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To use this sample z score calculator, follow the below steps.

- Choose the data type e.g Data points (s)
- Input the point separated by commas.
- Enter the population mean and population standard deviation.
- Click
**Calculate Button**.

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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.

Z-score is a statistical measure that helps evaluate the distance of each entry in a dataset from its mean value, expressed in terms of standard deviation. It allows us to understand how far away a particular value is from the sample's mean.

The Z-score calculation involves taking the difference between the value and the mean, and then dividing it by the standard deviation. A Z-score of 0 indicates that the value is equal to the mean, while a positive or negative Z-score indicates that the value is above or below the mean, respectively.

Think of standard deviation as a unit of measurement, similar to feet or inches, that provides context for the distances between individual data points and the mean.

**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.

The z-score formula for data points

**z = (x – µ)/σ **

The z-score formula for sample mean & size

**z = (x̄ – µ)/ (σ/ √n)**

The z-score formula for sample data

**z = (x̄ – µ)/ (σ/ √n)**

In the above equations,

**Z**represents the z-score.**x**is the raw score or any dataset entryrepresents the sample & population mean respectively.*x̄ & μ*represents the population standard deviation**σ****n**represents the total number of observations

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

**Example 1: For Sample Data**

What are the z-score values for the following inputs?

- Sample data = [42, 54, 65, 47, 59, 40, 53]
- Population mean = 50
- Population Standard Deviation = 16

**Solution:**

**Step 1:** Count the entries.

n = 7

**Step 2: **Take the formula for finding the z-score from the sample set of data values.

z = (x̄ – µ)/ (σ/ √n)

**Step 3:** Find the sample mean.

x̄ = [42 + 54 + 65 + 47 + 59 + 40 + 53] / 7

= 360 / 7

= **51.4**

**Step 4: **Put the values into the above formula.

z = (51.4 – 50)/ (16/√7)

z = (1.4)/(16/2.65)

z = (1.4)/(16/2.65)

z = 1.4/6.04

z = 0.2318

**Example 2: For Data Points**

What are the z-score values for the following inputs?

- Data points = 8
- Population mean = 20
- Population Standard Deviation = 10

**Solution:**

**Step 1:** Take the formula for finding the z-score from the data points.

z = (x – µ)/σ

**Step 2: **Put the values to the above formula.

z = (8 – 20)/10

z = -12/10

z = -.12

**Example 3: For sample mean and size**

What are the z-score values for the following inputs?

- Sample mean = 19
- Sample size = 9
- Population mean = 15
- Population Standard Deviation = 12

**Solution:**

**Step 1:**** **Take the formula for finding the z-score from the sample set of data values.

z = (x̄ – µ)/ (σ/ √n)

**Step 2:** Put the values into the above formula.

z = (19 - 15) / (12/√9)

z = (4) / (12/3)

z = 4/4

z = 1