#set document(title: "4.2 Values of the Pearson Correlation", author: "OpenStax") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 4.2#h(0.6em)Values of the Pearson Correlation #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Prerequisites] Introduction to Bivariate Data #linebreak() #linebreak() ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Learning Objectives] + Describe what Pearson's correlation measures + Give the symbols for Pearson's correlation in the sample and in the population + State the possible range for Pearson's correlation + Identify a perfect linear relationship ] The Pearson product-moment correlation coefficient is a measure of the strength of the linear relationship between two variables. It is referred to as Pearson's correlation or simply as the correlation coefficient. If the relationship between the variables is not linear, then the correlation coefficient does not adequately represent the strength of the relationship between the variables. The symbol for Pearson's correlation is "ρ" when it is measured in the population and "r" when it is measured in a sample. Because we will be dealing almost exclusively with samples, we will use r to represent Pearson's correlation unless otherwise noted. Pearson's r can range from -1 to 1. An r of -1 indicates a perfect negative linear relationship between variables, an r of 0 indicates no linear relationship between variables, and an r of 1 indicates a perfect positive linear relationship between variables. Figure 1 shows a scatter plot for which r = 1. #figure(figph[Scatterplot of y against x, both running from about -3 to 3 on x, in which every point lies exactly on one upward-sloping straight line — a perfect positive linear relationship, r = 1.], alt: "Scatterplot of y against x, both running from about -3 to 3 on x, in which every point lies exactly on one upward-sloping straight line — a perfect positive linear relationship, r = 1.", caption: [Figure 1. A perfect positive linear relationship, r = 1.]) #figure(figph[Scatterplot of y against x with every point lying exactly on one downward-sloping straight line, running from about (-2.9, 7) to (2.4, -5) — a perfect negative linear relationship, r = -1.], alt: "Scatterplot of y against x with every point lying exactly on one downward-sloping straight line, running from about (-2.9, 7) to (2.4, -5) — a perfect negative linear relationship, r = -1.", caption: [Figure 2. A perfect negative linear relationship, r = -1.]) #figure(figph[Scatterplot of y against x showing a formless cloud: for every x from -3 to 3 the y values scatter over roughly the same range of -3 to 4, with no upward or downward drift. Knowing x tells you nothing about y — r = 0.], alt: "Scatterplot of y against x showing a formless cloud: for every x from -3 to 3 the y values scatter over roughly the same range of -3 to 4, with no upward or downward drift. Knowing x tells you nothing about y — r = 0.", caption: [Figure 3. A scatter plot for which r = 0. Notice that there is no relationship between X and Y.]) #linebreak() With real data, you would not expect to get values of r of exactly -1, 0, or 1. The data for spousal ages shown in Figure 4 and described in the introductory section has an r of 0.97. #figure(figph[Scatterplot of Wife's Age (y, 30 to 85) against Husband's Age (x, 30 to 80) for about 280 couples. The points form a tight upward band running from about (32, 32) to about (78, 78): the older the husband, the older the wife. The relationship is strong, positive, and close to linear, with a handful of couples well off the band (for example about (53, 62) and (68, 75)).], alt: "Scatterplot of Wife's Age (y, 30 to 85) against Husband's Age (x, 30 to 80) for about 280 couples. The points form a tight upward band running from about (32, 32) to about (78, 78): the older the husband, the older the wife. The relationship is strong, positive, and close to linear, with a handful of couples well off the band (for example about (53, 62) and (68, 75)).", caption: [Figure 4. Scatter plot of spousal ages, r = 0.97.]) #figure(figph[Scatterplot of Arm Strength (y, 10 to 140) against Grip Strength (x, 20 to 200) for 147 subjects. The cloud rises to the right — stronger grip goes with stronger arm — but the points are scattered much more loosely about the trend than in the spousal-age plot, and one subject sits alone at the bottom left near (29, 19).], alt: "Scatterplot of Arm Strength (y, 10 to 140) against Grip Strength (x, 20 to 200) for 147 subjects. The cloud rises to the right — stronger grip goes with stronger arm — but the points are scattered much more loosely about the trend than in the spousal-age plot, and one subject sits alone at the bottom left near (29, 19).", caption: [Figure 5. Scatter plot of Grip Strength and Arm Strength, r = 0.63.]) The relationship between grip strength and arm strength depicted in Figure 5 (also described in the introductory section) is 0.63.