Question 1119164: A distribution has a standard deviation of ơ = 4. Find the z-score for each of the following locations in the distributions. Explain the logic and show the math processes for each score.
a. Above the mean by 4 points.
b. Above the mean by 12 points.
c. Below the mean by 2 points.
d. Below the mean by 8 points.
I cannot stand learning from the text book! If someone can show me a visual breakdown on how to solve the equation that would be fantastic!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! standard deviation is 4.
z-score is equal to (x-m)/s.
x is the raw score.
m is the mean.
s is the standard deviation.
when you're dealing with a z-score, the mean is equal to 0 and the standard deviation is equal to 1.
the z-score tells you how many standard deviations you are above or below the mean.
if your standard deviation is 4 points, then the formula becomes:
z = (x-m)/4
if your raw score is 4 points above the mean, then x = m + 4 and the formula becomes:
z = (m + 4 - m) / 4 which simplifies to z = 4/4 = 1.
z = 1 means the raw score of 4 points above the mean is 1 standard deviation above the mean.
if your raw score is 12 points above the mean, then x = m + 12 and the formula becomes:
z = (m + 12 - m) / 4 which simplifies to z = 12/4 = 3.
z = 3 means the raw score of 12 points above the mean is 3 standard deviations above the mean.
when the raw score is 2 points below the mean, then x = m - 2 and the formula becomes:
z = (m - 2 - m) / 4 which simplifies to z = -2/4 = -1/2.
z = -1/2 means the raw score of 2 points below the mean is 1/2 standard deviations below the mean.
when the raw score is 8 points below the mean, then x = m - 8 and the formula becomes:
z = (m - 8 - m) / 4 which simplifies to z = -2.
z = -2 means the raw score of 8 points below the mean is 2 standard deviations below the mean.
note that the z-score of the mean follows the same formula of z = (x-m)/s.
when x is equal to the mean, the formula becomes z = (m-m)/s, which becomes z = 0.
the z-score of the mean is always equal to 0.
the z-score tells you how many standard deviations you are above or below the mean.
graphically, there's a nice calculator that will show you what this looks like in terms of the normal distribution curve.
this calculator can be found at http://davidmlane.com/hyperstat/z_table.html
you can use this calculator with the mean of the raw score and the standard deviation of the raw score distribution.
you can also use this calculator with the mean of the z-score and the standard deviation of the z-score.
you still need the formula to translate from the raw score to the z-score and vice versa.
to confirm the calculations are correct, we can use the area to the left of the z-score and the raw score to confirm they are the same point under the normal distribution curve.
we can assume any mean for the raw score mean since all the measurements are relative to the mean and the standard deviation is fixed at 4 points.
we'll assume the raw score mean is 5000.
we'll do 4 points and 12 points above the mean and 2 points and 8 points below the mean in that order.
each one will be in pairs with the raw score first and the z-score following.
the graphs are shown below:
4 points above the mean:
12 points above the mean:
2 points below the mean:
8 points below the mean:
you can see that the area to the left of the raw score and the area to the left of the z-score are comparable for each pair.
|
|
|
