In this video we discuss how to find, or calculate the variance and standard deviation for a discrete probability distribution. We go through a detailed example and explain the formula step by step Transcript/notes
Variance and standard deviation for a probability distribution
In a past video we discussed what a probability distribution is. Real quick, you collect some data, say you are a personal trainer and you sell training packages for 1, 2, 3, 4 or 5 sessions. The number of sales for each of the session packages is listed in this frequency table. Next, you calculate out the probabilities by taking the number of sales of each session package and divide by the total number of sales. You then list those in a probability distribution table like this.
And now we want to find the variance and standard deviation for the probability distribution.
To calculate this, we first need to find the mean. And as a quick review, we take each of the possible values or outcomes for the variable x, from our probability distribution and multiply each of them by their probabilities as you see here. Then we sum them up which equals 2.82.
The formula for calculating the variance for a probability distribution is small sigma squared equals sum of the quantity x minus mew, or the mean, squared times probability of x.
In basic terms, what this says is you take each of the values that x can assume from the probability distribution, which is row 1, and subtract the mean. Then square that result and multiply it by the probability for that value, which is row 2, from the probability distribution. Again, you do this for each of the values in the table, and add them all up, and that result is the variance.
So, lets go through this for the example probability distribution I have. To make this clear as possible, I am going to mark these values that x can assume as x1 through x5.
Then I am going to make a table with columns for x’s, x minus mew, the mean, x minus mew squared, probability of x and then the result, which is x minus mew squared times probability of x.
The first column is just x1 through x5. For the x1 row, we have x1 minus the mean, which is 1 minus 2.82, which equals negative 1.82, x1 minus the mean squared is negative 1.82 squared, which equals 3.3124. Next we have probability of x1, which for x1 equals 0.10, and now for the final column we have 3.3124, x1 minus the mean squared times 0.10, the probability for x1, which equals 0.3312, and the first row is complete.
Now for the x2 row, x2 minus the mean, 2 minus 2.82 equals negative 0.82. x2 minus the mean squared is negative 0.82 squared, which equals 0.6724. The probability for x2 is 0.28, and finally for the last column, x2 minus the mean squared times probability of x2, 0.6724 times 0.28 equals 0.1882, and row 2 is complete.
And you would continue this process for the remaining x values, x3, x4 and x5, as you see in this completed table.
Now we have what we need, so, we just total up the last column, which equals 1.07 after rounding off, and that is the variance. And to find the standard deviation of the probability distribution, we simply take the square root of the variance, and that is 1.03.
One last note, there is a modified formula that can also be used to find the variance and standard deviation for a probability distribution and it is listed on the screen. And both of these yield the same result.
0 Comments