Z-Score Table

Z Score Table – Welcome to the Z-Score Table Guide! Under this Z-Score Table roof, you can easily find all the relevant information about the Z-Score Test & Z-Score Table. That’s why it is highly recommended to go through the whole article and find out the best information about Z-Score Table.

Definition of Z-Score

In simple words, a Z-Score is the number of standard deviations from the mean point of the data. But if we consider it more technically, it is the measurement to find out how many standard deviations are above or below the mean point of a raw score it. A Z-Score is also called the standard score and you can easily place it on the curve of normal distribution.

Ranges of Z-Score

Z-Score actually ranges from the -3 standard deviations which mean it falls to the far left of the normal distribution curve up to +3 standard deviations which mean it would fall to the far right of the Normal Distribution curve. That’s why for the proper use of the z-score you have to know the mean μ value and also the population of standard deviation σ.

Z-scores is the ultimate way to compare the results from any test to a normal population. We all know that the results from tests or survey have a plethora of possible results & units. But most of these results seems meaningless. For example, getting information about the weight of an individual person is good but if you want to compare the average weight of the population, looking at the table of data can be overwhelming especially if you measure & recorded the weights in the Kilogram. But in the case of Z-Score table, it can easily tell you where the person’s weight is compared to the average population’s mean weight.

Understanding of Z-Score

Z Score Formula

Z-Score Formula – One Sample

Here comes the basic Z-Score formula for a sample

z = (x – μ) / σ

x= Test Score

μ = Mean Value

σ = standard deviations.

For instance, let’s suppose the test score is 190. The mean and standard deviation of the test is 150 and 25 respectively. Assuming the distribution is the Normal distribution then your Z-Score will be

 

z = (x – μ) / σ
= 190 – 150 / 25 = 1.6.

The value 1.6 tells how many standard deviations from the means score of your test is. In the above example, your Z-Score is 1.6 standard deviation above the mean value.

Another Z-Score formula is shown below

Z Score Table

The above shown Z-Score formula is exactly the same as the Z Score formula shown above except the x̄ (Sample Mean) is used instead of μ (The population mean) and S (Standard Deviation) is used instead of σ (the population standard deviation). But solving both formulas are exactly the same as we defined.

Z Score Formula: Standard Error of the Mean

Here comes the Z Score Formula Standard Error of the Mean. If you have to find the Z Score with the multiple samples and want to describe the standard deviation of all of these sample means (the standard error) you have to use the z-score formula:

z = (x – μ) / (σ / √n)

x= Test Score

μ = Mean Value

σ = standard deviations.

n = Total numbers

Solved Example – If the mean height of men is 65 inches with the standard deviation of 3.5 inches then what is the probability of fining the random sample of 50 men with the 70 inches of mean height, assuming the heights of normal distributed?

By using the formula mentioned above.

z = (x – μ) / (σ / √n)
z = (70 – 65) / (3.5/√50) = 5 / 0.495 = 10.1

The important thing is here that we are ultimately dealing with the sampling distribution of the means that’s why it is necessary to include the standard error calculation in the Z Score formula. It is also a fact that 99% of the Z Score values fall in 3 standard deviations from the mean in the normal distribution of probability. That’s why there is only 1% or less than 1% probability which any sample f men will have a mean height of 70 inches.

Step by Step Guide about Z-Score Calculation

You can easily find out or calculate the Z-Score on the Scientific Calculator or in Excel. But in case if you are under strict constraint and have to calculate it by hand then here comes the step by step guide for you to calculate Z-Score Calculation.

Question – You appear in the GAT Test and the total score of the test is 1100. The mean value of the GAT score let’s say is 1026 and let’s say the standard deviation is just 209. How well you perform in the test as compared to the average test taker find out that?

Solution –

X = 1100

μ = 1026

σ = 209

Step # 1 – first of all you need to enter the value of x in the Z-Score Formula. For the question above the X-Value is the total score of your GAT Test.

Z-Score Formula Step 1

Step # 2 – Now you have to enter the mean value in the Z Score Formula or the equation above.

Z-Score Formula Step 2

Step # 3 – Now you have to enter the value of the standard deviation

Z-Score Formula Step 3

Step # 4 – Now you have to simply follow the BODMAS rule in order to solve the equation or simply you can use the calculator.

(1100 – 1026) / 209 = .354.

This means that your Z-Score of the test is .354 standard deviations above the mean because it is positive.

Step # 5 – (optional step) – It is the time to find out the z-value from the Z-Table in order to find out what percentage of test-takers score above or below the standard deviations. Your Z-Score is .354 is .1367+.5000* = .6368 or 63.68%.

Now you can ask why to add the .500 in the results? From the Z-table it has been shown that the scores from the right of the means. That’s why we had to add the .500 for all the area let of the mean. You can also find out more and more example on this website.

Standard Deviations & Z Scores

Technically, it is the measurement to find out how many standard deviations are above or below the mean point of a raw score it. For instance

  • The z-score of 1 is 1 standard deviation above the mean
  • The z-score of 2 is 2 standard deviations above the mean.
  • The z-score of -1.8 is -1.8 standard deviations below the mean.

The Z-Score Test will inform you about where you score of the test lies at the curve of Normal Distribution. The Zero Z-Score will tell you is exactly average while the +3 score of Z-test will tell you the value is much higher than the mean.

Real-Life Usage of Z-Score

The applications of Z-Score in real life is unlimited & beyond the scope. You can easily use the normal distribution graph and z-table in order to depict about how z-score of 2.0 means “higher than average”. For example, if the weight of the individual is about 240 pounds and you can familiar with the z-score and it is 2.0 but you want to find out how much above average this weight is?

The centre of the curve is zero. The z-scores which are right side to the means value are positive and the z scores which are to the left of the mean value are negative. If you look at the z-table you can easily find out what percentage of the population is just above or below the score.

The table shown below depicts the z-score 2.0 highlighted. It shows that .9772 which is actually 97.72%. It tells 97.76% of the sample population is lie below the particular score while rest of 2.28% of the score lies above the average score.

Z-Score Table Home

Z Score Table

On this website, you can also find out the values. The values on the left of the mean values are negative in the Z-Score table. Negative scores mean the values which are actually less than the mean value.

The values on the right of the mean values are positive in the Z-Score Table. Negative scores mean the values which are actually higher than the mean value. Find out the detailed information about the Z-Score Table.