How To Determine Z Score - GeorgeCook960/hello-world GitHub Wiki
Before we start explaining to you how to determine z-score, we believe that you need to know about this specific statistic value so that you can fully understand its calculation.
What Is A Z-Score?
Simply put, a z-score is just the number of standard deviations from the mean that a data point is. If you are looking for a more technical definition, then we can state that the z-score is the measure of how many standard deviations above and below the population a raw score is.
One of the things that you need to keep in mind is that a z-score is also referred to as a standard score and you can use the normal distribution curve to represent it graphically.
When you are trying to calculate the z-score, you will need to know two different parameters: the population mean (μ) and the population standard deviation (σ).
Why Do You Need A Z-Score?
In case you are wondering why you need to know all this about the z-score and why you need to understand how to calculate it, the reason is pretty simple. After all, the z-scores are a great way to compare results from a test to a "normal" population.
Just think of an exam or a survey that is being conducted. While you may get the results and represent them graphically, you need to have something to compare these results with. When you don't, exams and surveys may be pointless.
The Z-Score Formula - One Sample
When you want to calculate the z-score for one sample only, then you should use the following formula:
z = (x – μ) / σ
Let's just imagine that you got a test result and you got a score of 190. You also know that the test has a mean (μ) of 150 and a standard deviation (σ) of 25. Assuming a normal distribution, your z score would be:
z = (x – μ) / σ z = (190 – 150) / 25 z = 1.6
The z score tells you how many standard deviations from the mean your score is. In this example, your score is 1.6 standard deviations above the mean.
Just use this calculator to confirm your result.
The Z-Score Formula - Standard Error Of The Mean
While you may have just one sample and in that case, you should use the above formula to determine the z-score, the truth is that you may have multiple samples. So, in this case, you will want to describe the standard deviation of those sample means. The formula of the z-score that you should use in these situations is:
z = (x – μ) / (σ / √n)
Let's imagine that you know that the mean height of women is 65'' with a standard deviation of 3.5''. So, you want to determine the probability of finding a sample of 50 women who has a mean height of 70''. Assume that the heights are normally distributed.
You would then need to use the formula:
z = (x – μ) / (σ / √n)
And by replacing the values provided in the example:
z = (70 – 65) / (3.5 / √50) z = 5 / 0.495 z = 10.1
Since in this case, we are dealing with a sampling distribution of means, it is important that you include the standard error in the formula.
Considering that you know that 99% of all the values fall into 3 standard deviations from the mean in a normal probability distribution, then we can state that there’s less than 1% probability that any sample of women will have a mean height of 70″.