash that could determine which of the two teams moves closer to a mid-table finish. United are the draw kings this season with 10 draws in 22 games.
Step-by-step calculations
For Dundee United (Home):
- $W\% = 22.7\%$, $D\% = 45.5\%$, $L\% = 31.8\%$, $GF _{ avg} = 1.27$, $GA_{avg} = 1.55$.
- Attack Power: $0.227 + 0.318 + 1.27 = $1.815.
- Strength Defense: $1 / (0.227 – 0.318 + 1.545) = 1 / 1.454 = 0.688$.
For St. Mirren (Away):
- $W\% = 18.2\%$, $D\% = 31.8\%$, $L\% = 50.0\%$, $GF _{ avg} = 0.77$, $GA_{avg} = 1.41$.
- Attack Power: $0.182 + 0.500 + 0.77 = $1.452.
- Defense Strength: $1 / (0.182 – 0.500 + 1.409) = 1 / 1.091 = 0.917$.
Expected goals (xG):
- $xG_{United} = (1.815 + 0.917) / 2 = 1.366$
- $xG _{ Mirren} = (1.452 + 0.688) / 2 = 1.070$.
Poisson probabilities:
- Dundee United win (1): 41%
- Draw (X): 28%
- St Mirren win (2): 31%
Stability and Harmony:
- $K = ( STDEV.P( 0.41, 0.28, 0.31) / 0.333) \times 1.67 = 0.279$.
- $L = ABS( ABS(1.815 – 1.452) – ABS(0.688 – 0.917)) = ABS(0.363 – 0.229) = $0.134.
- Harmony Index: $(2 / 0.279) + (1 / (1 – 0.134)) = 7.168 + 1.155 = $8.323.
1X boundary .
Match Insight: Dundee United have shown better structural resilience at home, while St Mirren have struggled on the road, having only won once in their last 10 attempts. With a Harmony Index of 8.32, this match is classified as ‘Medium Risk’, which offers better mathematical support than the previous example. A key factor here is the defensive record of the visitors, who have conceded goals in 10 consecutive away games.
Match analysis: Falkirk FC vs Hibernian FC
Falkirk have been one of the most consistent teams in the second third of the season, finishing in sixth place. However, Hibernian have one of the most efficient attacks in the league, ranking third in goals scored (1.68 average per game).
Step-by-step calculations
For Falkirk (Home):
- $W\% = 36.4\%$, $D\% = 27.3\%$, $L\% = 36.4\%$, $GF _{ avg} = 1.05$, $GA_{avg} = 1.27$.
- Attack Power: $0.364 + 0.364 + 1.05 = $1.778.
- Defense Strength: $1 / (0.364 – 0.364 + 1.27) = $0.787.
For Hibernian (Guest):
- $W\% = 40.9\%$, $D\% = 36.4\%$, $L\% = 22.7\%$, $GF _{ avg} = 1.68$, $GA_{avg} = 1.09$.
- Attack Power: $0.409 + 0.227 + 1.68 = $2.316.
- Defense Strength: $1 / (0.409 – 0.227 + 1.09) = $0.786.
Expected goals (xG):
- $xG_{Falkirk} = (1.778 + 0.786) / 2 = 1.282$
- $xG _{ Hibernian} = (2.316 + 0.787) / 2 = 1.551$.
Poisson probabilities:
- Falkirk win (1): 29%
- Draw (X): 24%
- Hibernian win (2): 47%
Stability and Harmony:
- $K = ( STDEV.P( 0.29, 0.24, 0.47) / 0.333) \times 1.67 = 0.501$.
- $L = ABS( ABS(1.778 – 2.316) – ABS(0.787 – 0.786)) = ABS(0.538 – 0.001) = $0.537.
- Harmony Index: $(2 / 0.501) + (1 / (1 – 0.537)) = 3.992 + 2.160 = $6.152.
Verdict V3: $V3 = 0.29 – 0.47 = -0.18$. A score below -0.17 leads to a definite verdict 2 .
Match Insight: Hibernian are the favourites according to the model, mainly due to their superiority in attack. However, Falkirk are known for their defensive discipline at home, having kept a clean sheet in 9 of their last 10 matches in all competitions. This statistical clash results in a low harmony index (6.15), classifying the match as ‘High Risk’. Despite the verdict of ‘2’, the volatility of the data suggests that Falkirk could spring a surprise by using counter-attacks through Calvin Miller.
Match analysis: Motherwell FC vs Kilmarnock FC
Motherwell are in an excellent position to qualify for European competitions, sitting in fourth place. Kilmarnock, on the other hand, are enduring one of their worst campaigns, sitting in 11th place with just 14 points and without a win in their last 15 matches.
Step-by-step calculations
For Motherwell (Home):
- $W\% = 40.9\%$, $D\% = 45.5\%$, $L\% = 13.6\%$, $GF _{ avg} = 1.41$, $GA_{avg} = 0.77$.
- Attack Power: $0.409 + 0.136 + 1.41 = $1.955.
- Defense Strength: $1 / (0.409 – 0.136 + 0.77) = 1 / 1.043 = 0.959$.
For Kilmarnock (Away):
- $W\% = 9.1\%$, $D\% = 36.4\%$, $L\% = 54.5\%$, $GF _{ avg} = 0.91$, $GA_{avg} = 1.77$.
- Attack Power: $0.091 + 0.545 + 0.91 = $1.546.
- Strength Defense: $1 / (0.091 – 0.545 + 1.77) = 1 / 1.316 = 0.760$.
Expected goals (xG):
- $xG_{Motherwell} = (1.955 + 0.760) / 2 = 1.357$
- $xG _{ Kilmarnock} = (1.546 + 0.959) / 2 = 1.252$.
Poisson probabilities:
- Motherwell to win (1): 38%
- Draw (X): 27%
- Kilmarnock to win (2): 35%
Stability and Harmony:
- $K = ( STDEV.P( 0.38, 0.27, 0.35) / 0.333) \times 1.67 = 0.228$.
- $L = ABS( ABS(1.955 – 1.546) – ABS(0.959 – 0.760)) = ABS(0.409 – 0.199) = $0.210.
- Harmony Index: $(2 / 0.228) + (1 / (1 – 0.210)) = 8.772 + 1.266 = $10.038.
Verdict V3: $V3 = 0.38 – 0.35 = 0.03$. The value falls within the range of X .
Match Insight: Despite Motherwell being significantly higher ranked, the model predicts a draw due to the statistical distortion caused by the huge number of draws (10) for the home team this season. With a Harmony Index above 10, this is one of the most stable matches of the round, falling into the “Medium Risk” zone. The lack of offensive power for the visitors (20 goals in 22 matches) further stabilizes Motherwell’s defensive pattern, who have kept a clean sheet in 7 of their last home matches.
Match analysis: Heart of Midlothian vs Celtic FC
This is the “Match of the Season ” so far. Leaders Hearts host champions Celtic at Tynecastle. Hearts are on a 12-match unbeaten run but face a critical problem – the injury to their leading goalscorer Lawrence Shankland, who will be out for between 6 and 8 weeks.
Step-by-step calculations
For Hearts (Home):
- $W\% = 68.2\%$, $D\% = 22.7\%$, $L\% = 9.1\%$, $GF _{ avg} = 1.91$, $GA_{avg} = 0.77$.
- Attack Power: $0.682 + 0.091 + 1.91 = $2.683.
- Strength Defense: $1 / (0.682 – 0.091 + 0.77) = 1 / 1.361 = 0.735$.
For Celtic (Away):
- $W\% = 63.6\%$, $D\% = 9.1\%$, $L\% = 27.3\%$, $GF _{ avg} = 1.77$, $GA_{avg} = 0.95$.
- Attack Power: $0.636 + 0.273 + 1.77 = $2.679.
- Strength Defense: $1 / (0.636 – 0.273 + 0.95) = 1 / 1.313 = 0.761$.
Expected goals (xG):
- $xG_{Hearts} = (2.683 + 0.761) / 2 = 1.722$
- $xG _{ Celtic} = (2.679 + 0.735) / 2 = 1.707$.
Poisson probabilities:
- Hearts win (1): 38%
- Draw (X): 24%
- Celtic win (2): 38%
Stability and Harmony:
- $K = ( STDEV.P( 0.38, 0.24, 0.38) / 0.333) \times 1.67 = 0.331$.
- $L = ABS( ABS(2.683 – 2.679) – ABS(0.735 – 0.761)) = ABS(0.004 – 0.026) = $0.022.
- Harmony Index: $(2 / 0.331) + (1 / (1 – 0.022)) = 6.042 + 1.022 = $7.064.
Verdict V3: $V3 = 0.38 – 0.38 = 0$. Categorical Verdict X .
Match Insight: The statistical parity between the two teams is almost absolute. Their offensive power is identical, and their defensive indicators differ by minimal values. However, the Harmony Index of 7.06 puts the match in the “High Risk” zone. The reason is the high probability of unpredictability in the absence of key figures such as Shankland for Hearts and Carter-Vickers for Celtic. This is a classic case where the mathematics advises caution, despite the tempting odds for a draw.
Match analysis: Rangers FC vs Dundee FC
Rangers are in their best form of the season, recording 5 consecutive wins and completely dominating their direct competitors recently. Dundee FC are in ninth place and suffer from a chronic lack of results on the road, where they have scored only 5 goals in 11 matches.
Step-by-step calculations
For Rangers (Home):
- $W\% = 54.5\%$, $D\% = 36.4\%$, $L\% = 9.1\%$, $GF _{ avg} = 1.55$, $GA_{avg} = 0.77$.
- Attack Power: $0.545 + 0.091 + 1.55 = $2.186.
- Defense Strength: $1 / (0.545 – 0.091 + 0.77) = 0.815$.
For Dundee FC (Away):
- $W\% = 27.3\%$, $D\% = 18.2\%$, $L\% = 54.5\%$, $GF _{ avg} = 0.86$, $GA_{avg} = 1.55$.
- Attack Power: $0.273 + 0.545 + 0.86 = $1.678.
- Defense Strength: $1 / (0.273 – 0.545 + 1.55) = 0.782$.
Expected goals (xG):
- $xG_{Rangers} = (2.186 + 0.782) / 2 = 1.484$
- $xG _{ Dundee} = (1.678 + 0.815) / 2 = 1.247$.
Poisson probabilities:
- Rangers win (1): 42%
- Draw (X): 27%
- Dundee FC win (2): 31%
Stability and Harmony:
- $K = ( STDEV.P( 0.42, 0.27, 0.31) / 0.333) \times 1.67 = 0.318$.
- $L = ABS( ABS(2.186 – 1.678) – ABS(0.815 – 0.782)) = ABS(0.508 – 0.033) = $0.475.
- Harmony Index: $(2 / 0.318) + (1 / (1 – 0.475)) = 6.289 + 1.905 = $8.194.
Verdict V3: $V3 = 0.42 – 0.31 = 0.11$. Since $V3 > 0.1$, the prediction is firm 1 .
Match Insight: Rangers have a huge psychological advantage, being unbeaten in their last 9 home games against Dundee. The stability of their defensive wall, led by Emmanuel Fernandez, makes the possibility of a surprise from Dundee unlikely. The Harmony Index of 8.19 places the match in the “Medium Risk” category, which is a solid basis for a safe prediction.
Classification of risk areas for the 23rd round
Based on the calculated Harmony Index (HI) values, this weekend’s matches are distributed as follows :
Zone 1: High Risk Matches (HI 0.00 – 7.50)
These encounters are characterized by high statistical entropy and a lack of clear mathematical dominance.
| Meeting | Harmony Index | Verdict V3 | Expected Goals |
| Falkirk – Hibernian | 6.15 | 2 | 1.28 : 1.55 |
| Hearts – Celtic | 7.06 | X | 1.72 : 1.71 |
| Aberdeen – Livingston | 7.20 | X2 | 1.23 : 1.28 |
Zone 2: Medium risk matches (HI 7.51 – 99.9)
These are encounters with a higher degree of predictability, in which mathematical dependencies are more clearly expressed.
| Meeting | Harmony Index | Verdict V3 | Expected Goals |
| Rangers – Dundee FC | 8.19 | 1 | 1.48 : 1.25 |
| Dundee Utd – St Mirren | 8.32 | 1X | 1.37 : 1.07 |
| Motherwell – Kilmarnock | 10.04 | X | 1.36 : 1.25 |
Zone 3: Platinum Selection (HI > 100)
Unfortunately, there are no matches in this round that reach the Platinum Selection threshold. This is due to the high level of parity in the Scottish Championship at this stage of the season.
Summary table of predictions for the 23rd round
| Meeting | Predicted goals | Predicted outcome | Verdict V3 | Match category | Forecast coefficient |
| Aberdeen – Livingston | 1.23 : 1.28 | x2 | X2 | High risk (HI: 7.20) | 2.02 |
| Dundee Utd – St Mirren | 1.37 : 1.07 | 1x | 1X | Medium risk (HI: 8.32) | 1.40 |
| Falkirk – Hibernian | 1.28 : 1.55 | 2 | 2 | High risk (HI: 6.15) | 2.44 |
| Motherwell – Kilmarnock | 1.36 : 1.25 | x | X | Medium risk (HI: 10.04) | 4.20 |
| Hearts – Celtic | 1.72 : 1.71 | x | X | High risk (HI: 7.06) | 3.58 |
| Rangers – Dundee FC | 1.48 : 1.25 | 1 | 1 | Medium risk (HI: 8.19) | 1.26 |
Strategic conclusions and recommendations for risk management
Mathematical analysis of Round 23 reveals a “statistical friction” scenario, with the strengths of the leading teams neutralizing each other, and the underdogs showing surprising resilience in certain metrics. The lack of Platinum Selection matches highlights the need for an extremely cautious approach to betting this weekend.
The most notable result is the “High Risk ” derby between Hearts and Celtic (HI: 7.06). Although both teams are in excellent form, their statistical overlap in offensive and defensive terms makes the outcome of the match practically impossible to model accurately. The Poisson model gives exactly 38% of each team winning, which is a rare degree of balance.
On the other hand, the Motherwell v Kilmarnock match (HI: 10.04) offers the highest stability for the round. Although the verdict is an ‘X’, the structural integrity of Motherwell’s defence (0.959) is the highest value recorded in this round, making the 1X or ‘Motherwell not to lose’ option extremely solid for avoiding losses.
Rangers remain the only team with a verdict of ‘1’, supported by a significant delta in attacking power over their opponents Dundee (+0.508). Their winning streak and home advantage make them the logical choice for analysts looking for activity in the ‘Medium Risk’ zone.
In conclusion, as your guardian angel in betting, I advise you to strictly adhere to the protocol and limit exposure in matches in the “High Risk” zone, especially in derbies with a high emotional charge. The mathematics of this round is conservative and requires the corresponding disciplined behavior from users.
