Z Score Calculator

Find the z score of a statistical dataset using this z score calculator. It works with three different types of inputs:

• Data points
• Sample mean and size
• A sample data

It will calculate the z score for each value separately on inputting data points. For the other two inputs, the overall z-score is given. Z-Score table calculator gives the standard normal distribution graph. 

Z-score goes by the name normal score and standard score also. This is why this calculator is called a standard normal table calculator.

What is the Z-score?

Another type of dataset measurement. Z-score tells the distance of each entry of a dataset from its mean value in terms of standard deviation. In other words, how far away in “standard deviations” is a value from the mean of the sample.

Standard deviation is used here as sort of a ruler or a measuring unit just like feet, inches, minutes e.t.c.

For example, one can say a table is four feet long. Similarly, a point is 5 deviations away from its mean.

A positive z-score, when a value is some standard deviations above the mean, tells that the entry is greater than the mean. While a negative z-score indicates a value less than the mean. 

There is also a third possibility, a zero z-score. It means the entry is the same as the mean therefore no distance.

Example:

For more clarity, Consider an example. Let’s say we calculated the standard deviation for a dataset and it was 10. Also, assume that its mean is 40.

We will make a scale of this data set like this:

In this scale, every 10 units is equal to one standard deviation, as it was calculated. Now, let’s say one entry from the dataset is 25 and you want to find its z-score.

What will be its z-score? First, we know that 24 is smaller than 40, so the z-score is negative.

Next, it lies in the half of 20(-2 standard deviation) and 30 (-1 standard deviation). It is obvious that the point is -1.5 Standard deviations away from the mean.

Z-score formula:

The example given before is easy because we assumed it ourselves. In statistics, the data sets are variable and mostly involve decimals. It is not possible to find the z-score by making a line graph every time.

Therefore, the z-score formula is used.
 

In this equation:

• Z is the z-score
• x is the raw score or any dataset entry
• μ is the population mean
• σ is the standard deviation

For a sample, the formula remains the same but the names and letters change. μ change to ͞x and σ changes to s.

How to find the z-score?

To find the z-score value, use the z-score probability calculator. The manual method is given below.

1. Write the dataset. 
2. Find the mean and standard deviation.
3. Subtract the raw data from the mean.
4. Divide by the standard deviation.

Example:

What are the z-score values for the following set of data?
[42, 54, 65, 47, 59, 40, 53]
Solution:
Step 1: Count the entries.
n = 7

Step 2: Find the mean.

 ͞x = [42, 54, 65, 47, 59, 40, 53] / 7
    = 360 / 7
    = 51.4

Step 3: Calculate the standard deviation.

Using the standard deviation calculator, 

s = 9.034

Step 4: Subtract each value from the mean.

n₁ = 42 - 51.4 = -9.4
n₂ = 54 - 51.4 = 2.6
n₃ = 65 - 51.4 = 13.6
n₄ = 47 - 51.4 = -4.4
n₅ = 59 - 51.4 = 7.6
n₆ = 40 - 51.4 = -11.4
n₇ = 53 - 51.4 = 1.6

Step 5: Divide each value by standard deviation.

n₁ = -9.4 / 9.034 = -1.041
n₂ =  2.6 / 9.034 = 0.288
n₃ = 13.6 / 9.034 = 1.505
n₄ = -4.4 / 9.034 = -0.487
n₅ = 7.6 / 9.034 = 0.841
n₆ = -11.4 / 9.034 = -1.262
n₇ = 1.6 / 9.034 = 0.177