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Linear regressionStatistics and probabilityCollegeabout 1:15

Find the least-squares regression line for x = 2, 4, 6, 8 and y = 3, 7, 8, 12: animated lesson

A narrated, step-by-step linear regression video lesson with captions. Final answer: ŷ = 1/2 + (7/5)x; r ≈ 0.977802, r² ≈ 0.956098.

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    Find the least-squares regression line for x = 2, 4, 6, 8 and y = 3, 7, 8, 12.
  2. 2. Steps, part 1
    1. xˉ=204=5,yˉ=304=152≈7.5\bar x = \frac{20}{4} = 5,\quad \bar y = \frac{30}{4} = \frac{15}{2} \approx 7.5
      Means of x and y
    2. Sxx=∑(x−xˉ)2=20,Syy=41,Sxy=∑(x−xˉ)(y−yˉ)=28S_{xx} = \sum (x-\bar x)^2 = 20,\quad S_{yy} = 41,\quad S_{xy} = \sum (x-\bar x)(y-\bar y) = 28
      Sums of squares and cross-products about the means
    3. b=SxySxx=75≈1.4b = \frac{S_{xy}}{S_{xx}} = \frac{7}{5} \approx 1.4
      Slope
    4. a=yˉ−bxˉ=12≈0.5a = \bar y - b\bar x = \frac{1}{2} \approx 0.5
      Intercept: the line passes through the point of means
  3. 3. Steps, part 2
    1. y^=12+75x\hat y = \frac{1}{2} + \frac{7}{5}x
      Regression line
    2. r=SxySxxSyy≈0.977802,r2≈0.956098r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} \approx 0.977802,\quad r^2 \approx 0.956098
      Correlation and coefficient of determination
  4. 4. The answer
    y^=12+75x\hat y = \frac{1}{2} + \frac{7}{5}x

    b = 7/5 ≈ 1.4, a = 1/2 ≈ 0.5

  5. 5. Check
    • The residuals sum to 0, as they must for a least-squares line.

Transcript

The problem. Here is the problem: Find the least-squares regression line for x equals 2, 4, 6, 8 and y equals 3, 7, 8, 12. Fit the straight line ŷ equals a plus bx that minimises the sum of squared vertical distances to the data, and measure how well it fits with r and r squared.

Steps, part 1. Means of x and y. Sums of squares and cross-products about the means. Slope. Intercept: the line passes through the point of means.

Steps, part 2. Regression line. Correlation and coefficient of determination. Strong positive linear association; about 95.61 percent of the variation in y is explained by the line.

The answer. So the answer is ŷ equals 1 over 2 plus (7 over 5)x; r approximately 0.977802, r squared approximately 0.956098. The slope is the predicted change in y for each one-unit increase in x. Correlation measures the strength of the linear pattern, not whether x causes y.

Check. The residuals sum to 0, as they must for a least-squares line.

About linear regression

Fit a least-squares regression line and read its slope and intercept.

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