Quantiles

Quantiles are statistical measures that divide a set of data into equal-sized intervals. They help identify the position of a value within a dataset and show how data is spread out.

Quantiles include:

• Quartiles
• Deciles
• Percentiles

All are based on cumulative frequencies and are widely used in data analysis.

1. Quartiles

Quartiles divide the data into four equal parts.

• Q1 (First Quartile): 25% of data lies below it

• Q2 (Second Quartile): 50% of data lies below it

  • It is also the Median

• Q3 (Third Quartile): 75% of data lies below it

Formulas for Grouped Data

Where:

• $k = 1,2,3$
• $L$ = lower boundary of the quartile class
• $N$ = total frequency
• $cf$ = cumulative frequency before quartile class
• $f$ = frequency of quartile class
• $h$ = class width

2. Deciles

Deciles divide the data into ten equal parts.

• D1 = 10% of data below
• D2 = 20% of data below
• ...
• D9 = 90% of data below

Formula for Grouped Data

Where $k = 1,2,3,...,9$

3. Percentiles

Percentiles divide the data into one hundred equal parts.

• P1 = 1% of data below
• P50 = 50% (same as median)
• P90 = 90% below

Formula for Grouped Data

Where $k = 1,2,3,...,99$

Difference Between Quartiles, Deciles, and Percentiles

Uses of Quantiles

• To study dispersion and distribution of data
• To find relative position of data values
• Useful in education (percentile ranks), finance, health, income distribution, etc.
• Helps in constructing box plots (Q1, Median, Q3)

Quick Memory Trick

• Quartiles → 4 parts
• Deciles → 10 parts
• Percentiles → 100 parts