• Identify relations between variables
• Helps predict values
• Easy to interpret
• Low computational requirements

• Describes the line with the least predictive error
• Predictive error evaluated as least sum of squares
• “Squares” are the squared value of each data point’s displacement from the prediction line
• Generally the best fit line will run through the middle of the data

## `geom_smooth()` using formula = 'y ~ x'

• Find the means of both independent and dependent variables
• Find the difference between each point and the means
• Square these differences
• Plug into this equation: \(m = \sum_{}^{}(x - \bar{x}) * (y - \bar{y}) / (\sum_{}^{}(x-\bar{x})^2)\)
• Plug b into \(\bar{y} = m\bar{x}+b\) to find b

  Girth Volume     xbar     ybar xDifferences yDifferences xDiffSquared
1   8.3   10.3 13.24839 30.17097     4.948387     19.87097    24.486535
2   8.6   10.3 13.24839 30.17097     4.648387     19.87097    21.607503
3   8.8   10.2 13.24839 30.17097     4.448387     19.97097    19.788148
4  10.5   16.4 13.24839 30.17097     2.748387     13.77097     7.553632
5  10.7   18.8 13.24839 30.17097     2.548387     11.37097     6.494277
6  10.8   19.7 13.24839 30.17097     2.448387     10.47097     5.994599
   Product
1 98.32924
2 92.36795
3 88.83860
4 37.84795
5 28.97763
6 25.63698

We get a slope of 5.1

• In R it is very simple
• You can use code instead of calculating it all by hand
• For example:

model <- lm(Volume ~ Girth, data = trees)
print(model)

Call:
lm(formula = Volume ~ Girth, data = trees)

Coefficients:
(Intercept)        Girth  
    -36.943        5.066  

• It’s that easy!

Plug a value of your choosing into \(y = 5.066*x - 36.943\) If we have a tree with Girth of 14.0, we would expect the Volume to be about \(Volume = (5.066*14)-36.943\) That comes out to 33.981, which is similar to the number in our actual results (34.5)

h <- ggplot(data = trees, aes(x = Girth, y = Volume)) + geom_point()
h + geom_smooth(method='lm')

`geom_smooth()` using formula = 'y ~ x'

• Use Hooke’s Law and Best fit line to estimate a spring constant
• Experimentally estimate an object’s friction coefficient
• You can even use the log of a variable if the true relationship is exponential

Now you can do this yourself!