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SSE = sum of squared errors
https://openstax.org/books/introductory-statistics-2e/pages/12-formula-review
n= the number of data points
https://openstax.org/books/introductory-statistics-2e/pages/12-formula-review
Outlier : an observation that does not fit the rest of the data
https://openstax.org/books/introductory-statistics-2e/pages/12-key-terms
Analysis of variance extends the comparison of two groups to several, each a level of a categorical variable (factor). Samples from each group are independent, and must be randomly selected from normal populations with equal variances. We test the null hypothesis of equal means of the response in every group versus the alternative hypothesis of one or more group means being different from the others. A one-way ANOVA hypothesis test determines if several population means are equal. The distribution for the test is theFdistribution with two different degrees of freedom.
https://openstax.org/books/introductory-statistics-2e/pages/13-chapter-review
Each population from which a sample is taken is assumed to be normal.
https://openstax.org/books/introductory-statistics-2e/pages/13-chapter-review
All samples are randomly selected and independent.
https://openstax.org/books/introductory-statistics-2e/pages/13-chapter-review
The populations are assumed to have equal standard deviations (or variances).
https://openstax.org/books/introductory-statistics-2e/pages/13-chapter-review
Analysis of variance compares the means of a response variable for several groups. ANOVA compares the variation within each group to the variation of the mean of each group. The ratio of these two is theFstatistic from anFdistribution with (number of groups – 1) as the numerator degrees of freedom and (number of observations – number of groups) as the denominator degrees of freedom. These statistics are summarized in the ANOVA table.
https://openstax.org/books/introductory-statistics-2e/pages/13-chapter-review
The graph of theFdistribution is always positive and skewed right, though the shape can be mounded or exponential depending on the combination of numerator and denominator degrees of freedom. TheFstatistic is the ratio of a measure of the variation in the group means to a similar measure of the variation within the groups. If the null hypothesis is correct, then the numerator should be small compared to the denominator. A smallFstatistic will result, and the area under theFcurve to the right will be large, representing a largep-value. When the null hypothesis of equal group means is incorrect, then the numerator should be large compared to the denominator, giving a largeFstatistic and a small area (smallp-value) to the right of the statistic under theFcurve.
https://openstax.org/books/introductory-statistics-2e/pages/13-chapter-review
When the data have unequal group sizes (unbalanced data), then techniques from13.2 The F Distribution and the F-Rationeed to be used for hand calculations. In the case of balanced data (the groups are the same size) however, simplified calculations based on group means and variances may be used. In practice, of course, software is usually employed in the analysis. As in any analysis, graphs of various sorts should be used in conjunction with numerical techniques. Always look of your data!
https://openstax.org/books/introductory-statistics-2e/pages/13-chapter-review
TheFtest for the equality of two variances rests heavily on the assumption of normal distributions. The test is unreliable if this assumption is not met. If both distributions are normal, then the ratio of the two sample variances is distributed as anFstatistic, with numerator and denominator degrees of freedom that are one less than the samples sizes of the corresponding two groups. Atest of two varianceshypothesis test determines if two variances are the same. The distribution for the hypothesis test is theFdistribution with two different degrees of freedom.
https://openstax.org/books/introductory-statistics-2e/pages/13-chapter-review
The populations from which the two samples are drawn are normally distributed.
https://openstax.org/books/introductory-statistics-2e/pages/13-chapter-review
The two populations are independent of each other.
https://openstax.org/books/introductory-statistics-2e/pages/13-chapter-review
SSbetween=∑​[(sj)2nj]−(∑​sj)2nSSbetween=∑​[(sj)2nj]−(∑​sj)2n
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
SStotal=∑​x2−(∑​x)2nSStotal=∑​x2−(∑​x)2n
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
SSwithin=SStotal−SSbetweenSSwithin=SStotal−SSbetween
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
dfbetween=df(num) =k– 1
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
dfwithin=df(denom)=n–k
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
MSbetween=SSbetweendfbetweenSSbetweendfbetween
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
MSwithin=SSwithindfwithinSSwithindfwithin
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
F=MSbetweenMSwithinMSbetweenMSwithin
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
Fratio when the groups are the same size:F=nsx¯2s2poolednsx¯2s2pooled
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
Mean of theFdistribution:µ=df(num)df(denom)−2df(num)df(denom)−2
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
where:
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
k= the number of groups
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
nj= the size of thejthgroup
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
sj= the sum of the values in thejthgroup
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
n= the total number of all values (observations) combined
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
x= one value (one observation) from the data
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
sx¯2sx¯2= the variance of the sample means
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
s2pooleds2pooled= the mean of the sample variances (pooled variance)
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
Fhas the distributionF~F(n1– 1,n2– 1)
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
F=s12σ12s22σ22s12σ12s22σ22
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
Ifσ1=σ2, thenF=s12s22s12s22
https://openstax.org/books/introductory-statistics-2e/pages/13-formula-review
Analysis of Variance : also referred to as ANOVA, is a method of testing whether or not the means of three or more populations are equal. The method is applicable if:all populations of interest are normally distributed.the populations have equal standard deviations.samples (not necessarily of the same size) are randomly and independently selected from each population.The test statistic for analysis of variance is theF-ratio.
https://openstax.org/books/introductory-statistics-2e/pages/13-key-terms
One-Way ANOVA : a method of testing whether or not the means of three or more populations are equal; the method is applicable if:all populations of interest are normally distributed.the populations have equal standard deviations.samples (not necessarily of the same size) are randomly and independently selected from each population.there is one independent variable and one dependent variable.The test statistic for analysis of variance is theF-ratio.
https://openstax.org/books/introductory-statistics-2e/pages/13-key-terms
Variance : mean of the squared deviations from the mean; the square of the standard deviation. For a set of data, a deviation can be represented asx–x¯x¯wherexis a value of the data andx¯x¯is the sample mean. The sample variance is equal to the sum of the squares of the deviations divided by the difference of the sample size and one.
https://openstax.org/books/introductory-statistics-2e/pages/13-key-terms