Quick Answer: Phi Coefficient Result

The worked example in this report tests the association between sex and Pass_Status. The outcome variable is created from G3 final grade, where Pass = G3 ≥ 10 and Fail = G3 < 10. The predictor has two categories: Female and Male. Because both variables are binary, the correct effect-size measure for the 2×2 table is the Phi Coefficient.

The verified Excel calculation gives φ = 0.078222, χ²(1) = 3.971035, and p = 0.046289. At α = .05, the result is statistically significant. However, the effect size is very small. The practical conclusion is that sex and pass/fail status are associated in this sample, but the strength of association is negligible to small.

What Is the Phi Coefficient?

Phi Coefficient is a measure of association between two binary variables. A binary variable has only two categories, such as pass/fail, yes/no, male/female, treatment/control, exposed/not exposed, present/absent, success/failure, or correct/incorrect. When both variables have two categories, the data can be arranged in a 2×2 contingency table, and Phi Coefficient provides a signed effect-size measure for the association.

Phi behaves like a correlation coefficient. It can be positive, negative, or near zero depending on the arrangement of the 2×2 table. A positive value means the table has more weight along one diagonal. A negative value means the table has more weight along the opposite diagonal. A value near zero means the observed table is close to what would be expected under independence.

The Phi Coefficient is closely connected to the chi-square test of independence. For a 2×2 table, the relationship is:

This means that if you know the sample size and Phi, you can calculate the chi-square statistic. In this worked example, n = 649 and φ = 0.078222, so χ² = 649 × 0.078222² = 3.971035. That value gives the p-value of 0.046289 with 1 degree of freedom.

Phi Coefficient connects naturally with Cross Tabulation, Contingency Coefficient, Correlation Matrix, Correlation Assumptions, p-value, Effect Size, Confidence Interval, Null and Alternative Hypothesis and Parametric vs Nonparametric Tests.

When Should You Use Phi Coefficient?

Use the Phi Coefficient when both variables are binary and the data form a 2×2 table. It is especially useful when you need a compact effect size to accompany a chi-square test. The chi-square test tells whether the variables are independent; Phi tells how strong the association is.

For this example, sex has two categories and Pass_Status has two categories. Therefore, the 2×2 table is appropriate for Phi. If the outcome were the full numeric G3 score, Phi would not be the correct measure; a biserial or point-biserial correlation, t-test in Python, t-test in R, or t-test in SPSS would be more suitable.

Phi Coefficient Formula

For a 2×2 table, label the four cells as follows:

The signed Phi Coefficient formula is:

The numerator ad − bc compares the diagonal products of the table. If ad is larger than bc, Phi is positive. If bc is larger than ad, Phi is negative. If the two diagonal products are similar, Phi is near zero.

The positive sign appears because the diagonal product a × d is larger than b × c. In plain language, the observed pattern is slightly more concentrated in the Female-Pass and Male-Fail diagonal than the opposite diagonal. However, the size of the coefficient is still very small.

Null and Alternative Hypotheses

The Phi Coefficient is usually reported with the chi-square test of independence. The hypotheses are about whether the two binary variables are independent in the population.

Dataset and Variables Used

The analysis uses the student performance dataset with 649 valid cases. The predictor is sex, coded as Female and Male. The outcome is Pass_Status, created from G3 final grade. A student is classified as Pass if G3 is at least 10 and Fail if G3 is below 10.

This conversion is important. Phi is not used on the original numeric G3 score. It is used only after G3 is converted into a binary outcome. If the research question requires keeping G3 as a numeric score, then the correct method would be a different analysis, such as Biserial Correlation, Two Sample t-Test, t-Test in Excel or t-Test in Python.

2×2 Contingency Table

The 2×2 table below is the core of the Phi Coefficient calculation. Rows represent sex, and columns represent pass/fail status. The table has four observed cells and row/column totals.

The table shows that 86.95% of female students passed, compared with 81.20% of male students. The difference in pass proportions is 5.74 percentage points. This difference is enough to produce a statistically significant chi-square result at the .05 level, but it is not large enough to produce a strong effect size.

The risk ratio is 1.0707, meaning the pass proportion for female students is about 1.07 times the pass proportion for male students. The odds ratio is 1.5417, meaning the odds of passing are higher in the female group than the male group under this coding. However, because the Phi value is only 0.0782, all of these results should be described carefully as small or negligible in association strength.

Verified Phi Coefficient Results

The main calculation is shown below. This section is useful for students who need the exact values for interpretation, reporting, and checking formulas in Excel, Python, R, or SPSS.

The most important result is not only the p-value. The p-value tells whether the observed table differs enough from independence to be statistically detectable. The Phi Coefficient tells whether that difference is practically meaningful. Here, the result is statistically significant but very small.

Expected Counts and Standardized Residuals

Expected counts show what the table would look like if sex and pass/fail status were independent. Residuals show how far the observed counts are from those expected counts. Standardized residuals make the cell differences easier to compare across the table.

The largest standardized residual is in the Male + Fail cell, with a standardized residual of about 1.408. This means male failures are somewhat higher than expected under independence. The Female + Fail cell is lower than expected, and the Female + Pass cell is higher than expected. These cell-level patterns explain why Phi is positive under the selected table mapping.

Even so, the standardized residuals are not extremely large. This supports the same conclusion as the Phi value: there is a detectable association, but the pattern is not strong.

Python Chart-by-Chart Interpretation

The Python report includes five visuals: the 2×2 contingency heatmap, row percentages, observed versus expected counts, standardized residuals, and effect-size summary. Together, these charts explain the full Phi Coefficient result from raw counts to final interpretation.

Python Chart 1: 2×2 Contingency Heatmap

This heatmap displays the four observed counts: Female-Pass = 333, Female-Fail = 50, Male-Pass = 216, and Male-Fail = 50. It gives the simplest visual summary of the 2×2 table. The largest cell is Female-Pass because the female group has more students and most students passed overall.

The chart should be interpreted with row totals in mind. Female has 383 cases and Male has 266 cases, so raw counts alone can be misleading if read without percentages. That is why the row-percentage chart is also required. The heatmap tells where the counts are located; the percentage chart tells how the outcome rate differs within each group.

Python Chart 2: Row Percentages

The row-percentage chart shows the practical group comparison. Female students have a pass percentage of about 86.95%, while male students have a pass percentage of about 81.20%. The fail percentages are about 13.05% for female students and 18.80% for male students.

This chart explains the direction of the association. The female group has a higher pass percentage and a lower fail percentage than the male group. However, the percentage difference is only about 5.74 percentage points. This is why the association is statistically significant but practically small.

Python Chart 3: Observed vs Expected Counts

The observed-versus-expected chart compares each actual cell count with the count expected if sex and pass/fail status were independent. Female-Pass is observed at 333 and expected at about 323.99. Female-Fail is observed at 50 and expected at about 59.01. Male-Pass is observed at 216 and expected at about 225.01. Male-Fail is observed at 50 and expected at about 40.99.

The differences are consistent with a positive Phi value: more Female-Pass and Male-Fail cases than expected, and fewer Female-Fail and Male-Pass cases than expected. However, the observed counts are not extremely far from the expected counts, which supports the negligible effect-size interpretation.

Python Chart 4: Standardized Residuals

The standardized residual chart shows which cells contribute most to the chi-square statistic. The Male-Fail cell has the largest positive standardized residual, around 1.408. The Female-Fail cell has a negative standardized residual around -1.173. The Female-Pass and Male-Pass cells have smaller residuals.

This chart is important because the p-value alone does not show where the association comes from. The residuals show that the association is mainly driven by somewhat fewer Female-Fail cases than expected and somewhat more Male-Fail cases than expected. Still, the residual values are not large enough to suggest a strong cell-level departure.

Python Chart 5: Phi Effect Size Summary

The effect-size summary chart gives the final report values in one place: Phi = 0.078222, chi-square = 3.971035, p = 0.046289, odds ratio = 1.541667, risk ratio = 1.070714, and risk difference = 0.057422. This chart is the best visual to use when writing the final interpretation paragraph.

The key reporting lesson is that statistical significance and effect size must be separated. The p-value is just below .05, so the result is statistically significant. But Phi is below .10, so the association is negligible or very small in practical strength.

R Chart-by-Chart Interpretation

The R report validates the Python result using a separate workflow and colorful visuals. The same five chart types are repeated: 2×2 contingency heatmap, row percentages, observed versus expected counts, standardized residuals and effect-size summary.

R Chart 1: Colorful 2×2 Contingency Heatmap

The R contingency heatmap confirms the same observed 2×2 structure. Female-Pass is the largest cell, Female-Fail and Male-Fail are equal at 50, and Male-Pass is 216. This confirms that the R workflow is using the same binary coding and the same table as the Python and Excel workflows.

R Chart 2: Colorful Row Percentages

The R row-percentage chart confirms that the female group has a higher pass percentage and the male group has a higher fail percentage. The difference is visible, but not large. This is exactly the pattern that produces a significant p-value with a small Phi effect size.

R Chart 3: Colorful Observed vs Expected Counts

The R observed-versus-expected chart validates the same expected-count pattern. Female-Pass and Male-Fail are higher than expected, while Female-Fail and Male-Pass are lower than expected. The departures are enough to reject independence at the .05 level, but they remain modest in size.

R Chart 4: Colorful Standardized Residuals

The R residual chart confirms that the largest cell-level departure is the Male-Fail cell. This cell has more cases than expected under independence. The Female-Fail cell has fewer cases than expected. These two cells explain much of the chi-square contribution.

R Chart 5: Colorful Phi Effect Size Summary

The R effect-size summary confirms the same final conclusion. Phi is positive and statistically significant, but it is very small. The correct final wording is a statistically significant but negligible association, not a strong relationship.

SPSS Output Interpretation

The SPSS output PDF is included as the formal software output for the 2×2 table. In SPSS, Phi is usually reported in the Symmetric Measures table after running Crosstabs with Chi-square statistics selected. The Pearson chi-square table gives the test of independence, while the Phi and Cramer’s V row gives the effect-size magnitude.

Open the SPSS Phi Coefficient Output PDF

In a thesis or research report, the SPSS result should not be interpreted only from the chi-square significance row. The Phi row is necessary because it gives the strength of association. Here, the association is statistically significant but weak enough that the practical conclusion should remain cautious.

Excel Worked File Explanation

The Excel workbook provides a full formula-based Phi Coefficient analysis. It contains the dataset, binary coding, contingency table, full worked Phi calculation, expected counts, standardized residuals and chart source tables. This makes the file useful for students who want to understand each calculation step rather than relying only on SPSS, Python or R output.

Download the Phi Coefficient Full Excel Analysis File

Excel Main Calculation

The workbook calculates the denominator as 74,787.045670. The Phi formula then gives φ = 0.0782221031. The chi-square value is calculated as n × φ², giving 3.9710346255. The p-value is calculated using Excel’s right-tail chi-square function and equals 0.0462893224.

Excel Risk and Odds Measures

The workbook also adds practical 2×2 table measures. The female pass proportion is 0.869452, and the male pass proportion is 0.812030. The risk difference is 0.057422, and the risk ratio is 1.070714. The odds ratio is 1.541667. These measures give useful context, but the overall association strength is still summarized by the small Phi value.

Python, R, SPSS and Excel Workflows

The same Phi Coefficient analysis can be reproduced in Python, R, SPSS and Excel. The workflow is always the same: create two binary variables, build the 2×2 table, calculate Phi, run chi-square, inspect expected counts and interpret the effect size.

Code Blocks and Excel Formulas

Python Code for Phi Coefficient

import pandas as pd
import numpy as np
from scipy.stats import chi2_contingency, chi2

# Load data
df = pd.read_csv("dataset.csv")

# Create binary outcome
df["Pass_Status"] = np.where(df["G3"] >= 10, "Pass", "Fail")

# Build 2x2 table: rows = sex, columns = pass/fail status
table = pd.crosstab(df["sex"], df["Pass_Status"])
table = table.loc[["F", "M"], ["Pass", "Fail"]]

# Extract cell counts
a = table.loc["F", "Pass"]
b = table.loc["F", "Fail"]
c = table.loc["M", "Pass"]
d = table.loc["M", "Fail"]

# Phi coefficient
denominator = np.sqrt((a+b) * (c+d) * (a+c) * (b+d))
phi = ((a*d) - (b*c)) / denominator

# Chi-square relationship
n = table.to_numpy().sum()
chi_square_from_phi = n * phi**2
p_value = chi2.sf(chi_square_from_phi, df=1)

# Chi-square test using scipy for expected counts
chi2_stat, chi2_p, dof, expected = chi2_contingency(table, correction=False)

# Practical measures
risk_female = a / (a + b)
risk_male = c / (c + d)
risk_difference = risk_female - risk_male
risk_ratio = risk_female / risk_male
odds_ratio = (a * d) / (b * c)

print("2x2 table:")
print(table)
print("Phi:", phi)
print("Chi-square:", chi_square_from_phi)
print("p-value:", p_value)
print("Expected counts:")
print(expected)
print("Risk difference:", risk_difference)
print("Risk ratio:", risk_ratio)
print("Odds ratio:", odds_ratio)

R Code for Phi Coefficient

df <- read.csv("dataset.csv", stringsAsFactors = FALSE)

# Create binary pass/fail outcome
df$Pass_Status <- ifelse(df$G3 >= 10, "Pass", "Fail")

# Build 2x2 table
tab <- table(df$sex, df$Pass_Status)
tab <- tab[c("F", "M"), c("Pass", "Fail")]

# Extract cells
a <- tab["F", "Pass"]
b <- tab["F", "Fail"]
c <- tab["M", "Pass"]
d <- tab["M", "Fail"]

# Phi coefficient
denominator <- sqrt((a+b) * (c+d) * (a+c) * (b+d))
phi <- ((a*d) - (b*c)) / denominator

# Chi-square relationship
n <- sum(tab)
chi_square <- n * phi^2
p_value <- pchisq(chi_square, df = 1, lower.tail = FALSE)

# Chi-square test
chi_result <- chisq.test(tab, correct = FALSE)

# Practical measures
risk_female <- a / (a + b)
risk_male <- c / (c + d)
risk_difference <- risk_female - risk_male
risk_ratio <- risk_female / risk_male
odds_ratio <- (a * d) / (b * c)

print(tab)
cat("Phi =", phi, "\n")
cat("Chi-square =", chi_square, "\n")
cat("p-value =", p_value, "\n")
cat("Risk difference =", risk_difference, "\n")
cat("Risk ratio =", risk_ratio, "\n")
cat("Odds ratio =", odds_ratio, "\n")
print(chi_result)

SPSS Syntax for Phi Coefficient

* Phi Coefficient in SPSS.
* Predictor: sex.
* Outcome: Pass_Status from G3 where Pass = G3 >= 10.

OUTPUT CLOSE ALL.
OUTPUT NEW NAME=Phi_Coefficient_Output.

* Example assumes dataset is already open in SPSS.
RECODE G3 (Lowest thru 9=0) (10 thru Highest=1) INTO pass_status.
VARIABLE LABELS pass_status 'Pass status from G3: 1=Pass, 0=Fail'.
VALUE LABELS pass_status 0 'Fail (G3 < 10)' 1 'Pass (G3 >= 10)'.
EXECUTE.

CROSSTABS
  /TABLES=sex BY pass_status
  /FORMAT=AVALUE TABLES
  /STATISTICS=CHISQ PHI
  /CELLS=COUNT EXPECTED ROW COLUMN RESID SRESID
  /COUNT ROUND CELL.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE='Phi-Coefficient-SPSS-Output.pdf'.

Excel Formulas for Phi Coefficient

Assume the 2x2 table is:

            Pass    Fail
Female       a       b
Male         c       d

Female + Pass:
=COUNTIFS(sex_range,"F",pass_status_range,"Pass")

Female + Fail:
=COUNTIFS(sex_range,"F",pass_status_range,"Fail")

Male + Pass:
=COUNTIFS(sex_range,"M",pass_status_range,"Pass")

Male + Fail:
=COUNTIFS(sex_range,"M",pass_status_range,"Fail")

Denominator:
=SQRT((a+b)*(c+d)*(a+c)*(b+d))

Phi:
=((a*d)-(b*c))/SQRT((a+b)*(c+d)*(a+c)*(b+d))

Chi-square:
=N*Phi^2

p-value:
=CHISQ.DIST.RT(chi_square,1)

Expected count for each cell:
=(row_total*column_total)/grand_total

Residual:
=Observed-Expected

Standardized residual:
=(Observed-Expected)/SQRT(Expected)

Odds ratio:
=(a*d)/(b*c)

Risk in Female group:
=a/(a+b)

Risk in Male group:
=c/(c+d)

Risk difference:
=Risk_Female-Risk_Male

Risk ratio:
=Risk_Female/Risk_Male

Assumptions and Data Checks

Phi Coefficient is simple to calculate, but it still has important requirements. The most important requirement is that both variables must be binary. The observations should also be independent, and expected counts should be checked when using the chi-square p-value.

The expected-count check is especially strong here because the smallest expected cell count is about 40.99. This means the chi-square approximation is not threatened by tiny expected frequencies. The main caution is not the p-value calculation; it is interpretation. A statistically significant result with Phi below .10 should still be described as very weak.

How to Report Phi Coefficient

A good Phi Coefficient report should include the variables, sample size, 2x2 table result, chi-square statistic, degrees of freedom, p-value, Phi value, direction, and practical interpretation. If the table is meaningful for applied interpretation, also include row percentages, odds ratio, risk difference or risk ratio.

Common Mistakes in Phi Coefficient Interpretation

Downloads and Resources

External References

For additional learning, review documentation and references on chi-square tests of independence, 2x2 contingency tables, Phi Coefficient, Cramer's V, odds ratios, risk ratios and standardized residuals. These concepts are commonly taught together because they describe different parts of the same binary association problem.

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