Introduction to Z-Score in Hypothesis Testing & How to Evaluate it?

The z-score is very essential in hypothesis testing for dealing with various purposes. Hypothesis testing is a frequently used type of statistics that is used to get the outputs related to population factors based on a sample set of data.

The fields of medicine to social science use this branch of statistics to find the validity of assumptions and theories. In this post, we are going to explore the basics of z score along with solved examples. 

What is Z-score?

In statistics, the way to standardize the given set of data is known as the z-score. It is also known as standard scores. The standard form and means play an essential role in finding the z-score of the given set of data. 

In other words, a z-score is used to find how many STDs a data point is away from the expected value of a distribution. The comparisons and analysis across various datasets can be done using z-scores. It is helpful to transform raw data into a standardized form. 

Formulas for finding Z-score

There are three ways to find the z-score:

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

Here are the mathematical expressions for the above three kinds of data

Conditions of z-score result

Here are the three conditions of the result of the z-score.

How to Find the Z-Score?

Below are a few examples to understand how to evaluate z-score.

Example 1

Evaluate the z-score if the population data is 12, 14, 15, 18, 20, 24, 30, 36, 40, 42, 46

Solution 

Step 1: First of all, evaluate the population mean from the given set of data.

xi = 12, 14, 15, 18, 20, 24, 30, 36, 40, 42, 46

n = 11

µ = Σxi/n

µ = [12 + 14 + 15 + 18 + 20 + 24 + 30 + 36 + 40 + 42 + 46] / 11

µ = 297/11

µ = 27

Step 2: Now find the population standard deviation.

As the population mean is 27, find the difference of each data point from it and take the squares.

(xi – µ) = [(12 – 27), (14 – 27), (15– 27), (18 – 27), (20 – 27), (24 – 27), (30 – 27), (36 – 27), (40 – 27), (42 – 27), (46 – 27)]

(xi – µ) = [-15, -13, -12, -9, -7, -3, 3, 9, 13, 15, 19]

Now square them

(xi – µ)2 = [(-15)2, (-13)2, (-12)2, (-9)2, (-7)2, (-3)2, (3)2, (9)2, (13)2, (15)2, (19)2]

(xi – µ)2 = [225, 169, 144, 81, 49, 9, 9, 81, 169, 225, 361]

Now add the squared terms

Σ (xi – µ)2 = [225 + 169 + 144 + 81 + 49 + 9 + 9 + 81 + 169 + 225 + 361]

Σ (xi – µ)2 = 1522

Now use the formula of population STD 

√ [Σ (xi – µ)2 / n] = √[1522 / 11]

√ [Σ (xi – µ)2 / n] = √[138.36]

√ [Σ (xi – µ)2 / n] = 11.76

Step 3: Take the formula for finding the z-score from the data point.

z = (x – µ)/σ

Step 4: Now place the values to the above expression.

z = (11 – 27)/11.76

z = -16/11.76

z = -1.36

As the result of the z-score is negative, its mean data values of the given set are below the mean.

You can use a z score calculator to find the z score of the given data values with steps in no time. 

Example 2

Evaluate the z-score if the population mean is 12, the population STD is 13.6, the Sample mean is 15, and the sample size is 11

Solution

Step 1: Write the given terms

Sample mean = 15

Sample size = 11

Population mean = 12

Population STD = 13.6

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

z = (x̄ – µ)/ (σ/ √n)

Step 3: Put the values into the above formula.

z = (15 – 12) / (13.6/√11)

z = (15 – 12) / (13.6/3.32)

z = (3) / (4.096)

z = 0.73

As the result of the z-score is positive, its mean data values of the given set are above the mean.

Example 3: 

Evaluate the z-score if the population mean is 10, the population STD is 21.4, and the Sample data is 12, 18, 24, 26, 34, 35, 44, 48, and 49

Solution

Step 1: Write the given terms

Sample size = 9

Population mean = 10

Population STD = 21.4

Sample data = 12, 18, 24, 26, 34, 35, 44, 48, 49

Step 2: Now find the sample mean of the given set of data.

 x̄ = Σ xi /n

x̄ = [12 + 18 + 24 + 26 + 34 + 35 + 44 + 48 + 49] / 9

x̄ = 290/9

x̄ = 32.22

Step 3: Take the formula for finding the z-score from the sample set of data values.

z = (x̄ – µ)/ (σ/ √n)

Step 4: Put the values into the above formula.

z = (32.22 – 10) / (21.4/√9)

z = (32.22 – 10) / (21.4/3)

z = (22.22) / (7.13)

z = 3.12

As the result of the z-score is positive, its mean data values of the given set are above the mean.

Final Words

Z-score plays a vital role in hypothesis testing for dealing with various aspects. You can find the z-score from the population data and sample data based on different formulas. The solved examples in this post will help you to understand this topic accurately.