AP Statistics Curriculum 2007 NonParam VarIndep - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T17:52:23Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Dinov: /* Approach */

2014-10-20T19:23:56Z

‎Approach

Dinov: /* Approach */

2014-10-20T19:23:01Z

‎Approach

IvoDinov: /* See also */

2008-11-29T02:36:21Z

‎See also

IvoDinov: /* Computational Resources: Internet-based SOCR Tools */

2008-11-29T02:28:02Z

‎Computational Resources: Internet-based SOCR Tools

IvoDinov: /* Also see the SOCR FlignerKilleen Activity */

2008-10-20T20:27:27Z

‎Also see the SOCR FlignerKilleen Activity

IvoDinov at 20:25, 20 October 2008

2008-10-20T20:25:57Z

IvoDinov: /* Approach */

2008-10-07T19:30:21Z

‎Approach

IvoDinov: typos

2008-06-16T23:33:22Z

typos

IvoDinov at 05:35, 16 March 2008

2008-03-16T05:35:56Z

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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

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 ==Approach==
-The (modified) Fligner-Killeen test provides the means for studying the homogeneity of variances of ''k'' populations { <math>X_{i,j}</math>, for <math>1\leq i \leq n_j</math> and <math>1\leq j \leq k</math>}. The test jointly ranks the absolute values of <math>|X_{i,j}-\tilde{X_j}|</math> and assigns increasing '''scores''' <math>a_{N,i}=\Phi^{-1}\bigg \{{1 + {i\over N+1} \over 2} \bigg \}</math>, based on the ranks of all observations, see the [[AP_Statistics_Curriculum_2007_NonParam_VarIndep#References | Conover, Johnson, and Johnson (1981) reference below]].
+The (modified) Fligner-Killeen test provides the means for studying the homogeneity of variances of ''k'' populations { <math>X_{i,j}</math>, for <math>1\leq i \leq n_j</math> and <math>1\leq j \leq k</math>}. The test jointly ranks the absolute values of <math>|X_{i,j}-\tilde{X_j}|</math> and assigns increasing '''scores''' <math>a_{N,i}=\Phi^{-1}\bigg ({1 + {i\over N+1} \over 2} \bigg )</math>, based on the ranks of all observations, see the [[AP_Statistics_Curriculum_2007_NonParam_VarIndep#References | Conover, Johnson, and Johnson (1981) reference below]].
 In this test, <math>\tilde{X_j}</math> is the sample median of the ''j<sup>th</sup>'' population, and <math>\Phi(.)</math> is the [[AP_Statistics_Curriculum_2007_Normal_Std | cumulative distribution function for Normal distribution]]. The Fligner-Killeen test is sometimes also called the ''median-centering Fligner-Killeen test''.

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 ==Approach==
-The (modified) Fligner-Killeen test provides the means for studying the homogeneity of variances of ''k'' populations { <math>X_{i,j}</math>, for <math>1\leq i \leq n_j</math> and <math>1\leq j \leq k</math>}. The test jointly ranks the absolute values of <math>|X_{i,j}-\tilde{X_j}|</math> and assigns increasing '''scores''' <math>a_{N,i}=\Phi^{-1}({1 + {i\over N+1} \over 2})</math>, based on the ranks of all observations, see the [[AP_Statistics_Curriculum_2007_NonParam_VarIndep#References | Conover, Johnson, and Johnson (1981) reference below]].
+The (modified) Fligner-Killeen test provides the means for studying the homogeneity of variances of ''k'' populations { <math>X_{i,j}</math>, for <math>1\leq i \leq n_j</math> and <math>1\leq j \leq k</math>}. The test jointly ranks the absolute values of <math>|X_{i,j}-\tilde{X_j}|</math> and assigns increasing '''scores''' <math>a_{N,i}=\Phi^{-1}\bigg \{{1 + {i\over N+1} \over 2} \bigg \}</math>, based on the ranks of all observations, see the [[AP_Statistics_Curriculum_2007_NonParam_VarIndep#References | Conover, Johnson, and Johnson (1981) reference below]].
 In this test, <math>\tilde{X_j}</math> is the sample median of the ''j<sup>th</sup>'' population, and <math>\Phi(.)</math> is the [[AP_Statistics_Curriculum_2007_Normal_Std | cumulative distribution function for Normal distribution]]. The Fligner-Killeen test is sometimes also called the ''median-centering Fligner-Killeen test''.
 * '''Fligner-Killeen test statistics''':
-: <math>x_o^2 = {\sum_{j=1}^k {n_j(\bar{A_j} -\bar{a})^2} \over V^2}</math>,
+: <math>x_o^2 = {\sum_{j=1}^k {n_j(\bar{A_j} -\bar{a})^2} \over V^2},</math>
 : where <math>\bar{A_j}</math> is the mean score for the ''j<sup>th</sup>'' sample, <math>\bar{a}</math> is the overall mean score of all <math>a_{N,i}</math>, and <math>V^2</math> is the sample variance of all scores.
 That is:
-: <math>N=\sum_{j=1}^k{n_j}</math>,
+: <math>N=\sum_{j=1}^k{n_j},</math>
-: <math>\bar{A_j} = {1\over n_j}\sum_{i=1}^{n_j}{a_{N,m_i}}</math>, where <math>a_{N,m_i}</math> is the increasing rank score for the ''i<sup>th</sup>''-observation in the ''j<sup>th</sup>''-sample,
+: <math>\bar{A_j} = {1\over n_j}\sum_{i=1}^{n_j}{a_{N,m_i}},</math> where <math>a_{N,m_i}</math> is the increasing rank score for the ''i<sup>th</sup>''-observation in the ''j<sup>th</sup>''-sample,
-: <math>\bar{a} = {1\over N}\sum_{i=1}^{N}{a_{N,i}}</math>,
+: <math>\bar{a} = {1\over N}\sum_{i=1}^{N}{a_{N,i}},</math>
-: <math>V^2 = {1\over N-1}\sum_{i=1}^{N}{(a_{N,i}-\bar{a})^2}</math>.
+: <math>V^2 = {1\over N-1}\sum_{i=1}^{N}{(a_{N,i}-\bar{a})^2}.</math>
 * '''Fligner-Killeen probabilities''':
 For large sample sizes, the modified Fligner-Killeen test statistic has an asymptotic chi-square distribution with ''(k-1)'' degrees of freedom
-: <math>x_o^2 \sim \chi_{(k-1)}^2</math>.
+: <math>x_o^2 \sim \chi_{(k-1)}^2.</math>
 * '''Note''':

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 ==See also==
-[[SOCR_EduMaterials_AnalysisActivities_FlignerKilleen | SOCR Fligner-Killeen Activity]] provides more hands-on examples.
+* [[SOCR_EduMaterials_AnalysisActivities_FlignerKilleen | SOCR Fligner-Killeen Activity]] provides more hands-on examples.
 <hr>

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 ==Computational Resources: Internet-based SOCR Tools==
-Still under development.
+* See the [http://www.socr.ucla.edu/htmls/ana/FlignerKilleen_Analysis.html SOCR Fligner-Killeen Analysis applet].
 ==Examples==

@@ Line 32: / Line 32: @@
 Suppose we wanted to study whether the variances in certain time period (e.g., 1981 to 2006) of the consumer-price-indices (CPI) of several items were significantly different. We can use the [[SOCR_Data_Dinov_021808_ConsumerPriceIndex | SOCR CPI Dataset]] to answer this question for the '''Fuel''', '''Oil''', '''Bananas''', '''Tomatoes''', '''Orange Juice''', '''Beef''' and '''Gasoline''' items.
-==Also see the [[SOCR_EduMaterials_AnalysisActivities_FlignerKilleen | SOCR FlignerKilleen Activity]]==
+==See also==
 <hr>

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 ==Differences of Variances of Independent Samples==
-It is frequently necessary to test if ''k'' samples have equal variances. Equal variances across samples are called ''homogeneity of variances''. Some statistical tests, for example the [[AP_Statistics_Curriculum_2007_ANOVA_1Way | analysis of variance]], assume that variances are equal across groups or samples.
+It is frequently necessary to test if ''k'' samples have equal variances. ''Homogeneity of variances'' is often a reference to equal variances across samples. Some statistical tests, for example the [[AP_Statistics_Curriculum_2007_ANOVA_1Way | analysis of variance]], assume that variances are equal across groups or samples.
 ==Approach==
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 * '''Fligner-Killeen test statistics''':
 : <math>x_o^2 = {\sum_{j=1}^k {n_j(\bar{A_j} -\bar{a})^2} \over V^2}</math>,
-: where <math>\bar{A_j}</math> is the mean score for the ''j<sup>th</sup>'' sample, ''a'' is the overall mean score of all <math>a_{N,i}</math>, and <math>V^2</math> is the sample variance of all scores.
+: where <math>\bar{A_j}</math> is the mean score for the ''j<sup>th</sup>'' sample, <math>\bar{a}</math> is the overall mean score of all <math>a_{N,i}</math>, and <math>V^2</math> is the sample variance of all scores.
 That is: