The mean (average value) of the population to which the unstandardized value belongs.
|Population Standard Deviation:|
The standard deviation of the population to which the unstandardized value belongs.
The unstandardized raw value for which you want to compute a Z-score
Positive Z Score Table
When you get a positive z score, you should use the positive z score table to find the values. These are located on the right side of the mean as you can see in the above graph image. When you get a value that is greater than the mean, they are marked with a positive z score in the z table and they are shown in the area under the bell curve to the left of z.
Negative Z Score Table
When you get a negative z score, you should use the negative z score table to find the values. These are located on the left side of the mean as you can see in the above graph image. When you get a value that is less than the mean, they are marked with a negative z score in the z table and they are shown in the area under the bell curve to the left of z.
Understanding Z Scores
When you are learning statistics, one of the first concepts that you will learn is z score. But why do you need to use z scores?
Simply put, you need to use z scores to transform a given standard distribution into something that everyone can calculate probabilities on. This will allow you to know the likelihood os a specific event to occur.
One of the best things about z scores and when you are calculating them is the fact that you can easily display them on a reference chart – the normal distribution, which will be easier to withdraw conclusions. The values that you get correlate to the value under the normal distribution curve which tells you the chance of that even to occur. However, to be easier to interpret the values, it is better to use percentages. And this is why you will need to use a z table chart.
When To Use A Z Score?
One of the main questions that most statistics students usually have is regarding when they should use a z score calculator to determine the z score. To make things a bit easier for you, we can say that there are mainly 3 types of a score questions. And when you have these, you know that you will need to use a z score calculator or simply calculate the z score by hand.
- The chance in terms of the percentage of an event to occur beyond a certain point. In this case, you will be looking for a number that is under the curve but beyond the z value.
- The chance in terms of the percentage of an event to occur under a certain point. In this case, you will be looking for a number that is under the curve up to the z value.
- The chance in terms of the percentage of an event to occur between two points. In this case, you will be looking for a number that is under the curve bounded by two points. It’s important to notice that in some occasions, one of these points can be the mean (or the center of the distribution). In this particular case, z scores are used to determine how far off a specific point in a distribution is from the mean.
The best way to solve your z score problems depends on the information that the problem gives you. You may need to use the z score table to determine the z score or you may need to calculate the z score by hand or using a z score calculator like the one at the top of this page.
How To Calculate A Z Score
As we already said above, the z score is the number of standard deviations from the mean a data point is. In case you are looking for a more technical definition of the z score, we can say that the z score is the measure of how many standard deviations below or above the population mean a raw score is.
One of the things that you need to know about a z score is that it is also referred to as the standard score. And this is because it can be placed on a normal distribution curve.
When you are looking at the normal distribution curve, it is possible to see that z scores vary from -3 standard deviations that are located to the far left of the normal distribution curve, and up to +3 standard deviations that are located to the right of the normal distribution curve.
When you are determining the z score, you will need to know both the mean μ as well as the population standard deviation σ.
Before we get to the formulas of the z score, we want to take the time to tell you that z scores are a great way to compare results from a test to a normal population. When you are doing a survey about people’s height, for example, you may know that someone’s height is 5 feet. However, if you don’t have anything to compare this value with (the population), it won’t mean a thing. So, you will need to get the average person’s height. Then, the z score will be able to tell you where this person’s height can be compared to the average populations’ height.
The Z Score Formulas
As we already mentioned above, in order to determine the z score, you need to know the mean μ as well as the population standard deviation σ. However, there is something that you need to know about the z score formula – the formula is different depending on if you are using just one sample or multiple samples. Let’s check out each one in detail:
#1: The Z Score Formula For One Sample:
z = (x – μ) / σ
Let’s imagine that you just got your test result and you got 170. Knowing that the test has a mean (μ) of 150 and a standard deviation (σ) of 30. Assuming a normal distribution, your z score would be:
z = (x – μ) / σ
z = 170 – 150 / 25
z = 0.8
As you already know, the z score tells you how many standard deviations from the mean your score is. So, in this case, you can say that your score is 0.8 standard deviations above the mean.
#2: The Z Score Formula For Multiple Samples:
z = xi – x̄ / s
x̄ = the sample mean
s = the sample standard deviation
One of the things that you need to keep in mind is that when you have multiple samples and you need to describe the standard deviation os those samples – te standard error – you will need to use a slightly different formula:
z = (x – μ) / (σ / √n)
Again, the z score will be able to tell you how many standard errors there are between the sample mean and the population mean.
Let’s say that the mean height of women is 65″ with a standard deviation of 3.5″ and you want to determine the probability of finding a random sample of 50 women who have a height of 70”.
So, you will need to use the formula:
z = (x – μ) / (σ / √n)
z = (70 – 65) / (3.5/√50)
z = 5 / 0.495
z = 10.1
Since you are dealing with a sampling distribution of means, you will need to include the standard error in the formula. Besides, you already know that 99% of values fall within 3 standard deviations from the mean in a normal distribution. So, we can then state that there is less than a 1% probability that any sample of women will have a mean height of 70”.
Calculating A Z Score By Hand
In our opinion, there is nothing better to understand the z score than by calculating it by hand. So, let’s assume that you have decided to take the SAT and that you scored 1100. Knowing that the mean score for the SAT is 1026 and that the standard deviation is 209, you want to determine how well did you score on this test when compared to the average people who took the test.
As you know, in order to determine the z score, you will need to use its formula:
Z = (x – μ) / σ
By replacing the variables with the numbers that you got:
Z = (1100 – 1026) / 209
Z = 0.354.
This means that your score was 0.354 standard deviations above the mean.
If you want to take a step further, you can also take a look at the z score table to determine the percentage of the people who also took the test scored below you. You can either get a z score pdf with the tables or you can check them at the top of this page.
A z-score of 0.354 is 0.1368 + .500 = 0.6368 or 63.68%.
Calculating A Z Score By Using Our Z Score Calculator
No matter if you are in a hurry to get your z score or if you are only trying to check if you did all the calculations right, having a z score calculator in the nearby is always a good option.
If you check at the top of this page, we have created a free z score calculator that you can use anytime you want. Besides, it is pretty simple and straightforward which makes it perfect.
In case you are trying to determine the z value, all you need to do is to enter the population mean, the population standard deviation, and the raw value. In case you aren’t really sure about where you should place the information you got from the exercise, you can simply place your mouse over the question mark and you will get a basic definition of each one of these variables. This will allow you to know if you are filling the spaces with the right data.
As soon as you complete filling out the three blank spaces, just click on the Calculate button.
If you want to confirm the results of the previous calculation we made, you just need to enter 1027 on the population mean, 209 on the population standard deviation, and 1100 on the value. By hitting the Calculate button, you will get the result right below it: 0.3492822966507177.
Calculating A Z Score In Excel
Besides calculating the z score by hand and using our calculator, you can also use Excel to calculate it.
Let’s use a different example here. Just imagine that you took a test and you scored 650. Knowing that the population mean is 469 and that the population standard deviation is 119, you want to know how good your score is when compared to the general population of test-takers.
Step #1: Add The Population Mean
The first thing that you will need to do is to add the population mean into a blank cell. Let’s say that you are using the cell A2. Here, you’ll need to add 469.
Step #2: Add The Population Standard Deviation
Now, it is time to add the population standard deviation. Again, you will need to add this data to a different cell that is empty. In this example, you are going to add the value 119 into the cell B2.
Step #3: Add The X
In this case, the X is the score that you got on the test. So, you will need to add the value 650 into a blank cell again. In this example, you will use the cell C2.
Step #4: Add The Z Score Formula
As you can see, by now, you already have all the information that you need to calculate the z score. So, you just need to add the z score formula to the Excel spreadsheet that you are using. So, just pick one empty cell and add the formula: =(C2-A2)/B2
Step #5: Press Enter
In order to get the z score, you now need to press enter. As you can easily see, you will get a z score = 1.521008. This means that your test score was 1.521008.
What Is A Z Table?
A z table is also referred to as a z score table or as the standard normal z table.
One of the things that you need to know is that the standard normal model is used in hypothesis testing. And this occurs even in tests on proportions and on the difference between two means.
The area under the normal distribution curve is 100 percent or 1. And the z table chart will help you determine what percentage is under the curve at any specific point.
If you are wondering why we use z scores and then the z table, it is very easy to understand. As for you have read so far, especially through the examples that we have been showing to you, is that the values that you get are very different. So, it would be incredibly complicated to use such wide ranges when you are analyzing data. So, we need to standardize the normal curve by assuming that you always have a mean that is zero and a standard deviation that is one. And when you use the standardizes curve, you can then use the z table to discover the percentages under the curve.
As you can see in the above image, you have the standardized normal graph with the percentage of results (the data) that falls between standard deviations. If you take a quick look at the graph you can state that 68.27% of results will fall within one standard deviation of the mean.
How To Use The Z Table
As you have seen so far, calculating the z score is fairly simple. Besides, you can easily calculate the z score by hand, using our free calculator or even using Excel. However, how can you use the z table?
Let’s go through an example so that you can fully understand how to use the z table.
Just imagine that at the end of the semester 300 college students had to do a test. Mark scored 800 marks (X) in total out of 1000. You also know that the average score of these 300 students was 700 (µ) and that the standard deviation (σ) was 180. The goal is to discover how well Mark scored when compared to his college mates.
If you take a closer look at the exercise, you have all the values that you need to determine the z score by using the formula.
As you already know:
Z score = ( x – µ ) / σ
Z score = (800 – 700) / 150
Z score = 0.56
Up until now, you have Mark’s score. However, since you want to discover how good or how bad he did on the test, you will need to use the z table.
One of the things that you need to do first is to look at the value that you got for the z score. In case it is positive (as this one), you will need to look at the positive z score table. In case you had gotten a negative z score, you would need to check the negative z score table.
So, to determine how good or how bad Mark did on this test, you will need to find the corresponding value for the first two digits on the y-axis. According to Mark’s score, you’ll need to look for 0.5.
Then, you need to go alongside the x-axis to find the value for the second decimal. According to our example, you will need to look for 0.06. So, you will get the number 0.7123.
As you already know, you want to get a percentage. So, in order to get it, you just need to multiply the value that you got from the z table (0.7123) and multiply it by 100:
0.7123 X 100 = 71.23%.
This means that Mark did better than 71.23% of students.
Why Are There Two Z Tables?
One of the things that many statistics students wonder and ask about is related to the use of two different z tables. The reality is that these two tables could be combined into a giant table only. However, we believe that this would be overwhelming, especially for students who are just starting to learn statistics. Besides, there would probably be a lot more errors.
The truth is that the reason why you have 2 different z tables and not just one is to keep things simpler and easier. Simply put, you just need to think if you want to know the area from the mean for a positive value or for a negative value instead.
With this in mind, when you want to know the area between the mean and a negative value, you need to use the negative z score table. On the other hand, when you want to know the area between the mean and a positive value, you need to use the positive z score table.
How Is A Z Score Used In Real Life?
The truth is that the z score is very intuitive since you can use the z table as well as the graph. And these are just some of the reasons why so many people use the z score.
Simply put, the z score in the center of the curve is always zero. This means that when you have a z score to the right of the mean, it means that it is positive and when you have a z score to the left of the mean, it means that it is negative.
Then, by looking at the z table, you will be able to determine the percentage of the population that is above or below our score.