How do you calculate the mean and standard deviation of a set of numbers, for example

{1,1,1,2,2,3,5,7,7,7,9,10,10} The first way is the following:

1. Create a list of real numbers and store it in a list variable. For example {1,1,1,2,2,3,5,7,7,7,9,10,10}->L₁
2. Enter [Stat], select "Calc" then "1-Var Stats"
3. Enter L₁
4. The mean is given by x-bar (in this case 5.0)
5. There are two standard deviations, the first uses N-1 instead of N for computing the averages, use the second (in this case 3.374109569).

As an alternative, one can define a "frequency" vector which expresses how many times an entry is repeated. This is placed in L₂

1. Create a list of real numbers and frequencies, and store them as list variables. {1,2,3,5,7,9,10}->L₁
   {3,2,1,1,3,1,2}->L₂
2. Enter [Stat], select "Calc" then "1-Var Stats"
3. Enter L₁,L₂
4. The mean is given by x-bar (in this case 5.0)
5. There are two standard deviations, the first uses N-1 instead of N for computing the averages, use the second (in this case 3.374109569).

If you are given the probabilities p_i along with the values x_i, use the second method (placing x in L₁ and p in L₂).