Seven Sensors, One Number

Seven sensors placed around a greenhouse report the temperature at noon, in °C:

sensor	1	2	3	4	5	6	7
reading	$17$	$19$	$20$	$21$	$22$	$26$	$200$

Sensor 7 is broken.

The greenhouse controller summarises the readings with the simplest possible model: a single number $c$ , used as its prediction for every sensor. Each sensor's error is $e = c - \text{reading}$ , and $c$ is set to whatever value makes the average Huber loss over the seven sensors as small as possible. Huber's delta is $\delta = 3$ :

$\text{Huber}(e) = \begin{cases} \tfrac{1}{2}\,e^2 & \text{if } |e| \le 3 \\[4pt] 3\,\big(|e| - 1.5\big) & \text{if } |e| > 3 \end{cases}$

The two pieces meet smoothly at $|e| = 3$ : a curve like MSE for small errors, and a straight line like MAE for large ones.

What value of $c$ does the Huber fit choose?

Round to 2 decimal places.

Seven Sensors, One Number

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