Explanation:
the standard deviation is first of all based on the squared differences of every data point to the mean value (squared, so that the direction of the difference does not matter, each number is positive).
this number is normalized to the number of data points (divided by "N").
and from that number we calculate the positive square root.
we take the square root to bring things back from the squared dimension of the data differences to the mean value and make it better understandable in size compared to the sizes of the data points. otherwise there is benefit to the squared variance (the whole thing without the last square root).
we have 15 data points (8 + 7).
first we need the mean value, which is the sum of all data points divided by the number of data points :
(7+10+5+5+6+3+5+5+6+5+5+5+9+3+5)/15 = 84/15 =
= 5.6
the standard variance is now the sum of the squares of the difference of each data point to the mean value.
e.g. the squared difference of 7 to 5.6 is
(7-5.6)² =(-1.4)² = 1.96
(1.96+19.36+0.36+0.36+0.16+5.76+0.36+0.36+0.16+0.36+0.36+0.36+11.56+5.76+0.36) / 15
= 47.6 / 15 = 3.173333333...
and the standard deviation is
SD = sqrt(3.173333333...) = 1.781385229... ≈ 1.78