Percentiles, Quartiles, and the Five-Number Summary

Percentiles

Percentiles are statistical measures that indicate the relative standing of a value within a data set. Specifically, the p-th percentile is the value below which p% of the data falls.

• Definition: The percentile of a value is the percentage of data values in the set that are less than that value.

• Interpretation: If a test score is at the 90th percentile, it means 90% of the scores are lower than that score.

• Note: The percentile does not indicate the value itself, but the position relative to the rest of the data.

Example: If 90% of students scored below 70 on an exam, then a score of 70 is at the 90th percentile, even though the score itself is not 90%.

Quartiles

Quartiles are special percentiles that divide the data set into four equal parts. They are commonly used to summarize the spread and center of a data set.

• First Quartile (Q1): The 25th percentile; separates the lowest 25% of data from the highest 75%.

• Second Quartile (Q2): The 50th percentile; also known as the median; divides the data into two equal halves.

• Third Quartile (Q3): The 75th percentile; separates the lowest 75% from the highest 25%.

Steps to Find Quartiles

1. Sort the data in ascending order.

2. Find the median (Q2): Use the standard procedure for finding the median.

3. Divide the data:

   • If the number of values is even, split into lower and upper halves.

   • If the number of values is odd, include the median in both halves (write it twice) to ensure both halves have the same number of values.

4. Find Q1: The median of the lower half.

5. Find Q3: The median of the upper half.

Example 1: Even Number of Values

Data: Ages of children in a day care: 2, 5, 7, 7, 8, 8, 8, 12

• Sorted data: 2, 5, 7, 7, 8, 8, 8, 12

• Median (Q2): (7 + 8) / 2 = 7.5

• Lower half: 2, 5, 7, 7 (Q1: (5 + 7) / 2 = 6)

• Upper half: 8, 8, 8, 12 (Q3: (8 + 8) / 2 = 8)

Interpretation:

• 25% of children are younger than 6 years old (Q1).

• 50% are younger than 7.5 years old (Q2, median).

• 75% are 8 years old or younger (Q3).

Example 2: Odd Number of Values

Data: Number of pets per household: 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 28

• Sorted data: 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 28

• Median (Q2): 1 (the 7th value)

• To split, repeat the median: 0, 0, 0, 0, 1, 1, 1 | 1, 1, 2, 2, 2, 28

• Lower half: 0, 0, 0, 0, 1, 1, 1 (Q1: 0)

• Upper half: 1, 1, 2, 2, 2, 28 (Q3: 2)

Interpretation:

• 25% of households have 0 pets (Q1).

• 50% have 1 pet or fewer (Q2, median).

• 75% have 2 pets or fewer (Q3).

The Five-Number Summary

The five-number summary is a concise way to describe the distribution of a data set. It consists of:

• Minimum value

• First quartile (Q1)

• Median (Q2)

• Third quartile (Q3)

• Maximum value

This summary provides information about the center, spread, and range of the data, and is often used to construct boxplots.

Note: While it is technically correct to refer to the minimum as Q0 and the maximum as Q4, this is not standard practice.

Formulas

• Median (Q2): For an ordered data set of size n:

  • If n is odd:

  • If n is even:

• Quartiles: Use the median to split the data, then find the median of each half for Q1 and Q3 as described above.

Applications: The five-number summary is widely used in exploratory data analysis, especially for constructing boxplots and identifying outliers.