A percentile is a measure used in statistics that indicates the value below which a given percentage of observations in a group of observations falls. For example, the 85th percentile is the value below which 85% of the observations may be found. Percentiles are commonly used to report scores from standardized tests, assess weight and height measurements in growth charts, and in many other fields where understanding the relative standing of a value is essential.
Definition of Percentiles
Percentiles are denoted by p, where 0 < p < 100, and represent the p-th position in the data set when it is ordered from the smallest to the largest. Mathematically, percentiles can be defined using the rank (R) formula:
where:
- \( p \) is the desired percentile,
- \( N \) is the number of observations,
- \( R \) is the rank of the p-th percentile.
If \( R \) isn’t an integer, interpolation between the ranks is used to determine the exact percentile value.
Types of Percentiles
Quartiles
Quartiles are special percentiles:
- First Quartile (Q1): The 25th percentile.
- Second Quartile (Q2 or Median): The 50th percentile.
- Third Quartile (Q3): The 75th percentile.
Deciles
Deciles split the data into 10 parts:
- 1st Decile: The 10th percentile.
- 5th Decile (Median): The 50th percentile.
Example Calculation
Consider a dataset of scores: [50, 55, 60, 65, 70, 75, 80, 85, 90, 95].
To find the 85th percentile:
- The rank R is \(\frac{85}{100} \times (10 + 1) = 9.35\).
- The 85th percentile is then an interpolation between the 9th and 10th values in the sorted dataset (90 and 95):
$$ P_{85} = 90 + 0.35 \times (95 - 90) = 91.75 $$
Historical Context
The concept of percentiles dates back to the late 19th and early 20th centuries when mathematicians and statisticians sought better ways to describe the distribution of data. Francis Galton, a key figure in the development of statistics, helped popularize the use of percentiles through his work on regression and correlation.
Applicability
In Education
Percentiles are widely used to interpret standardized test scores. For example, a student’s score in the 85th percentile means that the student performed better than 85% of other test-takers.
In Health
Growth charts for children are often based on percentiles, indicating how a particular child’s height, weight, or head circumference compares to others of the same age and sex.
In Business and Economics
Percentiles are utilized to assess income distribution, measure economic inequality, and segment markets into different consumer groups.
Comparisons and Related Terms
- Quartile: Divides data into four equal parts.
- Decile: Divides data into ten equal parts.
- Median: The 50th percentile, or the midpoint of the data distribution.
FAQs
How is the percentile different from the percentage?
What is the difference between percentile and percentile rank?
References
- “Elementary Statistics” by Mario F. Triola.
- “Statistics for Business and Economics” by Paul Newbold, William L. Carlson, and Betty Thorne.
- Galton, Francis. “Regression Towards Mediocrity in Hereditary Stature.” The Journal of the Anthropological Institute of Great Britain and Ireland, 1886.
Summary
Percentiles are a powerful statistical tool for understanding and communicating the relative standing of data points within a distribution. They have broad applications across various fields, including education, health, business, and economics, aiding in the interpretation and decision-making processes. By providing a clear and quantifiable measure of position, percentiles enable insightful comparisons and analyses of data.
