Z-Scores and Standardization
In the previous lesson, we learned about the normal distribution and how data is spread around the mean. A natural next question is:
How far is a particular value from the average?
Z-scores help us answer this question in a precise and comparable way.
What Is a Z-Score?
A Z-score tells us how many standard deviations a data value is away from the mean.
It allows us to understand whether a value is:
- Above the average
- Below the average
- Close to or far from the average
Z-scores make values from different datasets comparable.
Z-Score Formula
The formula for calculating a Z-score is:
Z = (X − μ) ÷ σ
- X = data value
- μ = mean
- σ = standard deviation
Numerical Example
Suppose the average score on a test is 70 and the standard deviation is 10.
A student scores 85.
Step-by-step:
- X − μ = 85 − 70 = 15
- 15 ÷ 10 = 1.5
The Z-score is 1.5.
This means the student scored 1.5 standard deviations above the mean.
Interpreting Z-Scores
Z-scores can be interpreted as follows:
- Z = 0 → Value is exactly at the mean
- Z > 0 → Value is above the mean
- Z < 0 → Value is below the mean
The larger the absolute value of Z, the farther the value is from the average.
Real-World Example
Two students take different exams:
- Student A scores 85 on an exam with mean 70 and SD 10
- Student B scores 90 on an exam with mean 80 and SD 5
At first glance, Student B has a higher score. But Z-scores allow fair comparison.
Z-score for Student A:
(85 − 70) ÷ 10 = 1.5
Z-score for Student B:
(90 − 80) ÷ 5 = 2.0
Student B performed better relative to their group.
What Is Standardization?
Standardization is the process of converting data values into Z-scores.
After standardization:
- The new mean becomes 0
- The new standard deviation becomes 1
This transformed data follows the standard normal distribution.
Why Z-Scores Are Important
- They allow comparison across different scales
- They help identify unusual or extreme values
- They are widely used in exams, analytics, and research
Quick Check
If a value has a Z-score of −2, what does it mean?
Practice Quiz
Question 1:
What does a Z-score of 0 represent?
Question 2:
If X = 60, mean = 50, and standard deviation = 5,
what is the Z-score?
Question 3:
Can Z-scores be negative?
Mini Practice
A dataset has a mean of 100 and a standard deviation of 20.
- What is the Z-score for a value of 140?
- Is this value far from the mean?
What’s Next
In the next lesson, we will study Data Collection and Sampling Techniques, which explains how data is gathered in real-world studies.