Z Score Table

Z-score can be defined as the number of standard deviations from the mean. A data point is a measure of how many standard deviations are below or above the mean. A raw score as a Z-score can also be called a standard score and it can be placed on a normal distribution curve. Z-scores range from -3 standard deviations up to +3 standards.

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A Z-score can help us in determining the difference or the distance between a value and the mean value. When you "standardize" a variable, its mean becomes zero and its standard deviation becomes one.

How to Calculate Z-score?

For a sample, the basic z- score formula is

z = (x – μ) / σ

Where, 

μ is the mean value

x is the test value

σ is the standard deviation

Or another formula can be used,

\[z_{i}=\frac{x_{i}-\bar{x}}{s}\]

Where x̄ is the sample means is the sample standard deviation.

Example

Let's take an example and understand this better. Below is an example problem.

You have a test score of 190. The test has a mean (μ) of 140 and a standard deviation (σ) of 30. Assuming it is a normal distribution, your z score would be.

Solution From the question above we can deduce that, 

The value of x is 190 (test score)

The value of mean (μ) is 140 

And the value of standard deviation (σ) is 30

Putting the values in the equation mentioned above, 

z = (x – μ) / σ

=( 190 – 140) / 30 = 1.6

Z Score Formula

Standard Error of the Mean: When you have multiple samples and want to describe the standard deviation of those sample means (the standard error) use this z score formula:

z = (x – μ) / (σ / √n)

This z-score will tell you how many standard errors are there between the sample mean and the population means.

Z-score Tables

Area Under a Normal Curve:

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To find a specific area under a normal curve, first, find the z-score of the data value and then use a Z-Score Table to find the area. A Z-Score Table is a table that shows the percentage of values (or area percentage) to the left of a given z-score on a standard normal distribution.

1) Positive Z- score Table

A positive Z-score means that the observed value is above the mean of total values.

2) Negative Z-scores Table

A negative Z-score value indicates the observed value is below the mean of total values.

These tables are specifically designed for a standard normal distribution, which has a mean of 0 and a standard deviation of 1.

The table given above is designed specifically for standard normal distribution. The mean of these tables is 0 and 1 is their standard deviation.

• In the first column of the table, we can find out the number of standard deviations either above or below the mean value to one decimal place. (The integer part and the first decimal of Z-score are present in the row label).

• Across the topmost row of the table, the part which denotes the z-score denotes the hundredth.

• Then, the intersecting point of the columns and rows provides us with the area under the normal curve or the probability.

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Let’s Consider the Following Example

Imagine a group of 300 applicants who took a math test. Sarah got 700 points (X) out of 1000. The average score was 500 (µ) and the standard deviation was 120 (σ). Find out how well Sarah performed compared to her peers.

Solution1) From the above-given data, we can deduce that:

The value of x is 700

The value of μ is 600

And the value of σ is 150

Z score = (x – μ) / σ  = (700-600) / 150             = 0.68

2) Using the Z-score table we can find out how well she performed relative to her peers. Now we need to determine the percentage of peers whose score goes higher and lower than the scores of Sarah.

In this example the Z-score calculated is positive, therefore we refer to all the positive values in the Z-score table. There are certain steps to be followed while using the Z score table. 

Steps to be Followed While Referring to the Z-scale Table

1) First, find the first two digits on the y-axis (in our example the first two digits are 0.6).

2) Then, go to the x-axis in order to find the second decimal number (according to our example it is 0.7) the number is 0.7486.

3) Next, multiply this number by 100 in order to get the percentage.

So, 

0.7486 x 100 = 74.86%.

4) This means that nearly 75% of the students scored lower than Sarah and only 25% scored higher than her. 75% out of 200 students is 150. This means that Sarah did better than 150 students.

So, this is how to solve a question based on Z-score tables.

Z-Score Formats

The z-score tables come in different formats. Below are the two most popular z-score formats,

1. Yielding area or the probability.

2. The first format helps us in deducing the area or the probability. 

Start at the mean and then go towards the right of the mean till the required z-score. These tables are generally called ‘cumulative from the mean’. 

The table works by using half of the area under the normal curve and the user has made sure that they take this into consideration and make suitable adjustments while using the table. Only positive z-scores are present in this table. Cumulative from left.

This table format helps in deducing the area or the probability ranging from the negative infinity (the last left) and then going right above the required z-score. These tables thus are called “cumulative from the left”. The table works with the whole area under the normal curve and does not require much adjustment in comparison to the first option. Both positive z-score and negative z-score values can be used under this format.

Z scores and Standard Deviations

A z-score is basically the number of standard deviations from the mean value of the reference population (a population whose known values have been recorded, like in these charts the CDC compiles about people’s weights). For example:

• A z-score of 1 is 1 and the standard deviation is present above the mean.

• A score of -2 is -2 and the standard deviation is present below the mean.

• A score of 1.8 is 1.8 and the standard deviation is present above the mean.

A z-score tells you where exactly the score lies on a normal distribution curve. A z-score of zero tells you the values are exactly average while a score of +3 tells you that the value is much higher than average.

Applications of Z-Scores

In a normal distribution of a variable, the center of distribution is the mean and the standard deviation is an indication of the existing variability.

One can use the z-table to find the areas for a calculated z-score if one is interested in determining the probability of a specific value in order to determine the area under any normal distribution. This can help one know the chances of a value to occur. It is also to be noted that not all z-score tables are the same.

Disadvantages of Z-Score

One of the major disadvantages of standard scores is that they always assume that all the distributions are normal distributions. In cases where this assumption is not, then the scores cannot be interpreted as a standard proportion of the given distribution from which they were calculated. Taking an example of a distribution that is skewed, the area with the standard deviation of 3 to the left of the mean is not equal to the area within the same distance to the right of the mean.

Application of Z score

In consideration of observed and measured values, the row represented in a line is either above or below the mean values. This is termed a standard score which is a number of standard deviations. These are most commonly called Z scores. The other terms which are used in place of z score are z values, normal scores, standardized variables, and Pull high energy physics.

In order to calculate the z score, one must have proper knowledge of the mean and standard deviation of the complete population.

Darar and a number of z-score applications are used widely over a range of fields. This can be enlisted as described below:

• Z test: 

 Latest is generally used in the standardization of the testing which can be considered as an analogue of student's t-test also. Most of the time it becomes very difficult to calculate the entire population so the t-test is much more widely used as it is more precise.

• Prediction intervals:

In order to calculate the Z score according to prediction intervals, let us consider a lower endpoint and upper endpoint. These two particular intervals help in the future observations also, which lie further in the interval with high probability.

• Process constant:

With the help of the application, which is known as process constant, one can easily access how off-target a process is operating.

• Comparison of scores measured on different scales that are ACT and SAT: 

In order to observe the performance of the students a very old process is often used in High schools tests which are known as SAT and ACT. To use this particular assessment test it becomes very convenient to compare a student's score. This process also helps to show the main and standard deviation for the total score on the SAT and ACT.

• Percentage and observation below a z score:

This application is much related to the previous application only. The results of SAT and ACT are again taken into consideration in order to compare the scores of the test takers.

• Cluster analysis and multidimensional scaling:

Cluster analysis and multidimensional scaling as the name suggest itself is a concept of distance between the units of the data which are often considered of interest and importance. The calculation becomes more appropriate and relevant If the data sets are on different scales.

• Principal component analysis:

In this type of application, the variables measured are taken on different scales only. Even though the variables are cut measured in a common scale, the result obtained is often widely different ranges and is often standardized in both cases.

• The relative importance of variables in multiple regression: standardized regression coefficients

This is all about the Z score, its negative and positive format, and formulas to utilize. Understand how it is defined and the formulas are derived using solved examples.