Examples Of Z Score Calculations
When you are studying statistics, z score calculations are an important part. The truth is that the z score calculations are useful for many different things and you need to ensure that you understand the concepts well.
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One of the most basic things that you will learn in statistics is how to find the z score for some value of a normally distributed variable. So, today, we decided to show you some examples of z score calculations so you can better understand the concept. But before we begin with the examples, it is always useful to give you a brief insight into z scores.
Why We Use The Z Score
One of the most pertaining questions that many statistics students have is why do we need to learn and understand z scores.
Notice that while there is only one standard normal distribution, there is an infinite number of normal distributions. Therefore, the main goal of calculating the z score is to relate a specific normal distribution to the standard normal distribution.
As you probably already know, the standard normal distribution has been being studied for a very long time and there are tables that provide areas underneath the curve – the z score tables. These are the ones that you need to use when you are doing your z score calculations.
Check out the standard normal table.
Since there is an universal use of the standard normal distribution, it is incredibly important to standardize a normal variable. So, you can see the z score as the number of standard deviations that are away from the mean of your distribution.
The Z Score Formula
The formula that we use to calculate the z score is:
z = (x – μ)/ σ
Where:
- x is the value of our variable
- μ is the value of our population mean.
- σ is the value of the population standard deviation.
- z is the z-score.
Take a look at a z score table.
Examples Of Z Score Calculations
Now that we already took a look at the z score concept, it is time to do some z score calculations. This is the best thing you can do to ensure that you understand the concept of the z score.
Let’s say that you know about a population of a particular breed of cats having weights that are normally distributed. Furthermore, suppose that you also know that the mean of the distribution is 10 pounds and the standard deviation is 2 pounds.
In the first example, you want to know the z score for 13 pounds. How can you know this?
Simply put, you will simply need to replace the x = 13 into your z-score formula. With that said:
z = (13 – 10)/2 = 1.5
This means that 13 is one and a half standard deviations above the mean.
Check out the standard normal distribution table.
What about if you want to know the z score of 6 pounds?
As you can easily understand, the process is pretty similar to the previous one. All you need to do now is to replace the variable by 6 instead of 13.
The result is:
z = (6 – 10)/2 = -2
We can then state that 6 is two standard deviations below the mean.
What about if you now want to know how many pounds corresponds to a z score of 1.25?
While this may seem a bit trickier question, the reality is that you now know the z score. So, in this case, you will need to solve the formula for x:
1.25 = (x – 10)/2
Multiply both sides by 2:
2.5 = (x – 10)
Add 10 to both sides:
12.5 = x
So, you can see that 12.5 pounds correspond to a z-score of 1.25.