> s <- c(1.9,2.5,3.2,3.8,4.7,5.5, 5.9, 7.2)
> c <- c(22,33,30,42,38,49,42,55)
> plot(s,c)
> df <- data.frame(s,c)
> df
    s  c
1 1.9 22
2 2.5 33
3 3.2 30
4 3.8 42
5 4.7 38
6 5.5 49
7 5.9 42
8 7.2 55
> plot(df)

\begin{eqnarray*} b & = & \frac{\Sigma{(x-\overline(x))(y-\overline(y))}}{\Sigma{(x-\overline{x})^2}} \\ & = & \frac{\text{SP}}{\text{SS}_{x}} \\ a & = & \overline{y} - b \overline{x} \end{eqnarray*}

> mod <- lm(c~s, data=df)
> summary(mod)

Call:
lm(formula = c ~ s, data = df)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.2131 -3.0740 -0.9777  3.9237  5.9933 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)  15.7283     4.4372   3.545  0.01215 * 
s             5.3364     0.9527   5.601  0.00138 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.571 on 6 degrees of freedom
Multiple R-squared:  0.8395,	Adjusted R-squared:  0.8127 
F-statistic: 31.37 on 1 and 6 DF,  p-value: 0.001379

Textbook says:

\begin{align} r = & b * \frac {sd(x)}{sd(y)} \\ = & b * \sqrt{\frac {SS_X} {SS_Y}} \;\;\; \because \left(b = \frac {SP}{SS_X} \right) \nonumber \\ = & \frac {SP}{SS_X} * \frac {\sqrt{SS_X}} {\sqrt{SS_Y}} \nonumber \\ = & \frac {SP}{\sqrt{SS_X * SS_Y}} \nonumber \\ = & \frac {COV}{SD(X) SD(Y)} \end{align}

Regression page says:
\begin{eqnarray} R^2 & = & \frac{\text{SS}_{reg}}{\text{SS}_{total}} \\ & = & \frac{ \text{SS}_{total} - \text{SS}_{res}} {\text{SS}_{total}} \nonumber \\ & = & 1 - \frac{\text{SS}_{res}} {\text{SS}_{total}} \\ \end{eqnarray}

\begin{eqnarray*} & \text{Note that (1), (2) are the same.} \nonumber \\ & \text{Therefore,} \nonumber \\ & r = \sqrt{R^2} \\ \end{eqnarray*}

We, from correlation wiki page, also addressed that r (correlation coefficient value) can be obtained through
\begin{eqnarray*} r & = & \frac {Cov(x,y)} {sd(x)*sd(y)} \\ \end{eqnarray*}
see pearson_s_r

아래는 위의 세 가지 방법으로 R에서 r 값을 구해본 것이다.

# check the coefficient from the lm analysis
b <- summary(mod)$coefficients[2,1]
b
df
sd.x <- sd(df$s)
sd.y <- sd(df$c)

# 첫 번째
r.value1 <- b * (sd.x/sd.y)
r.value1

# 두 번째
# rsquared 
r.value2 <- sqrt(summary(mod)$r.squared)
r.value2

# or 
cov.sc <- cov(df$s, df$c)
r.value3 <- cov.sc/(sd.x*sd.y)

r.value3

아래는 위의 아웃풋

> # check the coefficient from the lm analysis
> b <- summary(mod)$coefficients[2,1]
> b
[1] 5.336411
> df
    s  c
1 1.9 22
2 2.5 33
3 3.2 30
4 3.8 42
5 4.7 38
6 5.5 49
7 5.9 42
8 7.2 55
> sd.x <- sd(df$s)
> sd.y <- sd(df$c)
> 
> # 첫 번째
> r.value1 <- b * (sd.x/sd.y)
> r.value1
[1] 0.9162191
> 
> # 두 번째
> # rsquared 
> r.value2 <- sqrt(summary(mod)$r.squared)
> r.value2
[1] 0.9162191
> 
> # or 
> cov.sc <- cov(df$s, df$c)
> r.value3 <- cov.sc/(sd.x*sd.y)
> 
> r.value3
[1] 0.9162191
>

re <- c(4, 4.5, 5, 5.5, 6, 6.5, 7)
we <- c(12, 10, 8, 9.5, 8, 9, 6)

mean.re <- mean(re)
mean.we <- mean(we)
sd.re <- sd(re)
sd.we <- sd(we)
var.re <- var(re)
var.we <- var(we)
ss.re <- sum((re-mean.re)^2)
ss.we <- sum((we-mean.we)^2)
sp <- sum((re-mean.re)*(we-mean.we))
df <- 7-1
cov.rewe <- sp/df
cov.rewe2 <- cov(re,we)

mean.re
mean.we
sd.re
sd.we
var.re
var.we
ss.re
ss.we
sp
df
cov.rewe
cov.rewe2

# r, b, r square 
r.val <- sp / (sqrt(ss.re*ss.we))
r.val
b <- sp / ss.re
b
a <- mean.we - (b * mean.re)
a
r.sq <- r.val^2
r.sq

cor(re,we)
m1 <- lm(we~re)
summary(m1)

########################
ss <- c(1.9, 2.5, 3.2, 3.8, 4.7, 5.5, 5.9, 7.2)
at <- c(22, 33, 30, 42, 38, 49, 42, 55)
mean.ss <- mean(ss)
mean.at <- mean(at)
ss.ss <- sum((ss-mean.sa)^2)
ss.at <- sum((at-mean.at)^2)
df <- 8-1
var.ss <- ss.ss/df
var.at <- ss.at/df
sd.ss <- sqrt(var.ss)
sd.at <- sqrt(var.at)
sp.ssat <- sum((ss-mean.ss)*(at-mean.at))
cov.ssat <- sp.ssat/df