The mean of ten different numbers is 11.8.

1. Find the total of the original ten numbers.

Mean of 10 numbers is $11.8$.

Total of the 10 numbers:
$10 \times 11.8 = 118$

2. Find the extra number needed so that the mean of 11 numbers is 12.

For 11 numbers to have mean $12$, required total:
$11 \times 12 = 132$

Number to add:
$132 - 118 = 14$

1. Identify how each adjustment affects distance from the mean.

For a value less than the mean, decreasing it by $2$ makes it further below the mean.

For a value greater than the mean, increasing it by $2$ makes it further above the mean.

A value equal to the mean is unchanged.

2. Conclude the effect on standard deviation.

Since the values are spread further away from the mean, the standard deviation increases.