{"id":232819,"date":"2025-06-12T08:30:50","date_gmt":"2025-06-12T08:30:50","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=232819"},"modified":"2025-06-12T08:30:52","modified_gmt":"2025-06-12T08:30:52","slug":"test-scores-consider-the-following-table-comparing-the-grade-averages-and-mathematics-sat-scores-of-high-school-students-in-1988-and-998-students-1988-1998-sat-score-1988-1998-grade-average-a-a-a","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/12\/test-scores-consider-the-following-table-comparing-the-grade-averages-and-mathematics-sat-scores-of-high-school-students-in-1988-and-998-students-1988-1998-sat-score-1988-1998-grade-average-a-a-a\/","title":{"rendered":"Test Scores Consider the following table comparing the grade averages and mathematics SAT scores of high school students in 1988 and [998, % Students 1988 1998 SAT Score 1988 1998 Grade Average A+ A A B C Overall average"},"content":{"rendered":"\n

Test Scores Consider the following table comparing the grade averages and mathematics SAT scores of high school students in 1988 and [998, % Students 1988 1998 SAT Score 1988 1998 Grade Average A+ A A B C Overall average Change 632 586 556 490 431 504 629 582 554 487 428 514 11 13 53 19 15 16 48 14 -3 +10 Source: Cited in Chance, Vol: 12, No. 2,1999, from data in New York Times_ September 2, 1999.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/strong>1<\/a><\/sup><\/p>\n\n\n\n\n\n

Correct Answer<\/strong><\/p>\n\n\n\n

The data reveals a statistical phenomenon known as Simpson’s Paradox. While the average mathematics SAT score for students within every specific grade average category (A+, A, A-, B, C) declined from 1988 to 1998, the overall average SAT score for all students increased by 10 points.<\/p>\n\n\n\n

This occurred because the distribution of students across the grade categories changed significantly. A much larger percentage of students achieved higher grades in 1998 than in 1988. This shift in weighting toward the higher-performing groups was substantial enough to raise the overall average, even though performance within each individual group slightly decreased.<\/p>\n\n\n\n

Explanation<\/h3>\n\n\n\n

The table presents a fascinating statistical puzzle: how can the overall average score increase when the average score for every single subgroup decreases? This counterintuitive result is a classic example of Simpson’s Paradox<\/strong>, which occurs when a trend that appears in different groups of data disappears or reverses when the groups are combined.<\/p>\n\n\n\n

The key to understanding this paradox lies in the changing distribution of students, as shown in the “% Students” columns. The overall SAT average is a weighted average<\/em>. The score of each grade group is weighted by the percentage of students who earned that grade.<\/p>\n\n\n\n

    \n
  1. Shift in Student Distribution:<\/strong>\u00a0Between 1988 and 1998, there was significant “grade inflation.” The percentage of students earning grades of A-, A, or A+ increased from 28% (4+11+13) to 38% (7+15+16). Consequently, the percentage of students earning B’s and C’s decreased.<\/li>\n\n\n\n
  2. The Power of Weighting:<\/strong>\u00a0Students with higher grades (like A+) have much higher SAT scores than students with lower grades (like C). In 1998, a larger proportion of the student population was in these high-scoring groups. Even though an “A” student in 1998 scored slightly lower than an “A” student in 1988 (582 vs. 586), the fact that there were\u00a0more<\/em>\u00a0“A” students in the 1998 cohort gave their high scores more weight in the overall calculation.<\/li>\n<\/ol>\n\n\n\n

    In essence, the increase in the overall average SAT score is not due to students becoming better prepared within their respective achievement levels. Instead, it is a compositional effect. The overall student body in 1998 was composed of a greater proportion of high-achieving (and thus higher-scoring) students, and this structural shift was powerful enough to pull the combined average up, masking the slight decline in scores within each fixed grade category.thumb_upthumb_down<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Test Scores Consider the following table comparing the grade averages and mathematics SAT scores of high school students in 1988 and [998, % Students 1988 1998 SAT Score 1988 1998 Grade Average A+ A A B C Overall average Change 632 586 556 490 431 504 629 582 554 487 428 514 11 13 53 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-232819","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232819","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=232819"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232819\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=232819"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=232819"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=232819"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}