#set document(title: "1.11 Summation Notation", 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")) == 1.11#h(0.6em)Summation Notation #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Prerequisites] None #linebreak() #linebreak() ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Learning Objectives] + Use summation notation to express the sum of all numbers + Use summation notation to express the sum of a subset of numbers + Use summation notation to express the sum of squares ] Many statistical formulas involve summing numbers. Fortunately there is a convenient notation for expressing summation. This section covers the basics of this #strong[summation notation]. Let's say we have a variable X that represents the weights (in grams) of 4 grapes. The data are shown in Table 1. Table 1. Weights of 4 grapes. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Grape], [X]), [1 #linebreak() 2 #linebreak() 3 #linebreak() 4], [4.6 #linebreak() 5.1 #linebreak() 4.9 #linebreak() 4.4], )) We label Grape 1's weight X#sub[1], Grape 2's weight X#sub[2], etc. The following formula means to sum up the weights of the four grapes: #math.equation(block: false, alt: "∑ sub i equals 1 to the power 4 X sub i")[$∑_(i = 1)^(4) X_(i)$] The Greek letter capital sigma (Σ) indicates summation. The "i = 1" at the bottom indicates that the summation is to start with X#sub[1] and the 4 at the top indicates that the summation will end with X#sub[4]. The "X#sub[i]" indicates that X is the variable to be summed as i goes from 1 to 4. Therefore, #math.equation(block: false, alt: "∑ sub i equals 1 to the power 4 X sub i")[$∑_(i = 1)^(4) X_(i)$] = X#sub[1] + X#sub[2] + X#sub[3] + X#sub[4] = 4.6 + 5.1 + 4.9 + 4.4 = 19.0. The symbol #math.equation(block: false, alt: "∑ sub i equals 1 cubed X sub i")[$∑_(i = 1)^(3) X_(i)$] indicates that only the first 3 scores are to be summed. The index variable i goes from 1 to 3. #linebreak() When all the scores of a variable (such as X) are to be summed, it is often convenient to use the following abbreviated notation: #math.equation(block: true, alt: "∑ X")[$∑ X$] Thus, when no values of i are shown, it means to sum all the values of X. Many formulas involve squaring numbers before they are summed. This is indicated as ΣX² = 4.6#super[2] + 5.1#super[2] + 4.9#super[2] + 4.4#super[2] = 21.16 + 26.01 + 24.01 + 19.36 = 90.54. Notice that: #math.equation(block: true, alt: "open parenthesis ∑ X close parenthesis squared not equal to ∑ X squared")[$attach(( ∑ X ), t: 2) ≠ ∑ X^(2)$] because the expression on the left means to sum up all the values of X and then square the sum (19² = 361), whereas the expression on the right means to square the numbers and then sum the squares (90.54, as shown). Some formulas involve the sum of cross products. Table 2 shows the data for variables X and Y. The cross products (XY) are shown in the third column. The sum of the cross products is 3 + 4 + 21 = 28. Table 2. Cross Products. #figure(table( columns: 3, align: left, inset: 6pt, table.header([X], [Y], [XY]), [1 #linebreak() 2 #linebreak() 3], [3 #linebreak() 2 #linebreak() 7], [3 #linebreak() 4 #linebreak() 21], )) In summation notation, this is written as: ΣXY = 28.