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Linear regressionStatistics and probabilityCollegeabout 1:24

Find the least-squares regression line for x = 1, 2, 3, 4, 5, 6 and y = 52, 58, 61, 67, 70, 78: animated lesson

A narrated, step-by-step linear regression video lesson with captions. Final answer: ŷ = 707/15 + (172/35)x; r ≈ 0.992161, r² ≈ 0.984383.

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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 (0.6 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
Find the least-squares regression line for x = 1, 2, 3, 4, 5, 6 and y = 52, 58, 61, 67, 70, 78.
2. Steps, part 1
1. $\bar x = \frac{21}{6} = \frac{7}{2} \approx 3.5,\quad \bar y = \frac{386}{6} = \frac{193}{3} \approx 64.3333$
  Means of x and y
2. $S_{xx} = \sum (x-\bar x)^2 = \frac{35}{2} \approx 17.5,\quad S_{yy} = \frac{1288}{3} \approx 429.333,\quad S_{xy} = \sum (x-\bar x)(y-\bar y) = 86$
  Sums of squares and cross-products about the means
3. $b = \frac{S_{xy}}{S_{xx}} = \frac{172}{35} \approx 4.91429$
  Slope
4. $a = \bar y - b\bar x = \frac{707}{15} \approx 47.1333$
  Intercept: the line passes through the point of means
3. Steps, part 2
1. $\hat y = \frac{707}{15} + \frac{172}{35}x$
  Regression line
2. $r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} \approx 0.992161,\quad r^2 \approx 0.984383$
  Correlation and coefficient of determination
4. The answer
$\hat y = \frac{707}{15} + \frac{172}{35}x$
b = 172/35 ≈ 4.91429, a = 707/15 ≈ 47.1333
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 1, 2, 3, 4, 5, 6 and y equals 52, 58, 61, 67, 70, 78. 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 98.44 percent of the variation in y is explained by the line.

The answer. So the answer is ŷ equals 707 over 15 plus (172 over 35)x; r approximately 0.992161, r squared approximately 0.984383. 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.

Linear regression: Find the least-squares regression line…