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I'm trying to find the rate of change and the error on that rate based on 7 measurements points and the assumption that the trend is linear. My calculations are below:

$$ \begin{array}{|c|c|c|c|} \hline \text{Position (mm)} & \text{Height (mm)} & \text{Slope} & \text{Slope}^2 \\ \hline 191 & 115 \\\ 256 & 110 & -0.077 & 0.006 \\\ 333 & 105 & -0.065 & 0.004 \\\ 391 & 100 & -0.086 & 0.007 \\\ 480 & 95 & -0.056 & 0.003 \\\ 528 & 90 & -0.104 & 0.011 \\\ 617 & 85 & -0.056 & 0.003 \\\hline \end{array} $$

$$ \begin{array}{|c|c|} \hline \left<s\right> & -0.07410 \\\ \left<s\right>^2 & 0.00549 \\\ \left<s^2\right> & 0.00579 \\\ σ^2 & 0.00030 \\\ σ & 0.01725 \\\ Δ & 0.00771 \\ \hline \end{array} $$

I've been told that my calculated standard deviation is not correct because the slopes are not independent of each other (they share points).

Is this a valid way to calculate error on a slope?

What other methods are available to calculate error on a slope based on limited points?

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You simply need to use linear regression. Using your data, this would give $$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\ a & 128.31945 & 0.80174 & \{126.09345,130.54545\} \\ b & -0.0708000 & 0.00189 & \{-0.0761557,-0.0656441\} \\ \end{array}$$

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