In statistics, a z score is very essential. Statistics is the branch of science that is used for developing and studying methods for presenting, collecting, and analyzing empirical data. Z score is used to determine the probability of the score that occurs within the normal distribution.
And also used for comparing two scores from different normal distributions. In this post, we will learn about the z score and how to solve the problems of z score with a lot of examples.
What is a Z score?
A raw score by measuring how much standard deviations below and above the population is known as the z score. Z score must be positive if the standard deviation lies above the population. And if the standard deviation lies below the population, the z score must be negative.
The z score is also named a standard score. Its basic work is to indicate a structure having how many standard deviations from the mean. And the equation used for the calculation of the z score is.
Z = x – µ / σ
Formulas of Z score
- The first formula for the z score is used to calculate the data points.
Z = x – µ / σ
In the above equation, x is the test values, µ is the population mean, and σ is the standard deviation. For the calculation of the z score population mean and the standard deviation are known.
- The second formula is used for the calculations of the sample mean and size. And is written as.
Z = (x̄ – µ)/ σ/√n
In the above equation, x̄ is the sample mean, µ is the population mean, n is the sample size, and σ is the standard deviation. For the calculation of the z score population mean and the standard deviation are known.
- And the third formula is used for the calculation of the sample data. And is written the same as the second formula but the sample mean is not given, we have to calculate the sample mean with the help mean formula.
x̄ = sum of all the numbers / total numbers
And then put in the second formula.
Z = (x̄ – µ)/ σ/√n
You can use the z score calculator for the calculation of the z score according to all of the above formulas.
How to solve z score problems using formulas?
For calculating the z score according to the formulas, let’s take some examples.
Example 1: for calculating the data points.
Find the z-score if the population mean is 30, the standard deviation is 20, and the test values are 19?
Solution
Step 1: Take the given information.
Test values = x = 19
Standard deviation = σ = 20
Population mean = µ = 30
Step 2: Now take the formula for calculating z-score by using data points.
Z = x – µ / σ
Step 3: Now put the given information in the above formula.
Z = 19 – 30 / 20
Z = -11 / 20
Z = -0.55
Example 2: For sample mean and size.
Find the z-score if the population mean is 30, the standard deviation is 20, the sample mean is 45, and the sample size is 9?
Solution
Step 1: Take the given information.
Sample mean = x = 45
Sample size = n = 9
Standard deviation = σ = 20
Population mean = µ = 30
Step 2: Now take the formula for calculating z-score by using the sample mean and sample size.
Z = (x̄ – µ)/ σ/√n
Step 3: Now put the given information in the above formula.
Z = 45 – 30 / 20/√9
Z = 15 / 20/3
Z = 15 x 3 / 20
Z = 45 / 20
Z = 9 / 4
Z = 2.25
Example 3: For sample data.
Find the z-score if the population mean is 30, standard deviation is 20, sample data is 12, 13, 15, 18, 21, 23, 25, 30, 31?
Solution
Step 1: Take the given information.
Sample data = 12, 13, 15, 18, 21, 23, 25, 30, 31
Sample size = n = 9
Standard deviation = σ = 20
Population mean = µ = 30
Step 2: First of all, find the sample mean.
Sample mean = x̄ = sum of all the numbers / total numbers
Sample mean = x̄ = (12 + 13 + 15 + 18 + 21 + 23 + 25 + 30 + 31) / 9
Sample mean = x̄ = 188/9
Sample mean = x̄ = 20.889
Step 3: Now take the formula of z-score.
Z = (x̄ – µ)/ σ/√n
Step 4: Now put the given information in the above formula.
Z = 20.889 – 30 / 20/√9
Z = -9.111 / 20/3
Z = -9.111 x 3 / 20
Z = -27.333 / 20
Z = -1.3667
If you want to find critical value of z, you can use z critical value calculator to get the left, right,
and two-tailed probability of z value.
Example 4
Find the z-score if the population mean is 20, standard deviation is 30, sample data is 2, 3, 5, 8, 11, 13, 15, 20, 21?
Solution
Step 1: Take the given information.
Sample data = 2, 3, 5, 8, 11, 13, 15, 20, 21
Sample size = n = 9
Standard deviation = σ = 30
Population mean = µ = 20
Step 2: First of all, find the sample mean.
Sample mean = x̄ = sum of all the numbers / total numbers
Sample mean = x̄ = (2 + 3 + 5 + 8 + 11 + 13 + 15 + 20 + 21) / 9
Sample mean = x̄ = 98/9
Sample mean = x̄ = 10.889
Step 3: Now take the formula of z-score.
Z = (x̄ – µ)/ σ/√n
Step 4: Now put the given information in the above formula.
Z = 10.889 – 20 / 30/√9
Z = -9.111 / 30/3
Z = -9.111 x 3 / 30
Z = -27.333 / 30
Z = -0.9111
Summary
Z-score is very essential for performing certain tasks of the statistics. Z score used three types of formulas for the calculation. One is by taking the data values, the second is by taking the sample mean and sample size, and the third is by using the sample data.
From the above examples, you can easily solve any kind of problem of z score by using any formula.