# 13.2 The f distribution and the f ratio

 Page 1 / 1
This module describes how to calculate the F Ratio and F Distribution based on the hypothesis test for the One-Way ANOVA.

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 forthe denominator.

For example, if $F$ follows an $F$ distribution and the degrees of freedom for the numerator are 4 and the degrees of freedom for the denominator are 10, then $F$ ~ ${F}_{4,10}$ .

The $F$ distribution is derived from the Student's-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.

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

1. Variance between samples: An estimate of ${\sigma }^{2}$ that is the variance of the sample means multiplied by n (when there is equal n). If the samples are different sizes, the variance between samples is weighted toaccount for the different sample sizes. The variance is also called variation due to treatment or explainedvariation.
2. Variance within samples: An estimate of ${\sigma }^{2}$ that is the average of the sample variances (also known as a pooled variance). When the sample sizes are different, thevariance within samples is weighted. The variance is also called the variation due to error or unexplained variation.
• ${\mathrm{SS}}_{\text{between}}=$ the sum of squares that represents the variation among the different samples.
• ${\mathrm{SS}}_{\text{within}}=$ 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 which, in some cases, may be weighted. We used sum of squares to calculate the sample variance andthe sample standard deviation in Descriptive Statistics .

$\mathrm{MS}$ means "mean square." ${\mathrm{MS}}_{\text{between}}$ is the variance between groups and ${\mathrm{MS}}_{\text{within}}$ is the variance within groups.

## Calculation of sum of squares and mean square

• $k$ = the number of different groups
• ${n}_{j}$ = the size of the $\text{jth}$ group
• ${s}_{j}$ = the sum of the values in the $\text{jth}$ group
• $n$ = total number of all the values combined. (total sample size: $\sum {n}_{j}$ )
• $x$ = one value: $\sum x=\sum {s}_{j}$
• Sum of squares of all values from every group combined: $\sum {x}^{2}$
• Between group variability: ${\text{SS}}_{\text{total}}=\sum {x}^{2}-\frac{{\left(\sum x\right)}^{2}}{n}$
• Total sum of squares: $\sum {x}^{2}-\frac{\left(\sum x{\right)}^{2}}{n}$
• Explained variation- sum of squares representing variation among the different samples ${\text{SS}}_{\text{between}}=\sum \left[\frac{\left(\text{sj}{\right)}^{2}}{{n}_{j}}\right]-\frac{\left(\sum {s}_{j}{\right)}^{2}}{n}$
• Unexplained variation- sum of squares representing variation within samples due to chance: ${\text{SS}}_{\text{within}}={\text{SS}}_{\text{total}}-{\text{SS}}_{\text{between}}$
• df's for different groups (df's for the numerator): ${\text{df}}_{\text{between}}=k-1$
• Equation for errors within samples (df's for the denominator): ${\text{df}}_{\text{within}}=n-k$
• Mean square (variance estimate) explained by the different groups: ${\text{MS}}_{\text{between}}=\frac{{\text{SS}}_{\text{between}}}{{\text{df}}_{\text{between}}}$
• Mean square (variance estimate) that is due to chance (unexplained): ${\text{MS}}_{\text{within}}=\frac{{\text{SS}}_{\text{within}}}{{\text{df}}_{\text{within}}}$

${\mathrm{MS}}_{\text{between}}$ and ${\mathrm{MS}}_{\text{within}}$ can be written as follows:

• ${\mathrm{MS}}_{\text{between}}=\frac{{\mathrm{SS}}_{\text{between}}}{{\mathrm{df}}_{\text{between}}}=\frac{{\mathrm{SS}}_{\text{between}}}{k-1}$
• ${\mathrm{MS}}_{\text{within}}=\frac{{\mathrm{SS}}_{\text{within}}}{{\mathrm{df}}_{\text{within}}}=\frac{{\mathrm{SS}}_{\text{within}}}{n-k}$

The One-Way ANOVA test depends on the fact that ${\mathrm{MS}}_{\text{between}}$ can be influenced by population differences among means of the several groups. Since ${\mathrm{MS}}_{\text{within}}$ compares values of each group to its own group mean, the fact that group means might be different doesnot affect ${\mathrm{MS}}_{\text{within}}$ .

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 samplegroups come from populations with different normal distributions. If the null hypothesis is true, ${\mathrm{MS}}_{\text{between}}$ and ${\mathrm{MS}}_{\text{within}}$ should both estimate the same value.

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

$F=\frac{{\mathrm{MS}}_{\text{between}}}{{\mathrm{MS}}_{\text{within}}}$

If ${\mathrm{MS}}_{\text{between}}$ and ${\mathrm{MS}}_{\text{within}}$ estimate the same value (following the belief that ${H}_{o}$ is true), then the F-ratio should be approximately equal to 1. Mostly just sampling errorswould contribute to variations away from 1. As it turns out, ${\mathrm{MS}}_{\text{between}}$ consists of the population variance plus a variance produced from the differences between thesamples. ${\mathrm{MS}}_{\text{within}}$ is an estimate of the population variance. Since variances are always positive, if the null hypothesis is false, ${\mathrm{MS}}_{\text{between}}$ will generally be larger than ${\mathrm{MS}}_{\text{within}}$ . Then the F-ratio will be larger than 1.However, if the population effect size is small it is not unlikely that ${\mathrm{MS}}_{\text{within}}$ will be larger in a give sample.

The above 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

$F=\frac{n\cdot {{s}_{\stackrel{_}{x}}}^{2}}{{{{s}^{2}}_{\text{pooled}}}^{}}$

## Where ...

• $n=$ the sample size
• ${\text{df}}_{\text{numerator}}=k-1$
• ${\text{df}}_{\text{denominator}}=n-k$
• ${{s}^{2}}_{\mathrm{pooled}}=$ the mean of the sample variances (pooled variance)
• ${{s}_{\overline{x}}}^{2}=$ the variance of the sample means

The data is 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

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 table is shown below.

Plan 1 Plan 2 Plan 3
5 3.5 8
4.5 7 4
4 3.5
3 4.5

One-Way ANOVA Table: The formulas for SS(Total), SS(Factor) = SS(Between) and SS(Error) = SS(Within) are shown above. This 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)
= 2.9792 + 20.8542
=23.1
n - 1
= 10 total data - 1
= 9

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 ${H}_{o}$ .

## Notation

The notation for the F distribution is $F$ ~ ${F}_{\text{df(num)},\text{df(denom)}}$

where $\text{df(num)}={\mathrm{df}}_{\text{between}}$ and $\text{df(denom)}={\mathrm{df}}_{\text{within}}$

The mean for the F distribution is $\mu =\frac{\mathrm{df\left(num\right)}}{\mathrm{df\left(denom\right)}-1}$

how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid.