{"id":1820,"date":"2020-10-31T11:47:09","date_gmt":"2020-10-31T15:47:09","guid":{"rendered":"http:\/\/josmfs.net\/?p=1820"},"modified":"2021-06-12T10:12:34","modified_gmt":"2021-06-12T14:12:34","slug":"the-percentage-trap","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2020\/10\/31\/the-percentage-trap\/","title":{"rendered":"The Percentage Trap"},"content":{"rendered":"

\"\"Twitter comments to the recently released GDP growth numbers for the Third Quarter reminded me of an old trap regarding percentages.\u00a0 The financial and technical articles were accurate, but the comments by Twitter users often reflected the pitfall.<\/p>\n

After a Second Quarter annualized US GDP fall of about 33%, the Third Quarter showed<\/a> a gain of about 33%.\u00a0 So some commenters thought the gain canceled the previous loss.<\/p>\n

<\/p>\n

But of course a Q2 loss of 33% or 1\/3, means the GDP is 2\/3 of what it was.\u00a0 So a 33% or 1\/3 gain means the new level will be 4\/3 of the previous 2\/3 level or 8\/9 \u2248 90% of the original level.\u00a0 So the gain does not<\/em> cancel the loss.\u00a0 Kevin Drum\u2019s subsequent diagram<\/a> makes that clear:<\/p>\n

\"\"<\/p>\n

To make the situation even clearer, suppose there were a 90% loss in Q2 leaving 10% of the original GDP.\u00a0 Then a 90% gain in Q3 would be 1.9 x 0.1 = .19 or 19% of the original GDP, reflecting a net loss of 81% for the two quarters.<\/p>\n

(Update 11\/5\/2020)<\/strong> Actually, this probably can be seen more clearly as follows.\u00a0 Let x<\/em> be the fraction of change in the two quarters.\u00a0 That is, x<\/em> is the fraction of loss in the first quarter and then x<\/em> is also the fraction of gain in the second.\u00a0\u00a0 Therefore, the total change over the two quarters will be (1 + x<\/em>)(1 \u2013 x<\/em>) = 1 \u2013 x<\/em>2<\/sup>.\u00a0\u00a0 In our first example x<\/em> = 1\/3, so the resulting change was 1 \u2013 1\/9 = 8\/9 \u2248 0.9.\u00a0 In the second example, x<\/em>\u00a0=\u00a0.9, so the resulting change was 1 \u2013 .81 = .19.\u00a0 Clearly, these changes will never restore the original amount.<\/p>\n","protected":false},"excerpt":{"rendered":"

Twitter comments to the recently released GDP growth numbers for the Third Quarter reminded me of an old trap regarding percentages.\u00a0 The financial and technical articles were accurate, but the comments by Twitter users often reflected the pitfall. After a Second Quarter annualized US GDP fall of about 33%, the Third Quarter showed a gain […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[209,188],"class_list":["post-1820","post","type-post","status-publish","format-standard","hentry","category-curiosities-and-questions","tag-innumeracy","tag-kevin-drum"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1820","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=1820"}],"version-history":[{"count":3,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1820\/revisions"}],"predecessor-version":[{"id":2187,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1820\/revisions\/2187"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=1820"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=1820"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=1820"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}