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 below or above mean. A raw score as a Z-score can also be called as a standard score and it can be placed on a normal distribution curve. Z-scores range from -3 standard deviations up to +3 standards.

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.

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 = (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,

Where,

x̄ is the sample mean

s is the sample standard deviation.

s is the sample standard deviation.

Example

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

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

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 = (x – μ) / (σ / √n)

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

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 which shows the percentage of values (or area percentage) to the left of a given z-score on a standard normal distribution.

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

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.

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.

SOLUTION

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

= 0.68

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.

So,

0.7486 x 100 = 74.86%.

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

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

This table format helps in deducing 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 as “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.

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.

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.

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 the 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.

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.