Without knowing the true slope there is no unique way of determining the error of the slope. So, all you can do is to select a method to determine the slope and then calculating the associated uncertainty. E.g. using the least square fit,
the uncertainty of the slope estimate $\hat \beta$ is usually taken to be the standard deviation
$$
Sd[\hat\beta_j] = \hat\sigma_\epsilon \sqrt{[(X^T X)^{-1}]_{jj}}
$$
where $X$ is the so called design matrix. I use the index $jj$ to indicate that we only take the diagonal elements. The point estimate of the "random error" $\hat\sigma_\epsilon$ is often taken to be
$\sqrt{\sum_{i=1}^N (y - y_{fit})^2/(N-p)}$, where $N$ is the number of data points and $p$ is the number of factors (including the intercept).
Often, I find it helpful to have an example to check my calculations. Therefore, here an example:
## Sample data:
x = 1:10
y = c(1.77954908388273, 2.44621066683337, -2.2018205040772, -0.764061711605929, -2.72481609322593, -3.51410342713565, -7.86973095833333, -8.78328718047611, -11.5217536084918, -6.80619245104001)
df = data.frame(x,y)
which look like this
library(MASS) # for inverse ginv()
## design matrix
X = c(rep(1, N), x)
dim(X) = c(N,2) # reshape the design matrix, so that its dimension is (N,2)
## Fit data (formulas not included above):
betaHat = ginv(t(X) %*% X) %*% t(X) %*% y
yFit = X %*% betaHat
## Estimate of the uncertainty:
res = y - yFit # residuals
sigma = sqrt( sum(res^2) / (N-2) ) # use p=2, because we fit slope and intercept
betaSD = sigma * sqrt( diag(ginv(t(X) %*% X)) )
> Coefficients:
> Estimate Std. Error t value Pr(>|t|)
> (Intercept) 3.6727 1.3757 2.670 0.028372 *
> beta -1.3943 0.2217 -6.289 0.000236 ***
Looking at your data, there are two regions where the data does not appear to be linear. This is at approx $x=15$ and $x = 115$. Thus, you should decide whether or not these regions are important. If they are not important you should consider omitting these regions, so that your fit is improved. Note that the data points at the "$x$-edge" carry more leverage. Hence, they are particularly dangerous to distort your fit.
