Percentiles: Values Dividing the Data Set into 100 Equal Parts

Percentiles are values that divide a data set into 100 equal parts, providing insights into the distribution of data by indicating the relative standing of specific data points.

Percentiles are values that divide a data set into 100 equal parts. Each percentile represents a specific point in the data distribution, showing the relative standing of a particular value within the dataset. For example, the 25th percentile (also known as the first quartile) represents the value below which 25% of the data falls.

Definition§

Mathematically, percentiles are defined as the data points below which a certain percentage of observations in a data set fall. Denoted generally as $P_k$, where $k$ ranges from 1 to 99, each percentile $k$ indicates that $k$% of the data values lie below it. The median is a special case, representing the 50th percentile.

$$P_k = \{ x \in X : F(x) \le \frac{k}{100} \}$$

Here, $F(x)$ is the cumulative distribution function of the data set.

Types of Percentiles§

Deciles§

Deciles are specific types of percentiles that divide the data into 10 equal parts (i.e., the 10th, 20th, …, 90th percentiles).

Quartiles§

Quartiles are another specific type of percentile. They divide the data into four equal parts:

• Q1 (25th percentile): This is the first quartile.
• Q2 (50th percentile): This is the median or second quartile.
• Q3 (75th percentile): This is the third quartile.

Special Considerations§

• Interpolation: When the data set is continuous, percentiles may need to be interpolated by estimating values between known data points.
• Data Distribution: The distribution of data affects percentile values; normal distributions differ from skewed distributions.
• Application Context: The choice of percentiles can depend on the specific questions being addressed (e.g., education, finance, health, etc.).

Examples§

Example 1: Test Scores§

Consider a dataset of test scores: [55, 60, 65, 70, 75, 80, 85, 90, 95, 100].

• The 25th percentile (P25) is 67.5, meaning 25% of the scores are below 67.5.
• The 50th percentile (P50) is 75, the median score.
• The 75th percentile (P75) is 87.5, meaning 75% of the scores are below 87.5.

Example 2: Salary Distribution§

If analyzing salary data, the 90th percentile can indicate the salary below which 90% of the employees earn, providing insights into earning distribution and inequalities.

Historical Context§

The concept of percentiles has been widely used in statistics since the early 20th century for data analysis and interpretation, prominently utilized in fields like education, health, and economics.

Applicability§

Percentiles are extensively used in various fields:

• Education: Assessing student performance.
• Finance: Analyzing income distribution and economic inequalities.
• Health: Growth charts and biomarker analysis.

Comparisons§

Percentiles vs. Quartiles:

• Quartiles are specific percentiles (25th, 50th, and 75th).
• Percentiles provide a broader view beyond quartiles, allowing finer distribution analysis.

Percentiles vs. Percentages:

• Percentages measure parts per hundred and are used in different contexts, such as efficiency or success rates.
• Percentiles specifically relate to data distribution within a dataset.

Related Terms§

• Quantiles: Points dividing data into equal intervals.
• Cumulative Distribution Function (CDF): A function representing the probability that the variable takes a value less than or equal to x.
• Median: The 50th percentile.

FAQs§

What is the 90th percentile?

The 90th percentile is the value below which 90% of the data falls. It identifies the point that separates the lowest 90% from the highest 10%.

How are percentiles used in standardized testing?

Percentiles rank students based on their test scores relative to peers, showing the percentage of students scored below a particular score.

How do you calculate percentiles?

Percentiles can be calculated using the rank formula:

$$P_k = x_{(k(N+1)/100)}$$

, where $x$ is the data value at rank $k$ within the dataset of size $N$.

References§

• Montgomery, D.C., & Runger, G.C. (2014). Applied Statistics and Probability for Engineers. John Wiley & Sons.
• Weiss, N.A. (2012). Introductory Statistics. Pearson Education.

Summary§

Percentiles play a crucial role in statistical analysis, providing valuable insights into how data is distributed. Whether used in education, finance, health, or various other fields, understanding and applying percentiles helps contextualize and interpret data accurately.