Scree Plot: Formula, Verified Results, Charts and Interpretation
A scree plot graphs ordered eigenvalues against component or factor number. The retention cue is the elbow where large substantive roots give way to a flatter tail of small roots. This guide uses the supplied real-data results, native MathML equations, matching charts, and separate Python, R, SPSS or AMOS, and Excel verification.
The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
For Scree Plot, eigenvalue 1 = 3.195831 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
What Scree Plot measures
The exact estimand and the result this method is allowed to support.
Scree Plot addresses one defined analytical target: A scree plot graphs ordered eigenvalues against component or factor number. The retention cue is the elbow where large substantive roots give way to a flatter tail of small roots.
Quantity estimated in this analysis
The ordered-root elbow graph is reconstructed from the exact variables, matrix, model, panel, or resampling design shown below. The primary output is Eigenvalue 1 = 3.195831; Eigenvalue 2 = 1.817089 supplies the first supporting check. Eigenvalue 1 = 3.195831 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
For Scree Plot, the calculation retains full precision until the final display. That matters because the software reports, spreadsheet formulas, chart labels, and narrative must refer to one identical result rather than separately rounded approximations.
Interpretation that is not permitted
The elbow is subjective and can be ambiguous. A scree plot does not simulate sampling noise, test factorability, or distinguish PCA roots from common-factor roots unless the source is labeled.
For Scree Plot, this boundary is substantive. A nearby coefficient may share the same data or model, yet it answers a different question. The article therefore names every supporting statistic instead of using broad labels such as “valid,” “good,” or “significant” without the object being evaluated.
When to use Scree Plot
Research scope, neighboring methods, and excluded claims.
Research question answered
The defensible question is whether the ordered-root elbow graph supports the result stated for the declared dataset and analytical specification. It is answered by mark the third-to-fourth boundary, followed by display exact eigenvalues in a companion table. The evidence is bounded by Eigenvalue 1 = 3.195831 and its named companion quantities.
For Scree Plot, changing the case set, expert panel, item block, estimator, factor count, rotation, baseline model, bootstrap design, or criterion definition changes the question. Such a change requires a new result rather than a revision of the wording around the old value.
Nearest methods that answer different questions
Parallel Analysis: Parallel analysis gives random-data cutoffs and reduces reliance on visual judgment.
Kaiser Criterion: The greater-than-one rule is numeric but ignores random sampling; the scree plot focuses on shape.
These distinctions determine which formula, output table, and chart can legitimately appear in a Scree Plot post.
Real data used for Scree Plot
Variables, coding, sample or panel size, and the role each input plays.
For Scree Plot, the factor-oriented analysis uses 649 complete records and nine ordered variables. TravelAccess is defined as 5 − traveltime so larger values represent easier travel. The correlation matrix, eigenvalues, communalities, loading matrices, rotation output, and simulation cutoffs all preserve the same variable order.
For the ordered-root elbow graph, the data are not merely background. A change in correlation type, standardization, missing-case rule, variable order, or retained dimension count changes the matrix on which Eigenvalue 1 = 3.195831 was obtained.
| Variable | Meaning | Mean | SD | Range | Construct |
|---|---|---|---|---|---|
| G1 | first-period grade | 11.3991 | 2.7453 | 0–19 | Academic Achievement |
| G2 | second-period grade | 11.5701 | 2.9136 | 0–19 | Academic Achievement |
| G3 | final grade | 11.9060 | 3.2307 | 0–19 | Academic Achievement |
| Medu | mother’s education | 2.5146 | 1.1346 | 0–4 | Educational Advantage |
| Fedu | father’s education | 2.3066 | 1.0999 | 0–4 | Educational Advantage |
| TravelAccess | reverse-coded travel accessibility | 3.4314 | 0.7487 | 1–4 | Educational Advantage |
| goout | frequency of going out | 3.1849 | 1.1758 | 1–5 | Social-Alcohol Exposure |
| Dalc | workday alcohol use | 1.5023 | 0.9248 | 1–5 | Social-Alcohol Exposure |
| Walc | weekend alcohol use | 2.2804 | 1.2844 | 1–5 | Social-Alcohol Exposure |
Scree Plot assumptions and design requirements
Six conditions checked before the coefficient or decision rule is interpreted.
1. Eigenvalues are ordered descending
This condition determines whether the input object matches the formula. In the current Scree Plot analysis, the check is to mark the third-to-fourth boundary while preserving Eigenvalue 1 = 3.195831.
For Scree Plot, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.
2. The matrix and extraction source are identified
This requirement controls whether the numerical estimate has the interpretation claimed. In the current Scree Plot analysis, the check is to display exact eigenvalues in a companion table while preserving Eigenvalue 2 = 1.817089.
For Scree Plot, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.
3. Axes are not distorted
This design condition prevents an attractive coefficient from being attached to the wrong population or model. In the current Scree Plot analysis, the check is to avoid truncating the y-axis to exaggerate an elbow while preserving Eigenvalue 3 = 1.393698.
For Scree Plot, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.
4. All relevant roots are displayed
This specification rule keeps the software routes numerically comparable. In the current Scree Plot analysis, the check is to compare independent reviewers’ elbow choices while preserving Eigenvalue 4 = 0.846560.
For Scree Plot, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.
5. The plot uses unrounded roots
This diagnostic requirement is checked before a benchmark is applied. In the current Scree Plot analysis, the check is to pair the plot with parallel analysis while preserving Three-dimension cumulative variance = 71.1846%.
For Scree Plot, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.
6. The visual decision is supplemented
This final condition governs whether the conclusion can survive replication or sensitivity analysis. In the current Scree Plot analysis, the check is to state whether roots come from PCA or common-factor extraction while preserving Horn retained factors = 3.
For Scree Plot, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.
Scree Plot hypotheses or decision rule
The statistical question is stated at the correct level for this method.
Statistical question
For Scree Plot, the retention decision asks whether each observed ordered root remains larger than its adjacent or simulated reference value; it is not a single omnibus null hypothesis.
For Scree Plot, uncertainty is evaluated at the retention boundary, especially the last retained and first rejected dimensions.
Decision for the worked analysis
For Scree Plot, the calculation yields Eigenvalue 1 = 3.195831 . Eigenvalue 1 = 3.195831 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Scree Plot formula and worked substitution
Native MathML preserves fractions, roots, summations, matrices, subscripts, and superscripts.
The equation below is the defining mathematical object for Scree Plot. Its symbols are connected to the saved inputs and to Eigenvalue 1 = 3.195831, Eigenvalue 2 = 1.817089, Eigenvalue 3 = 1.393698, Eigenvalue 4 = 0.846560.
The ordered-root curve is an ordered-root visualization; the elbow is interpreted with parallel analysis and substantive clarity.
The sharp flattening after the third root supports a three-dimension reading.
Symbol and denominator control
A scree plot graphs ordered eigenvalues against component or factor number. The retention cue is the elbow where large substantive roots give way to a flatter tail of small roots.
For Scree Plot, the numerator, denominator, matrix order, degrees of freedom, factor count, or panel size shown in the MathML card is retained exactly. A formula from a neighboring method is not substituted even when both produce values on a similar scale.
Full-precision substitution
For Scree Plot, the spreadsheet and software outputs retain unrounded inputs until the final displayed value. The arithmetic is then reconciled with Eigenvalue 1 = 3.195831 and Eigenvalue 2 = 1.817089 .
The elbow is subjective and can be ambiguous. A scree plot does not simulate sampling noise, test factorability, or distinguish PCA roots from common-factor roots unless the source is labeled.
Step-by-step Scree Plot calculation
Every stage is tied to a saved value and a method-specific condition.
The worked calculation follows six operations specific to the ordered-root elbow graph. Each operation produces a quantity used by the next step, so a discrepancy is resolved where it originates rather than hidden by rounding.
Establish the analytical object
Action: Mark the third-to-fourth boundary.
Numerical trace: Eigenvalue 1 = 3.195831; Eigenvalue 2 = 1.817089.
Condition: eigenvalues are ordered descending. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Reconstruct the first required quantity
Action: Display exact eigenvalues in a companion table.
Numerical trace: Eigenvalue 2 = 1.817089; Eigenvalue 3 = 1.393698.
Condition: the matrix and extraction source are identified. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Verify the companion quantity
Action: Avoid truncating the y-axis to exaggerate an elbow.
Numerical trace: Eigenvalue 3 = 1.393698; Eigenvalue 4 = 0.846560.
Condition: axes are not distorted. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Apply the decision rule
Action: Compare independent reviewers’ elbow choices.
Numerical trace: Eigenvalue 4 = 0.846560; Three-dimension cumulative variance = 71.1846%.
Condition: all relevant roots are displayed. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Inspect local evidence
Action: Pair the plot with parallel analysis.
Numerical trace: Three-dimension cumulative variance = 71.1846%; Horn retained factors = 3.
Condition: the plot uses unrounded roots. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Reconcile and report
Action: State whether roots come from PCA or common-factor extraction.
Numerical trace: Horn retained factors = 3; Parallel iterations = 500.
Condition: the visual decision is supplemented. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Scree Plot results and interpretation
Primary and supporting statistics are kept separate and precisely labeled.
Primary result
Eigenvalue 1
The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Why the result is internally coherent
For Scree Plot, eigenvalue 1 = 3.195831 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
For Scree Plot, eigenvalue 2 = 1.817089 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
For Scree Plot, the two quantities are reported together because one is primary and the other supplies context; neither is renamed as the other.
| Result item | Exact value | Interpretation restricted to this method |
|---|---|---|
| Eigenvalue 1 | 3.195831 | Eigenvalue 1 = 3.195831 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Eigenvalue 2 | 1.817089 | Eigenvalue 2 = 1.817089 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Eigenvalue 3 | 1.393698 | Eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Eigenvalue 4 | 0.846560 | Eigenvalue 4 = 0.846560 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Three-dimension cumulative variance | 71.1846% | Three-dimension cumulative variance = 71.1846% belongs to the declared matrix and retained dimensions and must not be relabeled as model fit. |
| Horn retained factors | 3 | Horn retained factors = 3 is retained as a distinct supporting quantity for the ordered-root elbow graph; it is not substituted for the primary result. |
| Parallel iterations | 500 | Parallel iterations = 500 documents simulation or convergence effort rather than substantive magnitude. |
| Observed eigenvalue 3 | 1.393698 | Observed eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Horn 95th percentile root 3 | 1.106209 | Horn 95th percentile root 3 = 1.106209 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Observed eigenvalue 4 | 0.846560 | Observed eigenvalue 4 = 0.846560 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Horn 95th percentile root 4 | 1.064106 | Horn 95th percentile root 4 = 1.064106 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Overall KMO | 0.713439 | Overall KMO = 0.713439 describes the balance between ordinary and partial correlations, so it informs factorability or item adequacy rather than factor retention. |
| Lowest item MSA | 0.588169 | Lowest item MSA = 0.588169 describes the balance between ordinary and partial correlations, so it informs factorability or item adequacy rather than factor retention. |
| Highest item MSA | 0.867806 | Highest item MSA = 0.867806 describes the balance between ordinary and partial correlations, so it informs factorability or item adequacy rather than factor retention. |
Scree Plot in Python
The Python route calculates or reconstructs the exact named result.
The Python workflow uses the explicit NumPy/Pandas calculation to calculate or extract the ordered-root elbow graph from the declared data and analytical specification. It must reproduce Eigenvalue 1 = 3.195831 and retain Eigenvalue 2 = 1.817089 as a separate supporting quantity.
The code is read as an executable analysis, not as a printed answer. Its critical verification is to mark the third-to-fourth boundary; the associated design condition is that eigenvalues are ordered descending. The elbow is subjective and can be ambiguous. A scree plot does not simulate sampling noise, test factorability, or distinguish PCA roots from common-factor roots unless the source is labeled.
import pandas as pd
import numpy as npdf = pd.read_csv("student-por.csv", sep=";")
df["TravelAccess"] = 5 - df["traveltime"]
vars9 = ["G1","G2","G3","Medu","Fedu","TravelAccess","goout","Dalc","Walc"]
X = df[vars9].dropna()
R=X.corr().to_numpy()
values,vectors=np.linalg.eigh(R)
order=np.argsort(values)[::-1]
values=values[order]; vectors=vectors[:,order]
loadings=vectors*np.sqrt(values)
print("eigenvalues",values)
print("explained",values/values.sum())
print("first three loadings",loadings[:,:3])
Scree Plot in R
The R route declares package, estimator, extraction, rotation, or resampling settings.
The R route uses base R and the displayed matrix operations and the displayed arguments to estimate the ordered-root elbow graph. Package defaults are made explicit because estimator, matrix type, extraction, rotation, baseline, or bootstrap choices can change the result.
R output is reconciled with Eigenvalue 1 = 3.195831 after the analyst display exact eigenvalues in a companion table. Agreement is expected only when the case set, variable order, and method settings match the Python and workbook calculations.
d <- read.csv2("student-por.csv")
d$TravelAccess <- 5 - d$traveltime
vars9 <- c("G1","G2","G3","Medu","Fedu","TravelAccess","goout","Dalc","Walc")
X <- d[vars9]
e <- eigen(cor(X))
print(e$values); print(e$values/sum(e$values))
loadings <- sweep(e$vectors,2,sqrt(e$values),"*")
print(loadings[,1:3])Scree Plot in SPSS or AMOS
The procedure is labeled honestly when base SPSS does not expose the coefficient.
The SPSS or AMOS section shows the procedure that is actually available for the ordered-root elbow graph. When base SPSS does not expose the coefficient, the syntax prepares the correct matrix or model and the coefficient is obtained through AMOS, MATRIX operations, or a validated integration rather than by renaming a different test.
The output must identify Eigenvalue 1 = 3.195831 and the settings needed to reproduce it. The software review specifically avoid truncating the y-axis to exaggerate an elbow, while preserving the requirement that axes are not distorted.
COMPUTE TravelAccess = 5 - traveltime.
EXECUTE.
FACTOR
/VARIABLES G1 G2 G3 Medu Fedu TravelAccess goout Dalc Walc
/MISSING LISTWISE
/PRINT INITIAL KMO EXTRACTION ROTATION
/PLOT EIGEN
/CRITERIA FACTORS(3) ITERATE(500)
/EXTRACTION PAF
/ROTATION OBLIMIN
/METHOD=CORRELATION.
* Read only the Scree Plot evidence identified in this post.Scree Plot in Excel
The workbook exposes source values, intermediate arithmetic, and the final formula.
The Excel workbook is an arithmetic audit for the ordered-root elbow graph. Named cells retain the inputs, intermediate components, and final formula leading to Eigenvalue 1 = 3.195831; no rounded constant is pasted over a formula cell.
Excel can verify visible calculations and cross-software agreement, but it does not replace estimation, optimization, rotation, or resampling that must occur in statistical software. The workbook therefore focuses on the check to compare independent reviewers’ elbow choices and documents Eigenvalue 2 = 1.817089 independently.
Data: 649 rows with documented coding.
Inputs: named cells or ranges required only by Scree Plot.
Calculation: Use the native MathML formula shown above with named ranges for every input
Audit: compare full-precision Excel output with the Python, R, and SPSS/AMOS values.
Decision: reference the exact result and diagnostics; never paste a rounded value over the formula cell.Scree Plot charts and visual diagnostics
Each supplied image is interpreted through its own values and analytical purpose.
Every image below is interpreted as part of the same Scree Plot analysis. The captions identify what the panel contributes, the exact values visible in the result set, and the condition that would invalidate the reading.
01 Scree-Plot Primary Metrics
This panel reconciles the headline estimate with its principal supporting values for Scree Plot. Read Eigenvalue 1 = 3.195831 beside Eigenvalue 2 = 1.817089; the first quantity is not replaced by the second.
The chart is used to mark the third-to-fourth boundary. Its interpretation remains valid only when eigenvalues are ordered descending. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
02 Scree-Plot Scree Coordinates
This panel places the ordered roots around the retention boundary for Scree Plot. Read Eigenvalue 2 = 1.817089 beside Eigenvalue 3 = 1.393698; the first quantity is not replaced by the second.
The chart is used to display exact eigenvalues in a companion table. Its interpretation remains valid only when the matrix and extraction source are identified. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
03 Scree-Plot Scree First Differences
This panel places the ordered roots around the retention boundary for Scree Plot. Read Eigenvalue 3 = 1.393698 beside Eigenvalue 4 = 0.846560; the first quantity is not replaced by the second.
The chart is used to avoid truncating the y-axis to exaggerate an elbow. Its interpretation remains valid only when axes are not distorted. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
04 Scree-Plot Scree Second Differences
This panel places the ordered roots around the retention boundary for Scree Plot. Read Eigenvalue 4 = 0.846560 beside Three-dimension cumulative variance = 71.1846%; the first quantity is not replaced by the second.
The chart is used to compare independent reviewers’ elbow choices. Its interpretation remains valid only when all relevant roots are displayed. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
05 Scree-Plot Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for Scree Plot. Read Three-dimension cumulative variance = 71.1846% beside Horn retained factors = 3; the first quantity is not replaced by the second.
The chart is used to pair the plot with parallel analysis. Its interpretation remains valid only when the plot uses unrounded roots. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
01 Scree-Plot Primary Metrics
This panel reconciles the headline estimate with its principal supporting values for Scree Plot. Read Horn retained factors = 3 beside Parallel iterations = 500; the first quantity is not replaced by the second.
The chart is used to state whether roots come from PCA or common-factor extraction. Its interpretation remains valid only when the visual decision is supplemented. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
02 Scree-Plot Scree Coordinates
This panel places the ordered roots around the retention boundary for Scree Plot. Read Parallel iterations = 500 beside Observed eigenvalue 3 = 1.393698; the first quantity is not replaced by the second.
The chart is used to mark the third-to-fourth boundary. Its interpretation remains valid only when eigenvalues are ordered descending. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
03 Scree-Plot Scree First Differences
This panel places the ordered roots around the retention boundary for Scree Plot. Read Observed eigenvalue 3 = 1.393698 beside Horn 95th percentile root 3 = 1.106209; the first quantity is not replaced by the second.
The chart is used to display exact eigenvalues in a companion table. Its interpretation remains valid only when the matrix and extraction source are identified. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
04 Scree-Plot Scree Second Differences
This panel places the ordered roots around the retention boundary for Scree Plot. Read Horn 95th percentile root 3 = 1.106209 beside Observed eigenvalue 4 = 0.846560; the first quantity is not replaced by the second.
The chart is used to avoid truncating the y-axis to exaggerate an elbow. Its interpretation remains valid only when axes are not distorted. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
05 Scree-Plot Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for Scree Plot. Read Observed eigenvalue 4 = 0.846560 beside Horn 95th percentile root 4 = 1.064106; the first quantity is not replaced by the second.
The chart is used to compare independent reviewers’ elbow choices. Its interpretation remains valid only when all relevant roots are displayed. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
Scree Plot verification and sensitivity analysis
Six failure modes are checked against the formula, data, output, and charts.
The following diagnostics are not a general checklist. Each one targets a failure mode that can change the calculation or interpretation of Scree Plot.
1. Mark the third-to-fourth boundary
Begin by mark the third-to-fourth boundary. For the ordered-root elbow graph, this operation directly connects Eigenvalue 1 = 3.195831 with Eigenvalue 3 = 1.393698. Eigenvalue 1 = 3.195831 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
The governing condition is that eigenvalues are ordered descending. If it fails, the primary coefficient may be attached to the wrong input object. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Parallel Analysis, because Parallel analysis gives random-data cutoffs and reduces reliance on visual judgment.
2. Display exact eigenvalues in a companion table
Next, display exact eigenvalues in a companion table. For the ordered-root elbow graph, this operation directly connects Eigenvalue 2 = 1.817089 with Eigenvalue 4 = 0.846560. Eigenvalue 2 = 1.817089 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
The governing condition is that the matrix and extraction source are identified. If it fails, the companion statistic may no longer describe the same model or sample. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Kaiser Criterion, because The greater-than-one rule is numeric but ignores random sampling; the scree plot focuses on shape.
3. Avoid truncating the y-axis to exaggerate an elbow
The third verification is to avoid truncating the y-axis to exaggerate an elbow. For the ordered-root elbow graph, this operation directly connects Eigenvalue 3 = 1.393698 with Three-dimension cumulative variance = 71.1846%. Eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
The governing condition is that axes are not distorted. If it fails, the decision boundary can move because the required quantity has changed. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Cumulative Variance, because Cumulative variance measures coverage but does not itself locate an elbow.
4. Compare independent reviewers’ elbow choices
After the core arithmetic is stable, compare independent reviewers’ elbow choices. For the ordered-root elbow graph, this operation directly connects Eigenvalue 4 = 0.846560 with Horn retained factors = 3. Eigenvalue 4 = 0.846560 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
The governing condition is that all relevant roots are displayed. If it fails, software agreement can be artificial if unlike definitions are compared. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Parallel Analysis, because Parallel analysis gives random-data cutoffs and reduces reliance on visual judgment.
5. Pair the plot with parallel analysis
A robustness review must pair the plot with parallel analysis. For the ordered-root elbow graph, this operation directly connects Three-dimension cumulative variance = 71.1846% with Parallel iterations = 500. Three-dimension cumulative variance = 71.1846% belongs to the declared matrix and retained dimensions and must not be relabeled as model fit.
The governing condition is that the plot uses unrounded roots. If it fails, a favorable average can conceal a local failure. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Kaiser Criterion, because The greater-than-one rule is numeric but ignores random sampling; the scree plot focuses on shape.
6. State whether roots come from PCA or common-factor extraction
The final reconciliation should state whether roots come from PCA or common-factor extraction. For the ordered-root elbow graph, this operation directly connects Horn retained factors = 3 with Observed eigenvalue 3 = 1.393698. Horn retained factors = 3 is retained as a distinct supporting quantity for the ordered-root elbow graph; it is not substituted for the primary result.
The governing condition is that the visual decision is supplemented. If it fails, the published conclusion can exceed the evidence actually reproduced. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Cumulative Variance, because Cumulative variance measures coverage but does not itself locate an elbow.
| # | Verification operation | Condition protected | Saved quantity traced |
|---|---|---|---|
| 1 | mark the third-to-fourth boundary | eigenvalues are ordered descending | Eigenvalue 1 = 3.195831 |
| 2 | display exact eigenvalues in a companion table | the matrix and extraction source are identified | Eigenvalue 2 = 1.817089 |
| 3 | avoid truncating the y-axis to exaggerate an elbow | axes are not distorted | Eigenvalue 3 = 1.393698 |
| 4 | compare independent reviewers’ elbow choices | all relevant roots are displayed | Eigenvalue 4 = 0.846560 |
| 5 | pair the plot with parallel analysis | the plot uses unrounded roots | Three-dimension cumulative variance = 71.1846% |
| 6 | state whether roots come from PCA or common-factor extraction | the visual decision is supplemented | Horn retained factors = 3 |
Scree Plot compared with related methods
Differences in estimand, formula, and conclusion determine the correct choice.
Method choice depends on the estimand, model, and data structure. These three comparisons explain why the post uses the Scree Plot formula and output rather than a nearby procedure.
Parallel Analysis
Parallel analysis gives random-data cutoffs and reduces reliance on visual judgment.
In the current analysis, Eigenvalue 2 = 1.817089 remains evidence for the ordered-root elbow graph; it is not relabeled as a Parallel Analysis result. Eigenvalue 2 = 1.817089 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
Kaiser Criterion
The greater-than-one rule is numeric but ignores random sampling; the scree plot focuses on shape.
In the current analysis, Eigenvalue 3 = 1.393698 remains evidence for the ordered-root elbow graph; it is not relabeled as a Kaiser Criterion result. Eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
Cumulative Variance
Cumulative variance measures coverage but does not itself locate an elbow.
In the current analysis, Eigenvalue 4 = 0.846560 remains evidence for the ordered-root elbow graph; it is not relabeled as a Cumulative Variance result. Eigenvalue 4 = 0.846560 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
How to report Scree Plot
A complete result paragraph includes the value, analytical object, settings, and limitation.
Results paragraph
Scree Plot was evaluated using the declared data, specification, and software settings. The primary result was Eigenvalue 1 = 3.195831; Eigenvalue 2 = 1.817089 and Eigenvalue 3 = 1.393698 supplied supporting context. The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
The report then states the limitation explicitly: The elbow is subjective and can be ambiguous. A scree plot does not simulate sampling noise, test factorability, or distinguish PCA roots from common-factor roots unless the source is labeled.
Settings that must accompany the result
eigenvalues are ordered descending; the matrix and extraction source are identified; axes are not distorted; all relevant roots are displayed.
For Scree Plot, these details identify the exact version of the analysis and make cross-software reconciliation possible.
Verification actions retained in the record
mark the third-to-fourth boundary; display exact eigenvalues in a companion table; avoid truncating the y-axis to exaggerate an elbow; compare independent reviewers’ elbow choices.
The final wording is revised only after those operations reproduce the saved values.
Scree Plot decision scenarios
For Scree Plot, worked conflicts show how the conclusion changes when an input, assumption, or supporting statistic fails.
Boundary-case interpretation: Mark the third-to-fourth boundary
Consider a review in which Eigenvalue 1 = 3.195831 is reproduced but Eigenvalue 2 = 1.817089 is not. For the ordered-root elbow graph, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to mark the third-to-fourth boundary and verify that eigenvalues are ordered descending.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis gives random-data cutoffs and reduces reliance on visual judgment. The published conclusion remains The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Input-definition sensitivity: Display exact eigenvalues in a companion table
Consider a review in which Eigenvalue 3 = 1.393698 is reproduced but Eigenvalue 4 = 0.846560 is not. For the ordered-root elbow graph, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to display exact eigenvalues in a companion table and verify that the matrix and extraction source are identified.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Kaiser Criterion only for method selection: The greater-than-one rule is numeric but ignores random sampling; the scree plot focuses on shape. The published conclusion remains The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Software-definition reconciliation: Avoid truncating the y-axis to exaggerate an elbow
Consider a review in which Three-dimension cumulative variance = 71.1846% is reproduced but Horn retained factors = 3 is not. For the ordered-root elbow graph, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to avoid truncating the y-axis to exaggerate an elbow and verify that axes are not distorted.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Cumulative Variance only for method selection: Cumulative variance measures coverage but does not itself locate an elbow. The published conclusion remains The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Local-chart conflict: Compare independent reviewers’ elbow choices
Consider a review in which Parallel iterations = 500 is reproduced but Observed eigenvalue 3 = 1.393698 is not. For the ordered-root elbow graph, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to compare independent reviewers’ elbow choices and verify that all relevant roots are displayed.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis gives random-data cutoffs and reduces reliance on visual judgment. The published conclusion remains The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Alternative-method challenge: Pair the plot with parallel analysis
Consider a review in which Horn 95th percentile root 3 = 1.106209 is reproduced but Observed eigenvalue 4 = 0.846560 is not. For the ordered-root elbow graph, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to pair the plot with parallel analysis and verify that the plot uses unrounded roots.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Kaiser Criterion only for method selection: The greater-than-one rule is numeric but ignores random sampling; the scree plot focuses on shape. The published conclusion remains The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Replication and reporting decision: State whether roots come from PCA or common-factor extraction
Consider a review in which Horn 95th percentile root 4 = 1.064106 is reproduced but Overall KMO = 0.713439 is not. For the ordered-root elbow graph, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to state whether roots come from PCA or common-factor extraction and verify that the visual decision is supplemented.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Cumulative Variance only for method selection: Cumulative variance measures coverage but does not itself locate an elbow. The published conclusion remains The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Boundary-case interpretation: Mark the third-to-fourth boundary
Consider a review in which Lowest item MSA = 0.588169 is reproduced but Highest item MSA = 0.867806 is not. For the ordered-root elbow graph, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to mark the third-to-fourth boundary and verify that eigenvalues are ordered descending.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis gives random-data cutoffs and reduces reliance on visual judgment. The published conclusion remains The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Input-definition sensitivity: Display exact eigenvalues in a companion table
Consider a review in which Bartlett chi-square = 3018.238 is reproduced but Eigenvalue 1 = 3.195831 is not. For the ordered-root elbow graph, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to display exact eigenvalues in a companion table and verify that the matrix and extraction source are identified.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Kaiser Criterion only for method selection: The greater-than-one rule is numeric but ignores random sampling; the scree plot focuses on shape. The published conclusion remains The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Software-definition reconciliation: Avoid truncating the y-axis to exaggerate an elbow
Consider a review in which Eigenvalue 2 = 1.817089 is reproduced but Eigenvalue 3 = 1.393698 is not. For the ordered-root elbow graph, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to avoid truncating the y-axis to exaggerate an elbow and verify that axes are not distorted.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Cumulative Variance only for method selection: Cumulative variance measures coverage but does not itself locate an elbow. The published conclusion remains The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Local-chart conflict: Compare independent reviewers’ elbow choices
Consider a review in which Eigenvalue 4 = 0.846560 is reproduced but Three-dimension cumulative variance = 71.1846% is not. For the ordered-root elbow graph, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to compare independent reviewers’ elbow choices and verify that all relevant roots are displayed.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis gives random-data cutoffs and reduces reliance on visual judgment. The published conclusion remains The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Alternative-method challenge: Pair the plot with parallel analysis
Consider a review in which Horn retained factors = 3 is reproduced but Parallel iterations = 500 is not. For the ordered-root elbow graph, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to pair the plot with parallel analysis and verify that the plot uses unrounded roots.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Kaiser Criterion only for method selection: The greater-than-one rule is numeric but ignores random sampling; the scree plot focuses on shape. The published conclusion remains The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Replication and reporting decision: State whether roots come from PCA or common-factor extraction
Consider a review in which Observed eigenvalue 3 = 1.393698 is reproduced but Horn 95th percentile root 3 = 1.106209 is not. For the ordered-root elbow graph, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to state whether roots come from PCA or common-factor extraction and verify that the visual decision is supplemented.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Cumulative Variance only for method selection: Cumulative variance measures coverage but does not itself locate an elbow. The published conclusion remains The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Boundary-case interpretation: Mark the third-to-fourth boundary
Consider a review in which Observed eigenvalue 4 = 0.846560 is reproduced but Horn 95th percentile root 4 = 1.064106 is not. For the ordered-root elbow graph, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to mark the third-to-fourth boundary and verify that eigenvalues are ordered descending.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis gives random-data cutoffs and reduces reliance on visual judgment. The published conclusion remains The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
Scree Plot downloads and reproducibility files
All linked files belong to the same analysis and remain on onlineinternetcafe.com.
The four files belong to one Scree Plot analysis. Their primary values, variable order, method settings, and chart labels must agree; a mismatch is resolved in the source calculation before the WordPress draft is published.
Scree Plot frequently asked questions
Answers use the worked result and the exact method boundary.
What does Scree Plot measure?
A scree plot graphs ordered eigenvalues against component or factor number. The retention cue is the elbow where large substantive roots give way to a flatter tail of small roots.
What is the main result in this Scree Plot analysis?
Eigenvalue 1 = 3.195831. The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.
What does the result not prove?
The elbow is subjective and can be ambiguous. A scree plot does not simulate sampling noise, test factorability, or distinguish PCA roots from common-factor roots unless the source is labeled.
Which supporting value should be reported with the primary result?
For Scree Plot, eigenvalue 2 = 1.817089 is the first companion quantity. Eigenvalue 2 = 1.817089 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
Which assumption is most likely to change the interpretation?
The first requirement is that eigenvalues are ordered descending. The result is recomputed if that condition is not satisfied.
What is the most important numerical verification?
The analyst must mark the third-to-fourth boundary. That operation traces Eigenvalue 1 = 3.195831 to the formula and saved inputs.
Why can software packages disagree on Scree Plot?
Disagreement can arise because the matrix and extraction source are identified or because the packages implement different estimators, matrices, baselines, rotations, standardizations, bootstrap rules, or coefficient definitions. Matching labels alone is not enough.
How is Scree Plot different from Parallel Analysis?
Parallel analysis gives random-data cutoffs and reduces reliance on visual judgment.
How should a chart be interpreted?
For Scree Plot, each chart is tied to a named output such as Eigenvalue 3 = 1.393698. It supports a local calculation or diagnostic and does not replace the full numerical result.
How should Scree Plot be reported?
Report Eigenvalue 1 = 3.195831, the required supporting quantities, sample or panel size, exact method settings, and this qualified conclusion: The sharp decline through the first roots and flattening after the third is consistent with retaining three dimensions. Parallel analysis confirms that the fourth root falls below its random-data reference.