Suppose you have a random sample of 
observations from 
and 
observations from 
Also suppose $S_1^2 = 244.8387,
S_2^2 = 370.9807,
F = S_1^2/S_2^2 = 0.659977.$ We know (because we simulated the data) that 
so that we hope
not to reject 
in favor of 
[Computations use R statistical software.]
y1 = rnorm(30, 100, 15); v1 = var(y1)
y2 = rnorm(40, 200, 15); v2 = var(y2)
f = v1/v2
v1; v2; f
[1] 244.8387 # 1st sample variance
[1] 370.9807 # 2nd
[1] 0.659977 # F-ratio
The critical values of a test at level 5% are 0.492 and 1.962, cutting 2.5%
of the probability from the left and right tails, respectively, of 
Because 
we do not reject 
qf(c(.025,.975), 29, 39) # 'qf' is an F quantile function
[1] 0.4919648 1.9618689
The density of 
the observed value of 
(solid black) and the
critical values (broken red) are shown below.
You can find the upper critical value in many printed F-distribution tables. However, to save space these printed tables typically do not show lower critical values (corresponding to quantiles below 0.5).
I believe that your second statement of critical values is incorrect, and guess it is based on an attempt to show how to find the lower critical values from printed tables.
By reversing the degrees of freedom, cutting 
from the upper tail,
and taking the reciprocal, you can find the lower critical value.
In R the procedure for finding quantile 0.025 of 
can be
illustrated as follows:
qf(.975, 39, 29); 1/qf(.975, 39, 29); qf(.025, 29, 39)
[1] 2.032666 # available in many printed tables
[1] 0.4919648 # reciprocal of above
[1] 0.4919648 # available directly in R, but not printed tables
Your second expression reverses the degrees of freedom but doesn't get
the rest correct. Part of the difficulty may be that printed tables
and books that rely on them use notation such as 
to represent quantile 0.975 of 
mentioning the
percentage above in the right tail instead of the percentage below
(as used for CDFs and quantiles). [One can only hope this '(upper) percentage
point' notation falls into richly deserved disuse as we rely increasingly
on software instead of printed tables.]
Addendum F-test in R:
F test to compare two variances
data: y1 and y2
F = 0.65998, num df = 29, denom df = 39, p-value = 0.2475
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.3364022 1.3415128
sample estimates:
ratio of variances
0.659977
Depauw
acad.depauw.edu › ~harvey › Chem351 › PDFs › F_TableTwoTail.pdf pdf
F-table for Two-Tailed Test at α = 0.05 (95% Confidence Level) ν2\ν1 1 2 3 4 5
F-table for Two-Tailed Test at α = 0.05 (95% Confidence Level)
Depauw
dpuadweb.depauw.edu › harvey_web › Chem351 › PDFs › F_TableTwoTail.pdf pdf
F-table for Two-Tailed Test at α = 0.05 (95% Confidence ...
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Videos
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F Test Two Tailed Example - YouTube
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F-test - 2 - Two Tailed Test of Two Variances - YouTube
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Using the F-Distribution Tables in Statistics - YouTube
University of Washington
faculty.washington.edu › heagerty › Books › Biostatistics › TABLES › F-Tables
F-Tables
Upper one-sided 0.10 significance levels; two-sided 0.20 significance levels; 90 percent percentiles. Tabulated are critical values for the distribution. The column headings give the numerator degrees of freedom and the row headings the demoninator degrees of freedom. Lower one-sided critical values may be found from these tables by reversing the degrees of freedom and using the reciprocal of the tabled value at the same significance level (100 minus the percent for the percentile).
UBalt
home.ubalt.edu › ntsbarsh › business-stat › StatistialTables.pdf pdf
STATISTICAL TABLES Cumulative normal distribution
TABLE A.2 · t Distribution: Critical Values of t · Significance level · Degrees of · Two-tailed test: 10% 5% 2% 1% 0.2% 0.1% freedom · One-tailed test: 5% 2.5% 1% 0.5% 0.1% 0.05% 1 · 6.314 · 12.706 · 31.821 · 63.657 · 318.309 · 636.619 · 2 · 2.920 ·
MedCalc
medcalc.org › en › manual › t-distribution-table.php
t-distribution table (two-tailed) - MedCalc Manual
October 13, 2025 - Statistical tables: values of the t-distribution (two tailed).
Statistics By Jim
statisticsbyjim.com › home › blog › f-table
F-table - Statistics By Jim
June 23, 2025 - Choose the F-table for your significance level. These three tables cover the most common significance levels of 0.10, 0.05, and 0.01. Columns specify the numerator degrees of freedom (DF1), while rows set the denominator’s (DF2).
Chemistry LibreTexts
chem.libretexts.org › learning objects › reference › reference tables › analytic references
Appendix 05: Critical Values for the F-Test - Chemistry LibreTexts
November 3, 2020 - The following tables provide values for F(0.05, νnum, νdenom) for one-tailed and for two-tailed F-tests.
Shark Bay Ecosystem Research Project
faculty.fiu.edu › ~mcguckd › F Distribution Tables.pdf pdf
F Table for α = 0.10 \ df1=1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 ∞
F Table · for α = 0.05 · / df1=1 · 2 · 3 · 4 · 5 · 6 · 7 · 8 · 9 · 10 · 12 · 15 · 20 · 24 · 30 · 40 · 60 · 120 · ∞ · df2=1 · 161.4476 · 199.5 · 215.7073 · 224.5832 · 230.1619 · 233.986 · 236.7684 · 238.8827 · 240.5433 · 241.8817 · 243.906 ·
Aleks
static.aleks.com › aleks › resources › gibbs › 9781259755330_TableF.pdf pdf
T A B L E F The t Distribution Confidence intervals 80% 90% 95% 98% 99%
One tail · Area · 훼 · t · Two tails · Area · Area · +t · ‒t · 2 · 2 · 훼 · 훼 · unded to 1.28 in the textbook. unded to 1.65 in the textbook. unded to 2.33 in the textbook.
Saylor
saylordotorg.github.io › text_introductory-statistics › s15-03-f-tests-for-equality-of-two-va.html
F-tests for Equality of Two Variances
In the context of Note 11.27 "Example 6", suppose Type A test strips are the current market leader and Type B test strips are a newly improved version of Type A. Test, at the 10% level of significance, whether the data given in Table 11.16 "Two Types of Test Strips" provide sufficient evidence to conclude that Type B test strips have better consistency (lower variance) than Type A test strips. ... Step 1. The test of hypotheses is now ... Step 2. The distribution is the F-distribution with degrees of freedom ... Step 3. The value of the test statistic is ... Step 4. The test is right-tailed.
University of Regina
uregina.ca › ~gingrich › f.pdf pdf
Appendix K The F Distribution
When the significance level is α = 0.05, use the second F table.
Omni Calculator
omnicalculator.com › statistics › critical-value
Critical Value Calculator
June 18, 2025 - Two-tailed test: critical value equals ±(1-α/2)-th quantile of the t distribution with N – 1 degrees of freedom. Open quantile tables for t-distribution. Look for the row corresponding to N – 1 degrees of freedom and for the column ...
T Table
tdistributiontable.com › home
T Table - T Table
April 23, 2021 - So we will choose the one-tail row to map our alpha level. 3. Next, we look for the alpha value along the above highlighted row. Our alpha level for this example is 0.05. Let us map the same on the table · 4. Once that is done, let us map the degrees of freedom under the leftmost column of the table under (df) 5. The intersection of these two presents us with the critical value we are looking for
UCLA Statistics
stats.oarc.ucla.edu › other › mult-pkg › faq › general › faq-what-are-the-differences-between-one-tailed-and-two-tailed-tests
FAQ: What are the differences between one-tailed and two-tailed tests?
If you are using a significance level of 0.05, a two-tailed test allots half of your alpha to testing the statistical significance in one direction and half of your alpha to testing statistical significance in the other direction. This means that .025 is in each tail of the distribution of your test statistic. When using a two-tailed test, regardless of the direction of the relationship you hypothesize, you are testing for the possibility of the relationship in both directions.
MedCalc
medcalc.org › manual › f-table.php
F Distribution critical values
April 4, 2025 - Values of the t-distribution (two-tailed) Values of the Chi-squared distribution · Logit transformation · Privacy Contact Site map · MedCalc Software Ltd Digimizer · © 2025 MedCalc Software Ltd · https://www.medcalc.org/manual/f-table.php© 2025 MedCalc Software Ltd
Crafton Hills College
craftonhills.edu › current-students › tutoring-center › mathematics-tutoring › distribution_tables_normal_studentt_chisquared.pdf pdf
Standard Normal Distribution Probabilities Table
Standard Normal Distribution Probabilities Table · one-tail area · 0.25 · 0.125 · 0.1 · 0.075 · 0.05 · 0.025 · 0.01 · 0.005 · 0.0005 · two-tail area · 0.5 · 0.25 · 0.2 · 0.15 · 0.1 · 0.05 · 0.02 · 0.01 · 0.001 · confidence level · 0.5 · 0.75 · 0.8 · 0.85 · 0.9 · 0.95 ...
Anindya Bhadra
stat.purdue.edu › ~lfindsen › stat503 › F_alpha_05.pdf pdf
Critical Values of the F-Distribution: α = 0.05 Denom.
Critical Values of the F-Distribution: α = 0.05 · Numerator Degrees of Freedom
Reddit
reddit.com › r/statistics › [q] double sided f-test
r/statistics on Reddit: [Q] Double sided F-test
October 28, 2024 -
If I have a significance level of alpha = 0.05, do I use 0.05 as the the alpha when choosing the critical value or alpha/2 since the test is double sided? Most sources I look at says that I should use 0.05 since the F-distribution is one sided, but how could that be? Is not the fact that it is either two or one sided completly neglected in that case?
Top answer 1 of 5
7
An F-test tests whether there is a difference between groups, not the directionality of the difference.
2 of 5
3
The F distribution itself isn't one or two 'sided'. That's a feature of an alternative hypothesis The F distribution has two ends of course, one at 0 and one going off to infinity. In some tests that have an F distribution under H0 the large end is consistent with H1 (e.g. ANOVA) and in some other tests, it's both ends (e.g. a test for equality of two variances under normality) But in terms of a hypothesis about equality of two means, a two sided alternative on the mean difference would both lead to a single tail of the F (were you to use anova rather than a t test) It's important to keep clear the distinction between a hypothesis (a statement sbout a parameter) and the statistic you use to test it. You might look in one tail for a two sided alternative, if the statistic puts samples corresponding to both 'sides' of the alternative into the same end of the null distribution of that statistic
Ucla
socr.ucla.edu › Applets.dir › F_Table.html
F-Distribution Tables
numerator degrees of freedom are ... F(10,12) does not equal F(12,10) ). For the four F tables below, the rows represent denominator degrees of freedom and the columns represent numerator degrees of freedom. The right tail area is given in the name of the table....