The mathematical formula for calculating the z-score is as following:
Z = (x-μ)/σ
Where,
X = raw score
μ = Population mean
σ = Population Standard Deviation
Thus, the z-score is the raw score minus the population mean, divided by the population standard deviation.
Whenever we come across situations where the population mean and the population standard deviation are unknown, the standard score can be calculated using the sample mean i.e. x- and the sample standard deviation as estimates of population values.
Now we will consider an example that will illustrate the use of z-score formula. Consider that we know about a population of group of kids having weights that are normally distributed. Further to this, consider that we know that the mean of the distribution is 10 kgs and the standard deviation is 2 kgs.
Now consider the below questions:
1. What is the z-score for 12 kgs?
2. What is the z-score for 5 kgs?
3. How many kgs corresponds to a z-score of 1.25?
For the first question, we simply plug x = 12 in our z-score formula.
The result is: (12-10)/2 = 1.
This means that 12 is one standard deviations above the mean.
The second question is also very similar. Simply put x = 5 into the formula. Thus, the result for this is:
(5-10)/2 = -2.5
The interpretation of this is that 5 is 2.5 standard deviations below the mean.
For the last question, we now know our z-score. For this problem we plug z = 1.25 into the formula and use basic algebra to solve for x
1.25 = (x-10)/2
Multiply both the sides by 2:
2.5 = (x-10)
Add 10 to both the sides:
12.5 = x
Hence, we see that 12.5 kgs corresponds to a z-score of 1.25.
