LearnChemE

.elementor-515 .elementor-element.elementor-element-4354ca8 > .elementor-container > .elementor-column > .elementor-widget-wrap{align-content:flex-start;align-items:flex-start;}.elementor-515 .elementor-element.elementor-element-4354ca8{transition:background 0.3s, border 0.3s, border-radius 0.3s, box-shadow 0.3s;margin-top:0px;margin-bottom:0px;padding:0px 0px 0px 0px;}.elementor-515 .elementor-element.elementor-element-4354ca8 > .elementor-background-overlay{transition:background 0.3s, border-radius 0.3s, opacity 0.3s;}.elementor-515 .elementor-element.elementor-element-18adb6d > .elementor-element-populated{transition:background 0.3s, border 0.3s, border-radius 0.3s, box-shadow 0.3s;margin:3px 0px 0px 0px;--e-column-margin-right:0px;--e-column-margin-left:0px;padding:0px 0px 0px 0px;}.elementor-515 .elementor-element.elementor-element-18adb6d > .elementor-element-populated > .elementor-background-overlay{transition:background 0.3s, border-radius 0.3s, opacity 0.3s;}.elementor-515 .elementor-element.elementor-element-79bb1c4{transition:background 0.3s, border 0.3s, border-radius 0.3s, box-shadow 0.3s;margin-top:0px;margin-bottom:0px;padding:0px 0px 0px 0px;}.elementor-515 .elementor-element.elementor-element-79bb1c4 > .elementor-background-overlay{transition:background 0.3s, border-radius 0.3s, opacity 0.3s;}.elementor-bc-flex-widget .elementor-515 .elementor-element.elementor-element-cf8bb55.elementor-column .elementor-widget-wrap{align-items:center;}.elementor-515 .elementor-element.elementor-element-cf8bb55.elementor-column.elementor-element[data-element_type="column"] > .elementor-widget-wrap.elementor-element-populated{align-content:center;align-items:center;}.elementor-515 .elementor-element.elementor-element-cf8bb55 > .elementor-widget-wrap > .elementor-widget:not(.elementor-widget__width-auto):not(.elementor-widget__width-initial):not(:last-child):not(.elementor-absolute){margin-bottom:0px;}.elementor-515 .elementor-element.elementor-element-cf8bb55 > .elementor-element-populated{margin:0px 0px 0px 10px;--e-column-margin-right:0px;--e-column-margin-left:10px;padding:0px 0px 0px 0px;}.elementor-515 .elementor-element.elementor-element-071cb4c .hfe-heading-text{color:var( --e-global-color-primary );}.elementor-515 .elementor-element.elementor-element-071cb4c .hfe-icon i{color:var( --e-global-color-primary );}.elementor-515 .elementor-element.elementor-element-071cb4c .hfe-icon svg{fill:var( --e-global-color-primary );}.elementor-bc-flex-widget .elementor-515 .elementor-element.elementor-element-5ff9494.elementor-column .elementor-widget-wrap{align-items:center;}.elementor-515 .elementor-element.elementor-element-5ff9494.elementor-column.elementor-element[data-element_type="column"] > .elementor-widget-wrap.elementor-element-populated{align-content:center;align-items:center;}.elementor-515 .elementor-element.elementor-element-5ff9494 > .elementor-element-populated{margin:0px 0px 0px 0px;--e-column-margin-right:0px;--e-column-margin-left:0px;padding:0px 0px 0px 0px;}.elementor-515 .elementor-element.elementor-element-99e6049 img{width:50px;max-width:50px;}.elementor-bc-flex-widget .elementor-515 .elementor-element.elementor-element-e3a08fe.elementor-column .elementor-widget-wrap{align-items:center;}.elementor-515 .elementor-element.elementor-element-e3a08fe.elementor-column.elementor-element[data-element_type="column"] > .elementor-widget-wrap.elementor-element-populated{align-content:center;align-items:center;}.elementor-515 .elementor-element.elementor-element-e3a08fe > .elementor-element-populated{margin:0px 0px 0px 0px;--e-column-margin-right:0px;--e-column-margin-left:0px;padding:0px 0px 0px 0px;}.elementor-515 .elementor-element.elementor-element-773020b{text-align:center;}.elementor-515 .elementor-element.elementor-element-773020b img{width:40px;max-width:40px;}.elementor-bc-flex-widget .elementor-515 .elementor-element.elementor-element-6bb885c.elementor-column .elementor-widget-wrap{align-items:center;}.elementor-515 .elementor-element.elementor-element-6bb885c.elementor-column.elementor-element[data-element_type="column"] > .elementor-widget-wrap.elementor-element-populated{align-content:center;align-items:center;}.elementor-515 .elementor-element.elementor-element-6bb885c > .elementor-element-populated{margin:0px 0px 0px 0px;--e-column-margin-right:0px;--e-column-margin-left:0px;padding:0px 0px 0px 0px;}.elementor-515 .elementor-element.elementor-element-544e1a8 img{width:100%;max-width:45px;}.elementor-bc-flex-widget .elementor-515 .elementor-element.elementor-element-8907359.elementor-column .elementor-widget-wrap{align-items:center;}.elementor-515 .elementor-element.elementor-element-8907359.elementor-column.elementor-element[data-element_type="column"] > .elementor-widget-wrap.elementor-element-populated{align-content:center;align-items:center;}.elementor-515 .elementor-element.elementor-element-8907359 > .elementor-element-populated{margin:0px 0px 0px 0px;--e-column-margin-right:0px;--e-column-margin-left:0px;padding:0px 0px 0px 0px;}.elementor-515 .elementor-element.elementor-element-ef0a165{text-align:center;}.elementor-515 .elementor-element.elementor-element-ef0a165 img{width:100%;max-width:55px;}.elementor-bc-flex-widget .elementor-515 .elementor-element.elementor-element-2acf5d2.elementor-column .elementor-widget-wrap{align-items:center;}.elementor-515 .elementor-element.elementor-element-2acf5d2.elementor-column.elementor-element[data-element_type="column"] > .elementor-widget-wrap.elementor-element-populated{align-content:center;align-items:center;}.elementor-515 .elementor-element.elementor-element-2acf5d2.elementor-column > .elementor-widget-wrap{justify-content:flex-end;}.elementor-515 .elementor-element.elementor-element-2acf5d2 > .elementor-element-populated{margin:0px 0px 0px 0px;--e-column-margin-right:0px;--e-column-margin-left:0px;padding:0px 0px 0px 08px;}.elementor-515 .elementor-element.elementor-element-5a21732{text-align:left;}.elementor-515 .elementor-element.elementor-element-5a21732 img{width:100%;max-width:350px;}.elementor-515 .elementor-element.elementor-element-54633e8 > .elementor-container{min-height:0px;}.elementor-515 .elementor-element.elementor-element-54633e8{transition:background 0.3s, border 0.3s, border-radius 0.3s, box-shadow 0.3s;margin-top:0px;margin-bottom:0px;padding:0px 0px 0px 0px;}.elementor-515 .elementor-element.elementor-element-54633e8 > .elementor-background-overlay{transition:background 0.3s, border-radius 0.3s, opacity 0.3s;}.elementor-515 .elementor-element.elementor-element-ff739e6 > .elementor-element-populated{margin:0px 0px 0px 0px;--e-column-margin-right:0px;--e-column-margin-left:0px;padding:0px 0px 0px 0px;}.elementor-515 .elementor-element.elementor-element-5b5f33e .hfe-nav-menu__toggle{margin:0 auto;}.elementor-515 .elementor-element.elementor-element-5b5f33e .menu-item a.hfe-menu-item{padding-left:12px;padding-right:12px;}.elementor-515 .elementor-element.elementor-element-5b5f33e .menu-item a.hfe-sub-menu-item{padding-left:calc( 12px + 20px );padding-right:12px;}.elementor-515 .elementor-element.elementor-element-5b5f33e .hfe-nav-menu__layout-vertical .menu-item ul ul a.hfe-sub-menu-item{padding-left:calc( 12px + 40px );padding-right:12px;}.elementor-515 .elementor-element.elementor-element-5b5f33e .hfe-nav-menu__layout-vertical .menu-item ul ul ul a.hfe-sub-menu-item{padding-left:calc( 12px + 60px );padding-right:12px;}.elementor-515 .elementor-element.elementor-element-5b5f33e .hfe-nav-menu__layout-vertical .menu-item ul ul ul ul a.hfe-sub-menu-item{padding-left:calc( 12px + 80px );padding-right:12px;}.elementor-515 .elementor-element.elementor-element-5b5f33e .menu-item a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e .menu-item a.hfe-sub-menu-item{padding-top:15px;padding-bottom:15px;}.elementor-515 .elementor-element.elementor-element-5b5f33e a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e a.hfe-sub-menu-item{font-size:1rem;}.elementor-515 .elementor-element.elementor-element-5b5f33e .menu-item a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e .sub-menu a.hfe-sub-menu-item{color:var( --e-global-color-primary );}.elementor-515 .elementor-element.elementor-element-5b5f33e .menu-item a.hfe-menu-item:hover, .elementor-515 .elementor-element.elementor-element-5b5f33e .sub-menu a.hfe-sub-menu-item:hover, .elementor-515 .elementor-element.elementor-element-5b5f33e .menu-item.current-menu-item a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e .menu-item a.hfe-menu-item.highlighted, .elementor-515 .elementor-element.elementor-element-5b5f33e .menu-item a.hfe-menu-item:focus{color:#CFB87C;}.elementor-515 .elementor-element.elementor-element-5b5f33e .menu-item.current-menu-item a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e .menu-item.current-menu-ancestor a.hfe-menu-item{color:#0377BD;}.elementor-515 .elementor-element.elementor-element-5b5f33e .sub-menu, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown-expandible, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown .menu-item a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown .menu-item a.hfe-sub-menu-item{background-color:#fff;} .elementor-515 .elementor-element.elementor-element-5b5f33e .sub-menu li a.hfe-sub-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown li a.hfe-sub-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown li a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown-expandible li a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown-expandible li a.hfe-sub-menu-item{font-size:0.9rem;}.elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-nav-menu__layout-horizontal .sub-menu, .elementor-515 .elementor-element.elementor-element-5b5f33e nav:not(.hfe-nav-menu__layout-horizontal) .sub-menu.sub-menu-open, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown .hfe-nav-menu, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown-expandible .hfe-nav-menu{border-style:solid;border-width:1px 1px 1px 1px;border-color:#C4C4C4;}.elementor-515 .elementor-element.elementor-element-5b5f33e .sub-menu{border-radius:2px 2px 2px 2px;}.elementor-515 .elementor-element.elementor-element-5b5f33e .sub-menu li.menu-item:first-child{border-top-left-radius:2px;border-top-right-radius:2px;overflow:hidden;}.elementor-515 .elementor-element.elementor-element-5b5f33e .sub-menu li.menu-item:last-child{border-bottom-right-radius:2px;border-bottom-left-radius:2px;overflow:hidden;}.elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown{border-radius:2px 2px 2px 2px;}.elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown li.menu-item:first-child{border-top-left-radius:2px;border-top-right-radius:2px;overflow:hidden;}.elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown li.menu-item:last-child{border-bottom-right-radius:2px;border-bottom-left-radius:2px;overflow:hidden;}.elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown-expandible{border-radius:2px 2px 2px 2px;}.elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown-expandible li.menu-item:first-child{border-top-left-radius:2px;border-top-right-radius:2px;overflow:hidden;}.elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown-expandible li.menu-item:last-child{border-bottom-right-radius:2px;border-bottom-left-radius:2px;overflow:hidden;}.elementor-515 .elementor-element.elementor-element-5b5f33e ul.sub-menu{width:262px;}.elementor-515 .elementor-element.elementor-element-5b5f33e .sub-menu li a.hfe-sub-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown li a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown-expandible li a.hfe-menu-item{padding-left:8px;padding-right:8px;}.elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown-expandible a.hfe-sub-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown li a.hfe-sub-menu-item{padding-left:calc( 8px + 20px );padding-right:8px;}.elementor-515 .elementor-element.elementor-element-5b5f33e .hfe-dropdown .menu-item ul ul a.hfe-sub-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e .hfe-dropdown-expandible .menu-item ul ul a.hfe-sub-menu-item{padding-left:calc( 8px + 40px );padding-right:8px;}.elementor-515 .elementor-element.elementor-element-5b5f33e .hfe-dropdown .menu-item ul ul ul a.hfe-sub-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e .hfe-dropdown-expandible .menu-item ul ul ul a.hfe-sub-menu-item{padding-left:calc( 8px + 60px );padding-right:8px;}.elementor-515 .elementor-element.elementor-element-5b5f33e .hfe-dropdown .menu-item ul ul ul ul a.hfe-sub-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e .hfe-dropdown-expandible .menu-item ul ul ul ul a.hfe-sub-menu-item{padding-left:calc( 8px + 80px );padding-right:8px;}.elementor-515 .elementor-element.elementor-element-5b5f33e .sub-menu a.hfe-sub-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown li a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown li a.hfe-sub-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown-expandible li a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown-expandible li a.hfe-sub-menu-item{padding-top:8px;padding-bottom:8px;}.elementor-515 .elementor-element.elementor-element-5b5f33e .sub-menu li.menu-item:not(:last-child), .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown li.menu-item:not(:last-child), .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown-expandible li.menu-item:not(:last-child){border-bottom-style:solid;border-bottom-color:#c4c4c4;border-bottom-width:1px;}.elementor-515 .elementor-element.elementor-element-6badb1b{margin-top:0px;margin-bottom:0px;padding:0px 0px 0px 0px;}.elementor-515 .elementor-element.elementor-element-954b0da > .elementor-widget-wrap > .elementor-widget:not(.elementor-widget__width-auto):not(.elementor-widget__width-initial):not(:last-child):not(.elementor-absolute){margin-bottom:0px;}.elementor-515 .elementor-element.elementor-element-954b0da > .elementor-element-populated{margin:0px 0px 0px 0px;--e-column-margin-right:0px;--e-column-margin-left:0px;padding:0px 0px 0px 0px;}.elementor-515 .elementor-element.elementor-element-15d435d > .elementor-widget-container{margin:0px 0px 0px 0px;padding:0px 0px 0px 0px;}@media(max-width:1024px){.elementor-515 .elementor-element.elementor-element-071cb4c .hfe-heading{text-align:left;}.elementor-515 .elementor-element.elementor-element-99e6049 img{max-width:45px;}.elementor-515 .elementor-element.elementor-element-773020b img{max-width:45px;}.elementor-515 .elementor-element.elementor-element-544e1a8 img{max-width:45px;}.elementor-515 .elementor-element.elementor-element-ef0a165 img{max-width:56px;}.elementor-bc-flex-widget .elementor-515 .elementor-element.elementor-element-2acf5d2.elementor-column .elementor-widget-wrap{align-items:center;}.elementor-515 .elementor-element.elementor-element-2acf5d2.elementor-column.elementor-element[data-element_type="column"] > .elementor-widget-wrap.elementor-element-populated{align-content:center;align-items:center;}.elementor-515 .elementor-element.elementor-element-54633e8 > .elementor-container{min-height:0px;}.elementor-515 .elementor-element.elementor-element-5b5f33e .menu-item a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e .menu-item a.hfe-sub-menu-item{padding-top:5px;padding-bottom:5px;}.elementor-515 .elementor-element.elementor-element-5b5f33e a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e a.hfe-sub-menu-item{font-size:0.95rem;}.elementor-515 .elementor-element.elementor-element-5b5f33e .sub-menu a.hfe-sub-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown li a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown li a.hfe-sub-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown-expandible li a.hfe-menu-item, .elementor-515 .elementor-element.elementor-element-5b5f33e nav.hfe-dropdown-expandible li a.hfe-sub-menu-item{padding-top:8px;padding-bottom:8px;}}@media(min-width:768px){.elementor-515 .elementor-element.elementor-element-cf8bb55{width:36%;}.elementor-515 .elementor-element.elementor-element-5ff9494{width:7%;}.elementor-515 .elementor-element.elementor-element-e3a08fe{width:7%;}.elementor-515 .elementor-element.elementor-element-6bb885c{width:7%;}.elementor-515 .elementor-element.elementor-element-8907359{width:7.33%;}.elementor-515 .elementor-element.elementor-element-2acf5d2{width:35%;}}@media(max-width:1024px) and (min-width:768px){.elementor-515 .elementor-element.elementor-element-cf8bb55{width:38%;}.elementor-515 .elementor-element.elementor-element-5ff9494{width:6%;}.elementor-515 .elementor-element.elementor-element-e3a08fe{width:7%;}.elementor-515 .elementor-element.elementor-element-6bb885c{width:7%;}.elementor-515 .elementor-element.elementor-element-8907359{width:9%;}.elementor-515 .elementor-element.elementor-element-2acf5d2{width:33%;}}@media(max-width:767px){.elementor-515 .elementor-element.elementor-element-cf8bb55{width:100%;}.elementor-515 .elementor-element.elementor-element-071cb4c .elementor-heading-title, .elementor-515 .elementor-element.elementor-element-071cb4c .hfe-heading a{font-size:2rem;}.elementor-515 .elementor-element.elementor-element-5ff9494{width:3%;}.elementor-515 .elementor-element.elementor-element-e3a08fe{width:3%;}.elementor-515 .elementor-element.elementor-element-6bb885c{width:3%;}.elementor-515 .elementor-element.elementor-element-8907359{width:3%;}.elementor-515 .elementor-element.elementor-element-2acf5d2{width:100%;}.elementor-515 .elementor-element.elementor-element-2acf5d2 > .elementor-element-populated{padding:5px 5px 5px 5px;}.elementor-515 .elementor-element.elementor-element-5a21732 img{max-width:237px;}.elementor-515 .elementor-element.elementor-element-54633e8 > .elementor-container{min-height:50px;}.elementor-515 .elementor-element.elementor-element-5b5f33e > .elementor-widget-container{margin:0px 0px 0px 0px;padding:0px 0px 0px 0px;}}
Visualizing R-Squared in Statistics
Description

In general, R2 is often referred to as the coefficient of determination and its verbal interpretation is that it is the fraction of variation explained by the model. This is illustrated in the case of simple linear regression. A random sample of size n is generated from a bivariate normal distribution with correlation parameter p, means 0, and variances 1. Graph 1 shows the data and the fitted regression line. Graph 2 shows the data and the fitted points. In graph 3, a rug is added on each of the y-axes. The axes on the left with the blue rug show the y values; the axes on the right with the red rug show the fitted y values. The plot label shows R2 = r2 = Sreg/Sy, where r is the correlation coefficient, Sy is the sum of the squared deviations of the y from their mean, and Sreg is the sum of the squared deviations of the \( \hat{ y } \) from their mean. In graph 3, the rugs provide a visualization of the spread of the \( y \) and \( \hat{ y } \). In statistics courses, it is often mentioned that R2 = r2 and that r2 is the fraction of variation explained by the model. Notice how the plot changes dramatically as the parameter p is changed. 

This simulation runs on desktop using the free Wolfram Player. Download the Wolfram Player here.
About

Author: Ian McLeod. Open content licensed under CC BY-NC-SA.

View the source code for this simulation