#set document(title: "7.3 Areas of Normal Distributions", 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")) == 7.3#h(0.6em)Areas of Normal Distributions #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Prerequisites] Distributions, Central Tendency, Variability, Introduction to Normal Distributions #linebreak() #linebreak() ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Learning Objectives] + State the proportion of a normal distribution within 1 and within 2 standard deviations of the mean + Use the calculator "Calculate Area for a given X" + Use the calculator "Calculate X for a given Area" ] Areas under portions of a normal distribution can be computed by using calculus. Since this is a non-mathematical treatment of statistics, we will rely on computer programs and tables to determine these areas. Figure 1 shows a normal distribution with a mean of 50 and a standard deviation of 10. The shaded area between 40 and 60 contains 68% of the distribution. #figure(figph[Normal curve with a mean of 50 and a standard deviation of 10 on an axis from 10 to 90. The region between 40 and 60 — one standard deviation either side of the mean — is shaded blue and contains 68% of the area.], alt: "Normal curve with a mean of 50 and a standard deviation of 10 on an axis from 10 to 90. The region between 40 and 60 — one standard deviation either side of the mean — is shaded blue and contains 68% of the area.", caption: [Figure 1. Normal distribution with a mean of 50 and standard deviation of 10. 68% of the area is within one standard deviation (10) of the mean (50).]) Figure 2 shows a normal distribution with a mean of 100 and a standard deviation of 20. As in Figure 1, 68% of the distribution is within one standard deviation of the mean. #figure(figph[Normal curve with a mean of 100 and a standard deviation of 20 on an axis from 20 to 180. The region between 80 and 120 is shaded blue: again exactly one standard deviation either side of the mean, and again 68% of the area, even though the numbers on the axis are quite different from Figure 1.], alt: "Normal curve with a mean of 100 and a standard deviation of 20 on an axis from 20 to 180. The region between 80 and 120 is shaded blue: again exactly one standard deviation either side of the mean, and again 68% of the area, even though the numbers on the axis are quite different from Figure 1.", caption: [Figure 2. Normal distribution with a mean of 100 and standard deviation of 20. 68% of the area is within one standard deviation (20) of the mean (100).]) The normal distributions shown in Figures 1 and 2 are specific examples of the general rule that 68% of the area of any normal distribution is within one standard deviation of the mean. Figure 3 shows a normal distribution with a mean of 75 and a standard deviation of 10. The shaded area contains 95% of the area and extends from 55.4 to 94.6. For all normal distributions, 95% of the area is within 1.96 standard deviations of the mean. For quick approximations, it is sometimes useful to round off and use 2 rather than 1.96 as the number of standard deviations you need to extend from the mean so as to include 95% of the area. #figure(figph[Normal curve with a mean of 75 and a standard deviation of 10 on an axis from 35 to 115. The shaded blue region runs from about 55.4 to 94.6 — 1.96 standard deviations either side of the mean — and contains 95% of the area.], alt: "Normal curve with a mean of 75 and a standard deviation of 10 on an axis from 35 to 115. The shaded blue region runs from about 55.4 to 94.6 — 1.96 standard deviations either side of the mean — and contains 95% of the area.", caption: [Figure 3. A normal distribution with a mean of 75 and a standard deviation of 10. 95% of the area is within 1.96 standard deviations of the mean.]) The #link("https://onlinestatbook.com/2/calculators/normal_dist.html")[normal calculator] can be used to calculate areas under the normal distribution. For example, you can use it to find the proportion of a normal distribution with a mean of 90 and a standard deviation of 12 that is above 110. Set the mean to 90 and the standard deviation to 12. Then enter "110" in the box to the right of the radio button "Above." At the bottom of the display you will see that the shaded area is 0.0478. See if you can use the calculator to find that the area between 115 and 120 is 0.0124. #figure(figph[Screenshot of the normal-area calculator. The plot shows a normal curve on an axis labelled 54 to 126, with the thin upper tail beyond about 110 shaded grey. The parameters read Mean 90, SD 12, with the Above radio button selected and 110 entered. The Results panel reads Area (probability) = 0.0478.], alt: "Screenshot of the normal-area calculator. The plot shows a normal curve on an axis labelled 54 to 126, with the thin upper tail beyond about 110 shaded grey. The parameters read Mean 90, SD 12, with the Above radio button selected and 110 entered. The Results panel reads Area (probability) = 0.0478.", caption: [Figure 4. Display from calculator showing the area above 110.]) Say you wanted to find the score corresponding to the 75th percentile of a normal distribution with a mean of 90 and a standard deviation of 12. Using the #link("https://onlinestatbook.com/2/calculators/inverse_normal_dist.html")[inverse normal calculator], you enter the parameters as shown in Figure 5 and find that the area below 98.09 is 0.75. #figure(figph[Screenshot of the inverse normal calculator. Area is set to 0.75, Mean to 90, SD to 12, and the answer appears in the Below field as 98.09; the plot shades everything left of about 98 on an axis from 54 to 126 — the 75th percentile of that distribution.], alt: "Screenshot of the inverse normal calculator. Area is set to 0.75, Mean to 90, SD to 12, and the answer appears in the Below field as 98.09; the plot shades everything left of about 98 on an axis from 54 to 126 — the 75th percentile of that distribution.", caption: [Figure 5. Display from normal calculator showing that the 75th percentile is 98.09.]) Normal and Inverse Normal Calculator