Extrapolate data points from a series of averages

Question

Given a series of data points:

$$\begin{array}{c|c|c|c|c|c|c|c|} & \text{Monday} & \text{Tuesday} & \text{Wednesday} & \text{Thursday} & \text{Friday} & \text{Saturday} & \text{Sunday} \\ \hline \text{Data Points} & 34 & 38&52&12&54&22&33 \\ \hline \end{array}$$

I can compute mean averages:

$$\begin{array}{c|c|c|c|c|c|c|c|} & \text{M-W} & \text{T-Th} & \text{W-F} & \text{Th-S} & \text{F-Su} \\ \hline \text{Data Points} & 41.33 & 34.00 & 39.33 & 29.33 & 36.33 \\ \hline \end{array}$$

Now say I only have the series of averages.

Two questions:

1. I think it is impossible to extrapolate the individual data points. Is there a proof of this?
2. Is it possible to extrapolate possible ranges for the individual data points? If so, how?

Edit: Assume all data points are in the range 0-100.

Answer 1

The table of averages gives $5$ linear equations in $7$ unknowns. It is unlikely that you can recover the data points from the averages.

Indeed, the general solution can be expressed in terms of $x_1$ and $x_2$: \begin{align} x_3 &=-x_1-x_2+124\\ x_4 &=x_1-22\\ x_5 &=x_2+16\\ x_6 &=-x_1-x_2+94\\ x_7 &=x_1-1 \end{align} where I've used fractions for the averages, that is, $124/3$ instead of $41.33$.

If you have limits for the ranges of $x_1$ and $x_2$, then these equations give you limits for the ranges of the other variables.