Differences between a persons head size and their intelligence - Assignment Example

? Statistics Research paper Superficial observation of data elements can sometimes lead to a wrongful presumption of correlation between thefactors in the data set. The misconception is especially common in controversial subjects with long histories of discussions and argumentations. However, through thorough statistical analysis, given a certain level of confidence, the claims usually come into careful and empirical examination. Even though, misrepresentation of data can lead to the wrong conclusions, which sometimes happens when there is an ulterior motive by the researcher to articulate a particular viewpoint. For instance, the debate on human intelligence derives its basis from many dimensions such as ethnicity, gender, age, head size, and talent. In a study of the relationship between social statuses of persons and their head sizes, a nineteenth century doctor, Paul Broca, came to the conclusion that brain size bears a direct correlation to a person’s intelligence, and by extension a person’s prestige in the society. The data however has gaping anomalies, which put to question the conclusion obtainable using the data. This research paper sets out to test whether there are any significant differences between a person’s head size and their intelligence, which according to the study presumably bears a correlation to the person’s vocation. Table of Contents Abstract 2 Table of Contents 3 Introduction 4 Discussion and Analysis 5 Test for at least one head-circumference mean differing from others 5 Test for equal variances in the data 5 The mislabeling of the Standard Error column as Standard Deviation 6 A recalculation of the correct standard deviation for Epstein’s data 7 Likeliness of rejection of the null hypothesis after recalculation of ANOVA with correct standard deviation 7 A calculation of revised sum of squared errors, mean square errors and F statistic 8 Squared sum of Squares 8 Degrees of Freedom 8 Mean Sum of Squares 8 F Statistic 8 A Retest of the hypothesis that head-circumference population means are equal at 0.05 significance level 9 ANOVA procedures requirement of equal variance is satisfied in light of new information 9 Tukey’s test for categories with significant difference in means 10 Recommendations and suggestions 10 Conclusion 11 Bibliography 12 Appendices/Supplementary Section 13 Introduction One of the primary assertions of Broca’s research is that ‘eminent’ men have significantly greater brain size than their ‘mediocre’ counterparts (Gould, 1981). The elucidation of the claim with data on Bostonian criminals from differing vocations articulated Broca’s argument that a person’s intelligence had a direct relation to the person’s brain size (Gould, 1981). Data on brain sizes with factorial categorizations on vocational backgrounds of the persons under study gave the impression that brain size and intelligence bear a relationship. The observation stirred considerable interest among researchers, who sought to establish how the human intelligence researchers had successfully pushed the possibility to the periphery in favor of intelligence tests (Gould, 1981). However, statistical tools, and in particular one-way ANOVA, which is a handy statistical tool for comparison of means and determining interrelations between data values, the data might lead to a different conclusion. Through tentative statistical analysis on the data purporting to illustrate Broca’s claim of brain correlation to intelligence, a number of importance issues come to notice. Firstly, what the researcher labels as standard deviation is in fact standard error, which are two different terms, and have differing impacts on the test of hypothesis for Broca’s argument. Secondly, it becomes apparent that there was omission of some groups from the final data, which may completely reverse the findings of the study (Gould, 1981). Finally, the listing of the data items suggests suspicious agenda, as it might very well lead a statistical analysis to lead to the acceptance of a contrary hypothesis. This report purposes to investigate, using statistical tools, the data articulating the case for a direct relation between head-circumference and intelligence, making provisions for the obvious manipulations, and investigating any variations in the conclusions given modifications. Discussion and Analysis In this study, we put to test the claim that head-circumference of the people of difference vocations are equal. The affirmation of the hypothesis would debunk Broca’s theory that intelligence has a direct relationship to head-circumference. The hypothesis is on a 0.05 level of significance. Test for at least one head-circumference mean differing from others From the results of the ANOVA table from Epstein’s statistics, it does appear that the means of the head-circumferences are different. A computed F value of 1,998.6275 evidences the fact that the means are markedly different, as it exceeds the table value of 2.010 at 0.05 level of significance. The conclusion implies that there is a significant difference between the means in head-circumferences for people of different vacations. The results affirm Broca’s claims that a person’s intelligence has a direct correlation to the brain size. Therefore, we reject the null hypothesis, which claims that the means of the head-circumferences are equal. Test for equal variances in the data The analysis of variance (ANOVA) hinges on many assumptions, one of which is that the data under analysis has equal variances (Akritas & Papadatos, 2004). When the variances differ, we talk of the sample groups showing heteroscedasticity, and when the variances do not differ, which is desirable, we have homoscedasticity (Akritas & Papadatos, 2004). Since variance is a factor of standard deviation, test for equal variances is similar to a test for equal standard deviations. However, the tests for determining the equality of variance, for instance Levene’s test and Bartlett’s test, are highly unreliable, and are very sensitive often resulting in type I error. Type I error is when we reject a null hypothesis when it is true. Levene’s test for equality of variance (Sturm-Beiss, 2005) Where W is the test result K is the number of groups, which contain the samples N is the total number in the samples W = {(1672 - 8) (7,865.359)} / {(8 - 1) (935.450)} = (13,088,108.80) / (6,548.15) = 1,998.7490 The F value for the Levene’s test is approximately equal to the value of the F-test (1,998.6275), which implies homogeneity in the variances of the samples. The mislabeling of the Standard Error column as Standard Deviation The relation between standard deviation and standard error is uncanny, as both refer are measures of variability. However, standard deviation specifically refers to variability of individual values from the sample mean; while standard error refers to the variability of the sample mean to the population mean (Brink, 2010). Mathematically, the relationship between standard deviation and standard error is: Standard Error = (Standard Deviation/Square Root of the Sample Size) A recalculation of the correct standard deviation for Epstein’s data The correct standard deviation value is obtained through multiplication of the given value, which represents the standard error, with the square root of the sample size. Sample Size (n) Square root n Table Value (Standard Error) Correct SD value 25 5 1.9 9.5 61 7.8 1.5 11.7 107 10.3 1.1 11.4 194 13.9 0.8 11.1 25 5 2.5 12.5 351 18.7 0.6 11.2 262 16.2 0.7 11.3 647 25.4 0.3 7.6 Likeliness of rejection of the null hypothesis after recalculation of ANOVA with correct standard deviation The new values of standard deviation are markedly larger than the previous values. The new values will have no effect on the value of treatment sum of squares (SSTR). However, the increase in standard deviations among sample groups will result in a higher value of the error sum of squares (SSE). A bigger sum of squares value for the error factor will translate in higher mean sum of squares value. Consequently, the division of MST by MSE will result in a lower value of computed F statistic. The lower the value of computed F, the more likely it is to satisfy the null hypothesis. Therefore, recalculation of the ANOVA with the new standard deviation makes the rejection of the null hypothesis less likely. A calculation of revised sum of squared errors, mean square errors and F statistic Squared sum of Squares Treatment sum of squares = 7,865.3595 Error Sum of Squares = 166,228.74 Degrees of Freedom Total = N – 1 = 1672 – 1 = 1671 Treatment = k – 1 = 8 – 1 = 7 Error = Total – Treatment = 1671 – 7 = 1664 Mean Sum of Squares Mean Treatment Sum of Squares = (Treatment Sum of Squares) / (Degrees of Freedom) = 7,865.3595 / 7 = 1,123.6228 Mean Error Sum of Squares = (Error Sum of Squares) / (Degrees of Freedom) = 166,228.74 / 1,664 = 99.8971 F Statistic Computed F (F statistic) = (Mean Treatment Sum of Squares) / (Mean Error Sum of Squares) = 1,123.6228 / 99.8971 = 11.2478 An ANOVA table for the analysis (StatPages, 2012) Source of Variation Sum of Squares Degrees of Freedom Mean Sum of Squares Computed F value Treatment 7,865.3595 7 1,123.6228 11.2478 Error 166,228.74 1,664 99.8971 Total 174,094.0995 1,671 A Retest of the hypothesis that head-circumference population means are equal at 0.05 significance level From the revised figure of the F statistic, the value has made a considerable drop from 1,998.6275 to 11.2478, which is much closer to the value necessary for the acceptance of the null hypothesis. However, the value of F at 0.05 level of significance (2.010) is still lower than the computed value (11.2478); hence, we still reject the null hypothesis, and conclude that significant differences exist in head-circumferences of people of difference intelligence levels. ANOVA procedures requirement of equal variance is satisfied in light of new information Levene’s formula for test of equality, as in the previous case Where W is the test result K is the number of groups, which contain the samples N is the total number in the samples W = {(1672 - 8) (7,865.359)} / {(8 - 1) (166,228.740)} = (13,088,108.80) / (1,163,601.18) = 11.2479 The F value for the Levene’s test is equal to the value of the F-test statistic, which implies homoscedasticity of the samples. The equal variances requirement is fully satisfied, the conclusions on the hypothesis remain as before, and the null hypothesis is rejected. Therefore the head-circumference means are different, which backs the initial claim that intelligence has a direct relationship to brain size. Tukey’s test for categories with significant difference in means The above ANOVA analysis on the data by Hooton has led to the rejection of the null hypothesis for equal population means. The rejection of a null hypothesis in a one-way ANOVA test means that at least one of the means in the experiment is different from the other means. Tukey’s test helps establish which among the means led to the rejection of the null hypothesis. Family wise error rate is the probability of making a type I error when using pairwise tests on data. After running Tukey’s test for the means on SPSS (University of Texas, 2005), the group sizes for the data set were unequal at 0.05 level of significance. However, the test did not guarantee the absence of family wise error for the pairwise test of the data. Recommendations and suggestions Further disclosures hinting at manipulation of the data need further details to provide more data. For instance, the revelations only disclose the ranks of the initially omitted vocational groups, and fail to expound on the ranks of the initial data on the Epstein’s table, which would have aided in further statistical analysis using rank statistics. With clear ranking of all the vocational classes, as the head-circumference ordering conceals this fact, would enable the use of rank correlation coefficients, which may bear a significant impact on the conclusion of the hypothesis. Failure to list Hooton’s prestige ranking effectively puts this analysis at a loss to find out this vital information. The additional information also conspicuously omits the standard deviation values of the vocations, which would have been critical in the generation of a new ANOVA table, which would have led to a new F statistic and quite possibly a new conclusion on the hypothesis. Finally, only summary statistics is available for analysis from the very onset, which limits our capacity to gain comprehensive insight into the nature of the raw data on the study. For instance, it would be impossible to recalculate values such as standard deviation, and the sample means if that were necessary. Conclusion Hooton’s data suggests a correlation between head size and intelligence. This report seeks to verify this assertion. In the analysis section tests, the null hypothesis that there is no difference in the means of the individuals of various intelligence levels against the alternative hypothesis that head-circumference affects a person’s intelligence. The analysis of initial data on relationship between head-circumference and intelligence backs Broca’s argument that the brain size and intelligence have a strong correlation. Even after correcting the ANOVA analysis error resulting from use of the standard error terms as the standard deviation, the hypothesis that the means of the head sizes are equal still does not hold. However, the F statistic drops significantly from a figure of 1,998 to a mere 11, which is markedly close to our hypothesis that head-circumference does not affect a person’s intelligence. Tukey’s test to determine the source of variation in the sample means gave the conclusion that the sample means were unequal, which was a further affirmation that the prior conclusion was right, and that difference in intelligence had a direct correlation to the head-circumference of the person. The analysis ends on a suspenseful note, especially in the light of the new information, which might have completely overturned the initial conclusions on the test hypothesis. The additional information is incomplete and further information is necessary to merge it into the ANOVA analysis with the prior data. In addition, the initial data is too summarized for running some statistical tests to prove the validity of the conclusion (Cohen, 2002). Therefore, the analysis of the data for the alleged claim is inconclusive, although as far as the given data goes, a direct positive correlation exists between head-circumference and intelligence. Generally, it appears the study and selective disclosure of the data had the ulterior agenda to lead any analysis to this conclusion, which confirms Broca’s theory that a person’s intelligence depends on the head-circumference. Bibliography Akritas, M, G and Papadatos, N, 2004, ‘Heteroscedastic One-way ANOVA and lack of fit tests’, Journal of the American Statistical Association, viewed 20 March 2012 < http://pubs.amstat.org/action/doSearch> Brink, David, 2010, Essentials of Statistics, Ventus Publishing ApS. Cohen, H, Barry, 2002, Calculating a Factorial ANOVA from means and standard Deviations, New York University, viewed 20 March 2012 < http://www.psych.nyu.edu/cohen/Calc_ANOVA.pdf> Gould, J, Stephen, 1981, The Mismeasure of Man, Norton and Company, New York. University of Texas, 2005, ‘SPSS’, University of Texas, Austin, viewed 20 March 2012 < http://ssc.utexas.edu/software/faqs/spss> StatPages, 2012, Analysis of Variance from Summary Data, statpages.com, viewed 20 March 2012 from < http://statpages.org/anova1sm.html> Sturm-Beiss, Rachel, 2005, ‘A visualization tool for one-way and two-way analysis of variance’, Journal of Statistics Education, viewed 20 March 2012 < http://www.amstat.org/publications/jse/v13n1/sturm-beiss.html> Appendices/Supplementary Section SPSS Output for Tukey’s Test Dependent Variable: SD Tukey HSD (I) MEAN (J) MEAN Mean Difference (I-J) Std. Error Sig. 95% Confidence Interval Lower Bound Upper Bound 560.70 562.70 -3.7000(*) .00000 .000 -3.7000 -3.7000 562.90 -3.6000(*) .00000 .000 -3.6000 -3.6000 564.10 -4.9000(*) .00000 .000 -4.9000 -4.9000 565.70 -3.5000(*) .00000 .000 -3.5000 -3.5000 566.20 -3.8000(*) .00000 .000 -3.8000 -3.8000 566.50 -4.1000(*) .00000 .000 -4.1000 -4.1000 569.90 -1.9000(*) .00000 .000 -1.9000 -1.9000 562.70 560.70 3.7000(*) .00000 .000 3.7000 3.7000 562.90 .1000(*) .00000 .000 .1000 .1000 564.10 -1.2000(*) .00000 .000 -1.2000 -1.2000 565.70 .2000(*) .00000 .000 .2000 .2000 566.20 -.1000(*) .00000 .000 -.1000 -.1000 566.50 -.4000(*) .00000 .000 -.4000 -.4000 569.90 1.8000(*) .00000 .000 1.8000 1.8000 562.90 560.70 3.6000(*) .00000 .000 3.6000 3.6000 562.70 -.1000(*) .00000 .000 -.1000 -.1000 564.10 -1.3000(*) .00000 .000 -1.3000 -1.3000 565.70 .1000(*) .00000 .000 .1000 .1000 566.20 -.2000(*) .00000 .000 -.2000 -.2000 566.50 -.5000(*) .00000 .000 -.5000 -.5000 569.90 1.7000(*) .00000 .000 1.7000 1.7000 564.10 560.70 4.9000(*) .00000 .000 4.9000 4.9000 562.70 1.2000(*) .00000 .000 1.2000 1.2000 562.90 1.3000(*) .00000 .000 1.3000 1.3000 565.70 1.4000(*) .00000 .000 1.4000 1.4000 566.20 1.1000(*) .00000 .000 1.1000 1.1000 566.50 .8000(*) .00000 .000 .8000 .8000 569.90 3.0000(*) .00000 .000 3.0000 3.0000 565.70 560.70 3.5000(*) .00000 .000 3.5000 3.5000 562.70 -.2000(*) .00000 .000 -.2000 -.2000 562.90 -.1000(*) .00000 .000 -.1000 -.1000 564.10 -1.4000(*) .00000 .000 -1.4000 -1.4000 566.20 -.3000(*) .00000 .000 -.3000 -.3000 566.50 -.6000(*) .00000 .000 -.6000 -.6000 569.90 1.6000(*) .00000 .000 1.6000 1.6000 566.20 560.70 3.8000(*) .00000 .000 3.8000 3.8000 562.70 .1000(*) .00000 .000 .1000 .1000 562.90 .2000(*) .00000 .000 .2000 .2000 564.10 -1.1000(*) .00000 .000 -1.1000 -1.1000 565.70 .3000(*) .00000 .000 .3000 .3000 566.50 -.3000(*) .00000 .000 -.3000 -.3000 569.90 1.9000(*) .00000 .000 1.9000 1.9000 566.50 560.70 4.1000(*) .00000 .000 4.1000 4.1000 562.70 .4000(*) .00000 .000 .4000 .4000 562.90 .5000(*) .00000 .000 .5000 .5000 564.10 -.8000(*) .00000 .000 -.8000 -.8000 565.70 .6000(*) .00000 .000 .6000 .6000 566.20 .3000(*) .00000 .000 .3000 .3000 569.90 2.2000(*) .00000 .000 2.2000 2.2000 569.90 560.70 1.9000(*) .00000 .000 1.9000 1.9000 562.70 -1.8000(*) .00000 .000 -1.8000 -1.8000 562.90 -1.7000(*) .00000 .000 -1.7000 -1.7000 564.10 -3.0000(*) .00000 .000 -3.0000 -3.0000 565.70 -1.6000(*) .00000 .000 -1.6000 -1.6000 566.20 -1.9000(*) .00000 .000 -1.9000 -1.9000 566.50 -2.2000(*) .00000 .000 -2.2000 -2.2000 * The mean difference is significant at the .05 level. 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