Quick Answer: F Distribution Result

The worked F Distribution example uses an ANOVA model where studytime is tested as a grouping factor for G3 final grade. The charted ANOVA result shows observed F = 15.876 and critical F = 2.619. The observed F line is far to the right of the critical F line, so the group mean difference is statistically significant.

The mean-square chart shows exactly where the F statistic comes from. The between-group mean square is 155.026, and the within-group mean square is 9.765. Dividing 155.026 by 9.765 gives an F statistic of about 15.876. This means between-group variation is much larger than the typical within-group variation.

What Is the F Distribution?

The F Distribution is a right-skewed probability distribution used when a test statistic is built from a ratio of two variance estimates. In ANOVA, the F statistic compares variation between groups with variation inside groups. If the between-group variation is much larger than the within-group variation, the F statistic becomes large.

The F distribution is not one single curve. Its shape changes depending on two degrees of freedom values. The first degrees of freedom value, df1, is linked to the numerator of the F statistic. The second degrees of freedom value, df2, is linked to the denominator. That is why the same observed F value can have a different p value under different df1 and df2 combinations.

In this worked example, the observed ANOVA F statistic is 15.876 and the critical F value is 2.619. The observed F is far beyond the rejection boundary, so the F distribution supports a statistically significant ANOVA result for the studytime effect on G3.

The F distribution should be interpreted together with ANOVA in Python, ANOVA in R, ANOVA in SPSS, ANOVA Assumptions, ANOVA Effect Size, P Value, Null and Alternative Hypothesis, and Effect Size.

F Distribution Formula and ANOVA F Statistic

In one-way ANOVA, the F statistic is calculated by dividing the between-group mean square by the within-group mean square.

In the supplied mean-square chart, MS Between = 155.026 and MS Within = 9.765. Dividing those values gives the observed ANOVA F statistic.

This ratio is large because the variation between studytime group means is much larger than the typical variation remaining inside the groups. The F distribution then tells whether a ratio this large is unusual under the null hypothesis.

Critical F Value

The observed F statistic is 15.876, and the critical value at the displayed .05 level is 2.619. Since 15.876 is greater than 2.619, the result is statistically significant.

Right-Tail P-Value Area

The right-tail p-value area is the area under the F distribution curve to the right of the observed F statistic. In the supplied chart, the observed F line is far to the right, so the remaining right-tail area is visually tiny.

Null and Alternative Hypothesis

For the ANOVA example shown in the F distribution charts, the null hypothesis says that the mean G3 score is the same across all studytime groups. The alternative hypothesis says that at least one studytime group mean differs.

Dataset and Variables Used

The worked example uses a student performance dataset. The dependent variable is G3 final grade, and the grouping factor is studytime. The F distribution is used after the ANOVA model calculates the between-group and within-group mean squares.

For supporting concepts, review Descriptive Statistics, Variance, Standard Deviation, Standard Error, Confidence Interval, Effect Size, Statistical Power, Type I and Type II Error and P Value.

Python Chart-by-Chart Interpretation

The Python charts below show the first F distribution workflow. They explain how F curves change by degrees of freedom, how the observed ANOVA F statistic is judged, how the right-tail p-value area works, how alpha changes the critical F value, and how mean squares produce the final F statistic.

Python Chart 1: F Distribution Curves by Degrees of Freedom

This chart compares several F distribution curves. The curve with df1 = 1 and df2 = 10 is sharply concentrated near zero and has a steep early drop. The curves with larger degrees of freedom become smoother and less extreme near zero.

The chart shows that the F distribution is always right-skewed and starts at nonnegative F values. An F statistic cannot be negative because it is a ratio of mean squares or variance estimates.

This chart explains why df1 and df2 matter. The same F statistic is judged against a different curve when the numerator and denominator degrees of freedom change.

Python Chart 2: Observed ANOVA F Distribution

This chart shows the ANOVA F distribution with two vertical decision lines. The dashed critical F line is at 2.619, and the observed F line is at 15.876.

The observed F statistic is far to the right of the critical value. This means the observed ratio of mean squares is much larger than the cutoff needed for statistical significance at the displayed decision level.

This chart gives the central decision for the ANOVA example. Since the observed F statistic is far beyond the critical F line, the null hypothesis of equal group means is rejected.

Python Chart 3: Right-Tail P-Value Area

This chart shows how the p value is located in an F test. The observed F statistic is placed far to the right on the horizontal axis, so the p-value area is the small remaining area to the right of that vertical line.

The right-tail area is visually tiny because the observed F value is much larger than the values where the density curve is concentrated. Most of the F distribution curve is near smaller F values.

This chart should be used to explain why ANOVA p values are right-tail probabilities. A large F statistic is evidence against the null hypothesis because it leaves very little probability in the right tail under the null distribution.

Python Chart 4: Critical F Values by Alpha

This chart shows how the critical F value changes when alpha changes. At alpha 0.10, the critical F value is 2.09. At alpha 0.05, it rises to 2.62. At alpha 0.025, it rises to 3.14. At alpha 0.01, it is 3.81, and at alpha 0.001, it is 5.49.

The pattern is clear: stricter alpha levels require larger F statistics before rejecting the null hypothesis. A smaller alpha leaves a smaller right-tail rejection area, so the cutoff moves farther to the right.

This chart is useful when explaining why the significance level must be selected before interpreting the test. The same observed F statistic is compared against a different critical value when alpha changes.

Python Chart 5: Critical F Changes as df1 Changes

This chart shows the critical F value at alpha .05 as df1 changes. The critical value is highest when df1 is small and then gradually decreases as df1 increases.

The curve falls quickly from the first few df1 values and then flattens. By df1 values near 8, 9 and 10, the critical F values are much closer together than they are at the beginning of the curve.

This chart explains why degrees of freedom cannot be ignored. When the numerator degrees of freedom change, the rejection boundary changes even if alpha stays at .05.

Python Chart 6: Critical F Changes as df2 Changes

This chart shows the critical F value at alpha .05 as df2 changes. The critical F value is high when df2 is small, then drops as df2 increases. The chart moves from a high value near df2 = 5 down toward about 2.62 when df2 reaches 645.

The largest decrease occurs at the smaller df2 values. After df2 becomes large, the curve flattens and the critical F values change only slightly.

This chart is important for ANOVA because df2 usually represents within-group or error degrees of freedom. Larger denominator degrees of freedom give a more stable estimate of within-group variation and change the F cutoff.

Python Chart 7: Mean Squares Used to Calculate F

This chart shows the two mean squares used to calculate the ANOVA F statistic. MS Between is 155.026, while MS Within is 9.765.

The between-group mean square is much larger than the within-group mean square. This means the studytime group means vary much more than the typical variation left inside the groups.

This chart gives the calculation behind the F distribution decision. Dividing 155.026 by 9.765 gives an F statistic of about 15.876, which is the same observed F value shown in the distribution charts.

R Chart-by-Chart Validation

The R validation charts repeat the same F distribution workflow in a second software environment. They confirm the same density-curve behavior, observed F decision, p-value area, alpha critical values, df1 and df2 critical-value patterns, and mean-square calculation.

R Chart 1: F Distribution Curves by Degrees of Freedom

This R chart confirms that the shape of the F distribution changes with df1 and df2. The curve with smaller degrees of freedom is sharper near zero, while curves with larger degrees of freedom spread more smoothly.

The distribution remains nonnegative and right-skewed in every curve. This matches the definition of the F statistic as a ratio of nonnegative variance estimates.

In reporting, this chart validates the concept that F distribution interpretation depends on degrees of freedom, not only on the observed F statistic.

R Chart 2: Observed ANOVA F Distribution

This R chart confirms the same ANOVA decision as Python. The critical F line is at 2.619, and the observed F line is at 15.876.

The observed F statistic is far beyond the critical boundary. This confirms that the studytime group effect is statistically significant in the F-test decision.

In the final article, this chart validates that the ANOVA decision is stable across software. Both Python and R show the observed statistic far into the right-tail rejection region.

R Chart 3: Right-Tail P-Value Area

This R chart confirms that the p value for an F test is found in the right tail. The observed F statistic is so far to the right that the remaining tail area is visually very small.

The chart supports the same ANOVA conclusion as the observed-F distribution chart. A very large F statistic leaves very little probability beyond it under the null distribution.

In reporting, this chart should be used to explain the logic behind the p value. It shows why the ANOVA result is statistically significant without relying only on a table.

R Chart 4: Critical F Values by Alpha

This R chart confirms the same alpha pattern. The critical F value increases as alpha becomes stricter. The bars move from 2.09 at alpha .10 to 5.49 at alpha .001.

The chart explains why a stricter significance level requires stronger evidence. When alpha becomes smaller, the rejection region moves farther into the right tail.

In reporting, this chart validates that the selected alpha level directly changes the critical value used for the F-test decision.

R Chart 5: Critical F Changes as df1 Changes

This R chart confirms that the critical F value decreases as df1 increases. The steepest decline appears at the smaller df1 values, and the line flattens at larger df1 values.

The chart shows that numerator degrees of freedom affect the right-tail cutoff. A test with df1 = 1 uses a different critical boundary from a test with df1 = 10.

In reporting, this chart supports a clear instruction: always report the degrees of freedom with an F statistic because the degrees of freedom determine the reference distribution.

R Chart 6: Critical F Changes as df2 Changes

This R chart confirms that critical F decreases as df2 increases. The line drops strongly from small df2 values and then becomes nearly flat as df2 becomes large.

The final point near df2 = 645 is close to the same critical value used in the observed ANOVA F chart. This connects the degrees-of-freedom chart with the actual ANOVA decision.

In the final explanation, this chart should be used to show that a large error degrees of freedom value stabilizes the denominator side of the F distribution.

R Chart 7: Mean Squares Used to Calculate F

This R chart confirms the same mean-square calculation. MS Between is 155.026, and MS Within is 9.765.

The large gap between the two bars explains why the observed F statistic is large. The model detects much more between-group variation than within-group mean-square variation.

In reporting, this chart verifies the computation behind the F value. The F statistic is not a separate mysterious number; it is the ratio of these two mean squares.

SPSS Output Interpretation for F Distribution

The SPSS output PDF is included as the downloadable software output for the F Distribution workflow. SPSS ANOVA tables normally show the sum of squares, degrees of freedom, mean squares, F statistic and significance value. The F distribution is the reference distribution used to interpret that F statistic.

Download F Distribution SPSS Output PDF

SPSS Output Items to Read

SPSS Reporting Summary

SPSS, R, Python and Excel Workflows for F Distribution

The same F Distribution workflow can be reproduced in SPSS, R, Python and Excel. The important steps are to calculate or extract the mean squares, compute the F statistic, identify df1 and df2, compare the observed F statistic with the critical F value and read the right-tail p-value area.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for F Distribution

SPSS Syntax for F Distribution Workflow

* F Distribution / ANOVA F statistic workflow in SPSS.
* Dependent variable: G3.
* Factor: studytime.

TITLE "F Distribution for ANOVA: G3 by Studytime".

ONEWAY G3 BY studytime
  /STATISTICS DESCRIPTIVES HOMOGENEITY
  /MISSING ANALYSIS.

UNIANOVA G3 BY studytime
  /METHOD=SSTYPE(3)
  /INTERCEPT=INCLUDE
  /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY
  /CRITERIA=ALPHA(.05)
  /DESIGN=studytime.

* F statistic = MS_between / MS_within.
* Interpret using df1, df2 and the right-tail p value.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="F-Distribution-SPSS-Output.pdf".

Python Code for F Distribution

import pandas as pd
import numpy as np
import scipy.stats as stats
import statsmodels.api as sm
from statsmodels.formula.api import ols

df = pd.read_csv("dataset.csv")

df["G3"] = pd.to_numeric(df["G3"], errors="coerce")
df["studytime"] = df["studytime"].astype("category")

df_model = df.dropna(subset=["G3", "studytime"]).copy()

model = ols("G3 ~ C(studytime)", data=df_model).fit()
anova_table = sm.stats.anova_lm(model, typ=2)

ss_between = anova_table.loc["C(studytime)", "sum_sq"]
df1 = anova_table.loc["C(studytime)", "df"]
ss_within = anova_table.loc["Residual", "sum_sq"]
df2 = anova_table.loc["Residual", "df"]

ms_between = ss_between / df1
ms_within = ss_within / df2
f_observed = ms_between / ms_within

alpha = 0.05
f_critical = stats.f.ppf(1 - alpha, df1, df2)
p_value = stats.f.sf(f_observed, df1, df2)

print(anova_table)
print("MS Between:", ms_between)
print("MS Within:", ms_within)
print("Observed F:", f_observed)
print("Critical F:", f_critical)
print("Right-tail p value:", p_value)

decision = "Reject H0" if f_observed > f_critical else "Fail to reject H0"
print("Decision:", decision)

R Code for F Distribution

df <- read.csv("dataset.csv")

df$G3 <- as.numeric(df$G3)
df$studytime <- as.factor(df$studytime)

df_model <- na.omit(df[, c("G3", "studytime")])

model <- aov(G3 ~ studytime, data = df_model)
anova_table <- summary(model)[[1]]

ss_between <- anova_table["studytime", "Sum Sq"]
df1 <- anova_table["studytime", "Df"]
ss_within <- anova_table["Residuals", "Sum Sq"]
df2 <- anova_table["Residuals", "Df"]

ms_between <- ss_between / df1
ms_within <- ss_within / df2
f_observed <- ms_between / ms_within

alpha <- 0.05
f_critical <- qf(1 - alpha, df1, df2)
p_value <- pf(f_observed, df1, df2, lower.tail = FALSE)

data.frame(
  ss_between = ss_between,
  ss_within = ss_within,
  df1 = df1,
  df2 = df2,
  ms_between = ms_between,
  ms_within = ms_within,
  f_observed = f_observed,
  f_critical = f_critical,
  p_value = p_value,
  decision = ifelse(f_observed > f_critical, "Reject H0", "Fail to reject H0")
)

Excel Formulas for F Distribution

Run ANOVA:
Data > Data Analysis > ANOVA: Single Factor

F statistic:
=MS_Between / MS_Within

Critical F value:
=F.INV.RT(alpha, df1, df2)

Right-tail p value:
=F.DIST.RT(F_observed, df1, df2)

Decision:
=IF(F_observed > F_critical, "Reject H0", "Fail to reject H0")

Example:
MS Between = 155.026
MS Within = 9.765
F = 155.026 / 9.765
F ≈ 15.876

APA Reporting Wording

When reporting an F distribution result, include the observed F statistic, numerator degrees of freedom, denominator degrees of freedom and p value. When possible, also report an effect size such as eta squared or omega squared.

Common Mistakes

When to Use F Distribution

Use the F Distribution when a statistical test compares variance estimates or mean-square ratios. It is most common in ANOVA, regression model comparison and tests involving variance ratios.

For related guides, see ANOVA in Python, ANOVA in R, ANOVA in SPSS, ANOVA Assumptions, ANOVA Effect Size, Balanced ANOVA, Brown Forsythe ANOVA, Cohen’s F Formula, T Test vs ANOVA, P Value and Statistical Power.

Downloads and Resources for F Distribution

Use these resources to reproduce the F Distribution workflow. The SPSS output PDF is included as the software output file, while the script and workbook placeholders can be replaced with final uploaded files after they are added to the WordPress Media Library.

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