A z-score gives us an idea of how far from the mean a data point is. It is an important topic in statistics. Z-scores are a method to compare results to a “normal” population. For example, we know someone’s weight is 70 kg, but if you want to compare it to the “average” person’s weight, looking at a vast table of data can be overwhelming. A z-score gives us an idea of where that person’s weight is compared to the average population’s mean weight. In this article, we will learn what is z score.
What is z Score in Statistics?
A measure of how many standard deviations below or above the population mean a raw score is called z score. It will be positive if the value lies above the mean and negative if it lies below the mean. It is also known as standard score. It indicates how many standard deviations an entity is, from the mean. In order to use a z-score, the mean μ and also the population standard deviation σ should be known. A z score helps to calculate the probability of a score occurring within a standard normal distribution. It also enables us to compare two scores that are from different samples. A table for the values of ϕ, indicating the values of the cumulative distribution function of the normal distribution is termed as a z score table.
Formula
The equation is given by z = (x – μ)/ σ.
μ = mean
σ = standard deviation
x = test value
When we have multiple samples and want to describe the standard deviation of those sample means, we use the following formula:
z = (x – μ)/ (σ/√n)
Interpretation
1. If a z-score is equal to -1, then it denotes an element, which is 1 standard deviation less than the mean.
2. If a z score is less than 0, then it denotes an element less than the mean.
3. If a z score is greater than 0, then it denotes an element greater than the mean.
4. If the z score is equal to 0, then it denotes an element equal to the mean.
5. If the z score is equal to 1, it denotes an element, which is 1 standard deviation greater than the mean; a z score equal to 2 signifies 2 standard deviations greater than the mean; etc.
Example 1
The test score is 190. The test has a mean of 130 and a standard deviation of 30. Find the z score. (Assume it is a normal distribution)
Solution:
Given test score x = 190
Mean, μ = 130
Standard deviation, σ = 30
So z = (x – μ)/ σ
= (190 – 130)/ 30
= 60/30
= 2
Hence, the required z score is 2.
Example 2: You score 1100 for an exam. The mean score for the exam is 1026 and the standard deviation is 209. How well did you score on the test compared to the average test taker?
Solution:
Given test score x = 1100
Mean, μ = 1026
Standard deviation, σ = 209
So z = (x – μ)/ σ
= (1100-1026)/209
= 0.354
This means that your score was 0.354 standard deviation above the mean.
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Frequently Asked Questions
What do you mean by z-score?
A z score is a statistical measure which gives an idea of how far a raw score is from the mean of a distribution.
Give the formula for z-score.
The formula for z-score is given by z = (x -μ)/σ. Here μ is the mean and σ is the standard deviation.
Can z-score be zero?
Yes. It can be zero, negative, or positive.
What do you understand by z-score zero?
If a z-score is 0, it denotes that the data point’s score is identical to the mean score.
Why is z-score used in statistics?
We use z score because it helps us to determine the probability of a score occurring within our normal distribution and also helps us to compare two scores that are from different normal distributions.
Give two real life applications of z- score.
Z-score is used in a medical field to find how a certain new born baby’s weight compares to the mean weight of all babies. It is used to find how a certain shoe size compares to the mean population size.
What is the meaning of z-score value equals 1?
A z-score of 1 denotes a value that is one standard deviation from the mean.
How do you calculate z-score?
We subtract the mean from each of the individual data points. Then we have to divide the result by the standard deviation. This gives the z-score.
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