In order to get mean and standard deviation for a given dataset with probabilities do the following

when i get the mean, thats M_x and when i get the standard deviation that is σ_x

If i multiple all the dataset by 100, then the mean and standard deviation are both multipled by 100.

multiplication affects both.

if i add 50 to every item on the dataset, the mean goes up by 50, but standard deviation stays the same, because they aren’t getting any more distant

Introducing a constant “C”

If you multiply or divide each value of a distribution by C
- Mean is multiplied / divided by C
- Standard Deviation is also multiplied / divided by C
If you add or subtract each value a distribution by C
- You add or subtract C from/to the mean
- Standard Deviation is unchanged ( distances stay the same )
When you multiply or add or subtract from things, shape stays the same. ( It may spread out, or it may squish together, but it will still be symmetric or left skewed or right skewed etc. )

Variance is σ

Combining Multiple Distributions

Usually its just 2 but it can be more than 2
if its more than 2 just do it more times
We are level 1 statistics, so when i say combined i mean adding them or subtracting them, we are not gonna try to multiply or divide distributions, that is a level 2 statistics question.
Taking 2 totally seperate distributions and trying to squish them together
multiple distributions all merging into one
Add
- Mean of our sum is equal to first mean + second mean
  - M_x+y = M_x + M_y
- σ^2_x+y = sqrt(σ^2_x + σ^2_y)

$σ_{x + Y}^{2} = σ_{x}^{2} + σ_{y}^{2}$

Subtract
- Mean of a difference in distribution is mean of the first - mean of the second
  - M_x-y = M_x + M_y
- σ^2_x-y = sqrt(σ^2_x + σ^2_y)
Whether you are adding or subtracting, that interaction between the standard deviations is always addition *