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Introductory Statistics

13.2 The F Distribution and the F-Ratio

Introductory Statistics13.2 The F Distribution and the F-Ratio
Table of contents
  1. Preface
  2. 1 Sampling and Data
    1. Introduction
    2. 1.1 Definitions of Statistics, Probability, and Key Terms
    3. 1.2 Data, Sampling, and Variation in Data and Sampling
    4. 1.3 Frequency, Frequency Tables, and Levels of Measurement
    5. 1.4 Experimental Design and Ethics
    6. 1.5 Data Collection Experiment
    7. 1.6 Sampling Experiment
    8. Key Terms
    9. Chapter Review
    10. Practice
    11. Homework
    12. Bringing It Together: Homework
    13. References
    14. Solutions
  3. 2 Descriptive Statistics
    1. Introduction
    2. 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
    3. 2.2 Histograms, Frequency Polygons, and Time Series Graphs
    4. 2.3 Measures of the Location of the Data
    5. 2.4 Box Plots
    6. 2.5 Measures of the Center of the Data
    7. 2.6 Skewness and the Mean, Median, and Mode
    8. 2.7 Measures of the Spread of the Data
    9. 2.8 Descriptive Statistics
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. Bringing It Together: Homework
    16. References
    17. Solutions
  4. 3 Probability Topics
    1. Introduction
    2. 3.1 Terminology
    3. 3.2 Independent and Mutually Exclusive Events
    4. 3.3 Two Basic Rules of Probability
    5. 3.4 Contingency Tables
    6. 3.5 Tree and Venn Diagrams
    7. 3.6 Probability Topics
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Bringing It Together: Practice
    13. Homework
    14. Bringing It Together: Homework
    15. References
    16. Solutions
  5. 4 Discrete Random Variables
    1. Introduction
    2. 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable
    3. 4.2 Mean or Expected Value and Standard Deviation
    4. 4.3 Binomial Distribution
    5. 4.4 Geometric Distribution
    6. 4.5 Hypergeometric Distribution
    7. 4.6 Poisson Distribution
    8. 4.7 Discrete Distribution (Playing Card Experiment)
    9. 4.8 Discrete Distribution (Lucky Dice Experiment)
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. References
    16. Solutions
  6. 5 Continuous Random Variables
    1. Introduction
    2. 5.1 Continuous Probability Functions
    3. 5.2 The Uniform Distribution
    4. 5.3 The Exponential Distribution
    5. 5.4 Continuous Distribution
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  7. 6 The Normal Distribution
    1. Introduction
    2. 6.1 The Standard Normal Distribution
    3. 6.2 Using the Normal Distribution
    4. 6.3 Normal Distribution (Lap Times)
    5. 6.4 Normal Distribution (Pinkie Length)
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  8. 7 The Central Limit Theorem
    1. Introduction
    2. 7.1 The Central Limit Theorem for Sample Means (Averages)
    3. 7.2 The Central Limit Theorem for Sums
    4. 7.3 Using the Central Limit Theorem
    5. 7.4 Central Limit Theorem (Pocket Change)
    6. 7.5 Central Limit Theorem (Cookie Recipes)
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. References
    13. Solutions
  9. 8 Confidence Intervals
    1. Introduction
    2. 8.1 A Single Population Mean using the Normal Distribution
    3. 8.2 A Single Population Mean using the Student t Distribution
    4. 8.3 A Population Proportion
    5. 8.4 Confidence Interval (Home Costs)
    6. 8.5 Confidence Interval (Place of Birth)
    7. 8.6 Confidence Interval (Women's Heights)
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. References
    14. Solutions
  10. 9 Hypothesis Testing with One Sample
    1. Introduction
    2. 9.1 Null and Alternative Hypotheses
    3. 9.2 Outcomes and the Type I and Type II Errors
    4. 9.3 Distribution Needed for Hypothesis Testing
    5. 9.4 Rare Events, the Sample, Decision and Conclusion
    6. 9.5 Additional Information and Full Hypothesis Test Examples
    7. 9.6 Hypothesis Testing of a Single Mean and Single Proportion
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. References
    14. Solutions
  11. 10 Hypothesis Testing with Two Samples
    1. Introduction
    2. 10.1 Two Population Means with Unknown Standard Deviations
    3. 10.2 Two Population Means with Known Standard Deviations
    4. 10.3 Comparing Two Independent Population Proportions
    5. 10.4 Matched or Paired Samples
    6. 10.5 Hypothesis Testing for Two Means and Two Proportions
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. Bringing It Together: Homework
    13. References
    14. Solutions
  12. 11 The Chi-Square Distribution
    1. Introduction
    2. 11.1 Facts About the Chi-Square Distribution
    3. 11.2 Goodness-of-Fit Test
    4. 11.3 Test of Independence
    5. 11.4 Test for Homogeneity
    6. 11.5 Comparison of the Chi-Square Tests
    7. 11.6 Test of a Single Variance
    8. 11.7 Lab 1: Chi-Square Goodness-of-Fit
    9. 11.8 Lab 2: Chi-Square Test of Independence
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. Bringing It Together: Homework
    16. References
    17. Solutions
  13. 12 Linear Regression and Correlation
    1. Introduction
    2. 12.1 Linear Equations
    3. 12.2 Scatter Plots
    4. 12.3 The Regression Equation
    5. 12.4 Testing the Significance of the Correlation Coefficient
    6. 12.5 Prediction
    7. 12.6 Outliers
    8. 12.7 Regression (Distance from School)
    9. 12.8 Regression (Textbook Cost)
    10. 12.9 Regression (Fuel Efficiency)
    11. Key Terms
    12. Chapter Review
    13. Formula Review
    14. Practice
    15. Homework
    16. Bringing It Together: Homework
    17. References
    18. Solutions
  14. 13 F Distribution and One-Way ANOVA
    1. Introduction
    2. 13.1 One-Way ANOVA
    3. 13.2 The F Distribution and the F-Ratio
    4. 13.3 Facts About the F Distribution
    5. 13.4 Test of Two Variances
    6. 13.5 Lab: One-Way ANOVA
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. References
    13. Solutions
  15. A | Review Exercises (Ch 3-13)
  16. B | Practice Tests (1-4) and Final Exams
  17. C | Data Sets
  18. D | Group and Partner Projects
  19. E | Solution Sheets
  20. F | Mathematical Phrases, Symbols, and Formulas
  21. G | Notes for the TI-83, 83+, 84, 84+ Calculators
  22. H | Tables
  23. Index

The distribution used for the hypothesis test is a new one. It is called the F distribution, named after Sir Ronald Fisher, an English statistician. The F statistic is a ratio (a fraction). There are two sets of degrees of freedom; one for the numerator and one for the denominator.

For example, if F follows an F distribution and the number of degrees of freedom for the numerator is four, and the number of degrees of freedom for the denominator is ten, then F ~ F4,10.

Note

The F distribution is derived from the Student's t-distribution. The values of the F distribution are squares of the corresponding values of the t-distribution. One-Way ANOVA expands the t-test for comparing more than two groups. The scope of that derivation is beyond the level of this course. It is preferable to use ANOVA when there are more than two groups instead of performing pairwise t-tests because performing multiple tests introduces the likelihood of making a Type 1 error.

To calculate the F ratio, two estimates of the variance are made.

  1. Variance between samples: An estimate of σ2 that is the variance of the sample means multiplied by n (when the sample sizes are the same.). If the samples are different sizes, the variance between samples is weighted to account for the different sample sizes. The variance is also called variation due to treatment or explained variation.
  2. Variance within samples: An estimate of σ2 that is the average of the sample variances (also known as a pooled variance). When the sample sizes are different, the variance within samples is weighted. The variance is also called the variation due to error or unexplained variation.
  • SSbetween = the sum of squares that represents the variation among the different samples
  • SSwithin = the sum of squares that represents the variation within samples that is due to chance.

To find a "sum of squares" means to add together squared quantities that, in some cases, may be weighted. We used sum of squares to calculate the sample variance and the sample standard deviation in Descriptive Statistics.

MS means "mean square." MSbetween is the variance between groups, and MSwithin is the variance within groups.

Calculation of Sum of Squares and Mean Square

  • k = the number of different groups
  • nj = the size of the jth group
  • sj = the sum of the values in the jth group
  • n = total number of all the values combined (total sample size: ∑nj)
  • x = one value: ∑x = ∑sj
  • Sum of squares of all values from every group combined: ∑x2
  • Between group variability: SStotal = ∑x2 – ( ∑ x 2 ) n ( ∑ x 2 ) n
  • Total sum of squares: ∑x2 – ( ∑x ) 2 n ( ∑x ) 2 n
  • Explained variation: sum of squares representing variation among the different samples: SSbetween = ∑[ ( s j ) 2 n j ]− (∑ s j ) 2 n ∑[ ( s j ) 2 n j ]− (∑ s j ) 2 n
  • Unexplained variation: sum of squares representing variation within samples due to chance: S S within =S S total –S S between S S within =S S total –S S between
  • df's for different groups (df's for the numerator): df = k – 1
  • Equation for errors within samples (df's for the denominator): dfwithin = n – k
  • Mean square (variance estimate) explained by the different groups: MSbetween = S S between df between S S between df between
  • Mean square (variance estimate) that is due to chance (unexplained): MSwithin = S S within d f within S S within d f within

MSbetween and MSwithin can be written as follows:

  • M S between = S S between d f between = S S between k−1 M S between = S S between d f between = S S between k−1
  • M S within = S S within d f within = S S within n−k M S within = S S within d f within = S S within n−k

The one-way ANOVA test depends on the fact that MSbetween can be influenced by population differences among means of the several groups. Since MSwithin compares values of each group to its own group mean, the fact that group means might be different does not affect MSwithin.

The null hypothesis says that all groups are samples from populations having the same normal distribution. The alternate hypothesis says that at least two of the sample groups come from populations with different normal distributions. If the null hypothesis is true, MSbetween and MSwithin should both estimate the same value.

Note

The null hypothesis says that all the group population means are equal. The hypothesis of equal means implies that the populations have the same normal distribution, because it is assumed that the populations are normal and that they have equal variances.

F-Ratio or F Statistic F= M S between M S within F= M S between M S within

If MSbetween and MSwithin estimate the same value (following the belief that H0 is true), then the F-ratio should be approximately equal to one. Mostly, just sampling errors would contribute to variations away from one. As it turns out, MSbetween consists of the population variance plus a variance produced from the differences between the samples. MSwithin is an estimate of the population variance. Since variances are always positive, if the null hypothesis is false, MSbetween will generally be larger than MSwithin.Then the F-ratio will be larger than one. However, if the population effect is small, it is not unlikely that MSwithin will be larger in a given sample.

The foregoing calculations were done with groups of different sizes. If the groups are the same size, the calculations simplify somewhat and the F-ratio can be written as:

F-Ratio Formula when the groups are the same size F= n⋅ s x ¯ 2 s 2 pooled F= n⋅ s x ¯ 2 s 2 pooled

where ...
  • n = the sample size
  • dfnumerator = k – 1
  • dfdenominator = n – k
  • s2 pooled = the mean of the sample variances (pooled variance)
  • s x ¯ 2 s x ¯ 2 = the variance of the sample means

Data are typically put into a table for easy viewing. One-Way ANOVA results are often displayed in this manner by computer software.

Source of Variation Sum of Squares (SS) Degrees of Freedom (df) Mean Square (MS) F
Factor
(Between)		 SS(Factor) k – 1 MS(Factor) = SS(Factor)/(k – 1) F = MS(Factor)/MS(Error)
Error
(Within)		 SS(Error) n – k MS(Error) = SS(Error)/(n – k)
Total SS(Total) n – 1
Table 13.1

Example 13.1

Three different diet plans are to be tested for mean weight loss. The entries in the table are the weight losses for the different plans. The one-way ANOVA results are shown in Table 13.2.

Plan 1: n1 = 4 Plan 2: n2 = 3 Plan 3: n3 = 3
5 3.5 8
4.5 7 4
4 3.5
3 4.5
Table 13.2

s1 = 16.5, s2 =15, s3 = 15.5

Following are the calculations needed to fill in the one-way ANOVA table. The table is used to conduct a hypothesis test.

SS(between)=∑[ ( s j ) 2 n j ]− ( ∑ s j ) 2 n  SS(between)=∑[ ( s j ) 2 n j ]− ( ∑ s j ) 2 n 
=  s 1 2 4 + s 2 2 3 + s 3 2 3 − ( s 1 + s 2 + s 3 ) 2 10 =  s 1 2 4 + s 2 2 3 + s 3 2 3 − ( s 1 + s 2 + s 3 ) 2 10

where n1 = 4, n2 = 3, n3 = 3 and n = n1 + n2 + n3 = 10

  = ( 16.5 ) 2 4 + ( 15 ) 2 3 + ( 15.5 ) 2 3 − ( 16.5+15+15.5 ) 2 10   = ( 16.5 ) 2 4 + ( 15 ) 2 3 + ( 15.5 ) 2 3 − ( 16.5+15+15.5 ) 2 10
SS(between)=2.2458 SS(between)=2.2458
S(total)=∑ x 2 − ( ∑x ) 2 n S(total)=∑ x 2 − ( ∑x ) 2 n
 =( 5 2 + 4.5 2 + 4 2 + 3 2 + 3.5 2 + 7 2 + 4.5 2 + 8 2 + 4 2 + 3.5 2 )  =( 5 2 + 4.5 2 + 4 2 + 3 2 + 3.5 2 + 7 2 + 4.5 2 + 8 2 + 4 2 + 3.5 2 )
− (5+4.5+4+3+3.5+7+4.5+8+4+3.5) 2 10 − (5+4.5+4+3+3.5+7+4.5+8+4+3.5) 2 10
=244− 47 2 10 =244−220.9 =244− 47 2 10 =244−220.9
SS(total)=23.1 SS(total)=23.1
SS(within)=SS(total)−SS(between) SS(within)=SS(total)−SS(between)
= 23.1−2.2458 = 23.1−2.2458
SS(within)=20.8542 SS(within)=20.8542

Using the TI-83, 83+, 84, 84+ Calculator

One-Way ANOVA Table: The formulas for SS(Total), SS(Factor) = SS(Between) and SS(Error) = SS(Within) as shown previously. The same information is provided by the TI calculator hypothesis test function ANOVA in STAT TESTS (syntax is ANOVA(L1, L2, L3) where L1, L2, L3 have the data from Plan 1, Plan 2, Plan 3 respectively).

Source of Variation Sum of Squares (SS) Degrees of Freedom (df) Mean Square (MS) F
Factor
(Between)		 SS(Factor)
= SS(Between)
= 2.2458		 k – 1
= 3 groups – 1
= 2		 MS(Factor)
= SS(Factor)/(k – 1)
 = 2.2458/2
= 1.1229		 F =
 MS(Factor)/MS(Error)
= 1.1229/2.9792
= 0.3769
Error
(Within)		 SS(Error)
= SS(Within)
= 20.8542		 n – k
= 10 total data – 3 groups
= 7		 MS(Error)
= SS(Error)/(n – k)
= 20.8542/7
= 2.9792
Total SS(Total)
= 2.2458 + 20.8542
= 23.1		 n – 1
= 10 total data – 1
= 9
Table 13.3

Try It 13.1

As part of an experiment to see how different types of soil cover would affect slicing tomato production, Marist College students grew tomato plants under different soil cover conditions. Groups of three plants each had one of the following treatments

  • bare soil
  • a commercial ground cover
  • black plastic
  • straw
  • compost

All plants grew under the same conditions and were the same variety. Students recorded the weight (in grams) of tomatoes produced by each of the n = 15 plants:

Bare: n1 = 3 Ground Cover: n2 = 3 Plastic: n3 = 3 Straw: n4 = 3 Compost: n5 = 3
2,625 5,348 6,583 7,285 6,277
2,997 5,682 8,560 6,897 7,818
4,915 5,482 3,830 9,230 8,677
Table 13.4


Create the one-way ANOVA table.

The one-way ANOVA hypothesis test is always right-tailed because larger F-values are way out in the right tail of the F-distribution curve and tend to make us reject H0.

Notation

The notation for the F distribution is F ~ Fdf(num),df(denom)

where df(num) = dfbetween and df(denom) = dfwithin

The mean for the F distribution is μ= df(denom) df(denom)–2 μ= df(denom) df(denom)–2

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