The problem is asking to find the value of the following summation:

$\sum_{i=1}^{6} (x_i - M)$

where $M$ is the mean of the six values $x_1, x_2, x_3, x_4, x_5, x_6$.

Solution:

The mean $M$ of the six values is given by:

$M = \frac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6}$

Now, consider the summation:

$\sum_{i=1}^{6} (x_i - M) = (x_1 - M) + (x_2 - M) + (x_3 - M) + (x_4 - M) + (x_5 - M) + (x_6 - M)$

Since $M$ is the average of the $x_i$'s, subtracting the mean from each individual value and adding them up will result in zero. This is because the deviations of the values from the mean always sum to zero by definition of the mean.

$\sum_{i=1}^{6} (x_i - M) = 0$

Final Answer:

$\sum_{i=1}^{6} (x_i - M) = 0$

Would you like more details or have any questions?

Here are 5 related questions to deepen understanding:

1. Why does the sum of deviations from the mean always equal zero?
2. How can you interpret the sum of squared deviations from the mean?
3. How would the result change if the summation used absolute deviations instead of regular deviations?
4. What role does the mean play in measures of spread like variance and standard deviation?
5. How would this summation change if the set of numbers had weights or frequencies associated with them?

Tip: When calculating sums involving deviations from the mean, the result is always zero, which is a property of the mean in statistics.