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Linear regressionStatistics and probabilityCollegeabout 1:18

Find the least-squares regression line for x = 0, 1, 2, 3, 4 and y = 1, 3, 7, 9, 11: animated lesson

A narrated, step-by-step linear regression video lesson with captions. Final answer: ŷ = 1 + (13/5)x; r ≈ 0.991241, r² ≈ 0.982558.

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 = 0, 1, 2, 3, 4 and y = 1, 3, 7, 9, 11.
  2. 2. Steps, part 1
    1. xˉ=105=2,yˉ=315=315≈6.2\bar x = \frac{10}{5} = 2,\quad \bar y = \frac{31}{5} = \frac{31}{5} \approx 6.2
      Means of x and y
    2. Sxx=∑(x−xˉ)2=10,Syy=3445≈68.8,Sxy=∑(x−xˉ)(y−yˉ)=26S_{xx} = \sum (x-\bar x)^2 = 10,\quad S_{yy} = \frac{344}{5} \approx 68.8,\quad S_{xy} = \sum (x-\bar x)(y-\bar y) = 26
      Sums of squares and cross-products about the means
    3. b=SxySxx=135≈2.6b = \frac{S_{xy}}{S_{xx}} = \frac{13}{5} \approx 2.6
      Slope
    4. a=yˉ−bxˉ=1a = \bar y - b\bar x = 1
      Intercept: the line passes through the point of means
  3. 3. Steps, part 2
    1. y^=1+135x\hat y = 1 + \frac{13}{5}x
      Regression line
    2. r=SxySxxSyy≈0.991241,r2≈0.982558r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} \approx 0.991241,\quad r^2 \approx 0.982558
      Correlation and coefficient of determination
  4. 4. The answer
    y^=1+135x\hat y = 1 + \frac{13}{5}x

    b = 13/5 ≈ 2.6, a = 1

  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 0, 1, 2, 3, 4 and y equals 1, 3, 7, 9, 11. 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 98.26 percent of the variation in y is explained by the line.

The answer. So the answer is ŷ equals 1 plus (13 over 5)x; r approximately 0.991241, r squared approximately 0.982558. 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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