Percentiles and Z-Scores
Position of a value relative to the data.
Key idea
Position of a value relative to the data.
In depth
A percentile tells the percentage of values below a given point. A z-score z = (x − μ)/σ says how many standard deviations a value is from the mean, putting different scales on a common footing.
Worked example
A score of 85 with μ = 70, σ = 10 has z = 1.5, above about 93% of a normal distribution.
Formula
Real-world uses
- Explaining a child’s growth chart result
- Comparing exam scores from different tests
Common mistake
Confusing percentile with percent correct.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Percentiles and Z-Scores?
Position of a value relative to the data. It is a topic at the advanced level in Probability & Statistics.
What is the formula for Percentiles and Z-Scores?
The key formula is z = (x−μ)/σ.
Where is Percentiles and Z-Scores used in real life?
Real-world examples: Explaining a child’s growth chart result; Comparing exam scores from different tests.
What should I learn before Percentiles and Z-Scores?
Make sure you are comfortable with Weighted Mean, the previous topic in Probability & Statistics.
What is a common mistake with Percentiles and Z-Scores?
Confusing percentile with percent correct.
How can I practise Percentiles and Z-Scores?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Percentiles and Z-Scores?
Move on to Skewness and Kurtosis next.
Related advanced topics in Probability & Statistics
- Quartiles and IQR – Quartiles split data into four parts; IQR = Q3 − Q1.
- Box Plots – Five-number summary visualised.
- Outliers – Values beyond 1.5 IQR from the quartiles.
- Variance – Average squared deviation from the mean.
- Standard Deviation – Square root of variance; typical spread.
- Sample vs Population Variance – Divide by n−1 for sample variance.