Notes on Percentiles and Standard Normal Distribution
Definition of Percentile
- A percentile is a measure used in statistics that indicates the value below which a given percentage of observations in a group of observations fall.
Understanding Percentiles
- Example Scenario: If a student scores 91 at the 89th percentile on a standardized test:
- Interpretation: 89% of all test-takers scored lower than 91, while 11% scored higher.
- The 89th percentile signifies that at this value, 89% lies below and 11% lies above that point on the distribution curve.
Steps to Calculate Percentiles Using Z-Score
- To illustrate finding the 89th percentile using the normal curve, follow these steps:
- Convert the Percentage to Probability: 89% can be expressed as 0.8900.
- Consult the Z-Table: Identify the area closest to 0.8900 in the Z-table (Cumulative Distribution Function [CDF] of the Standard Normal Curve).
- Evaluate the Z-scores:
- The area of 0.8900 lies between 0.8888 (Z-score = 1.22) and 0.8907 (Z-score = 1.23).
- Therefore, since 0.8888 is the closest, it indicates that the distribution lies below Z = 1.22.
- Visual Representation: Draw a normal curve and shade the area to the left of Z = 1.22 to visualize the 89th percentile.
Cumulative Distribution Function (CDF) of the Standard Normal Curve
- The Z-scores and corresponding probabilities from Z-tables provide critical data points necessary for statistical analysis, such as:
| Z | Probability | |
|---|---|---|
| 0.0 | 0.5000 | |
| 0.1 | 0.5398 | |
| 0.2 | 0.5793 | |
| 0.3 | 0.6179 | |
| 0.4 | 0.6554 | |
| 0.5 | 0.6915 | |
| 0.6 | 0.7257 | |
| 0.7 | 0.7580 | |
| 0.8 | 0.7881 | |
| 0.9 | 0.8413 | |
| 1.0 | 0.8413 | |
Finding Other Percentiles |
Example: Determine the 97.5th percentile in a standard normal distribution:
- Interpretation of this percentile means 97.5% of values lie below Z = 1.96 in the standard normal distribution.
Example: Determine the 67th percentile in a standard normal distribution:
- Interpretation of this percentile means 67% of the values lie below Z = 0.44 in the standard normal distribution.
Application of Percentiles in Various Disciplines
- Percentiles are widely used in education, psychology, finance, and many other fields for evaluating and comparing performances, scores, or other measurements in relation to a population. They help in determining rankings and understanding distributions within a dataset.