Determine Median, Quartiles, Percentiles, and Interquartile Range

Introduction

Key Concepts

Median

The median is the middle value in a data set when the numbers are arranged in ascending or descending order. It divides the data into two equal halves. For an odd number of observations, the median is the central number. For an even number, it is the average of the two central numbers.

Formula:

For an ordered data set with an odd number of observations:

For an even number of observations:

Example:

Consider the data set: 3, 7, 8, 5, 12, 14, 21, 13, 18

Arranged in order: 3, 5, 7, 8, 12, 13, 14, 18, 21

Median = 12 (the fifth value in a nine-number data set)

Quartiles

Quartiles divide a ranked data set into four equal parts. The three quartiles (Q1, Q2, Q3) represent the 25th, 50th, and 75th percentiles, respectively.

First Quartile (Q1): The median of the lower half of the data (25th percentile).

Second Quartile (Q2): The median of the data set (50th percentile).

Third Quartile (Q3): The median of the upper half of the data (75th percentile).

Example:

Using the previous data set: 3, 5, 7, 8, 12, 13, 14, 18, 21

Q1 = 5 (median of 3, 5, 7, 8)

Q2 = 12 (median of the entire data set)

Q3 = 14 (median of 12, 13, 14, 18, 21)

Percentiles

Percentiles indicate the relative standing of a value within a data set. The nth percentile is the value below which n percent of the data fall.

Formula to find the percentile rank:

Where:

• P = Percentile rank
• b = Number of values below the target value
• c = Number of values equal to the target value
• d = Total number of values
• N = Total number of observations

Example:

Find the 40th percentile in the data set: 3, 5, 7, 8, 12, 13, 14, 18, 21

N = 9

P = 40

Position = $\frac{40}{100} \times (9 + 1) = 4$

The 4th value is 8, so the 40th percentile is 8.

Interquartile Range (IQR)

The interquartile range measures the spread of the middle 50% of the data. It is the difference between the third quartile (Q3) and the first quartile (Q1).

Formula:

Example:

Using the previous quartiles: Q3 = 14 and Q1 = 5

IQR = 14 - 5 = 9

Determining Median, Quartiles, Percentiles, and IQR

To determine these measures, follow these steps:

1. Organize the data in ascending order.
2. Find the median (Q2) of the entire data set.
3. Determine Q1 by finding the median of the lower half.
4. Determine Q3 by finding the median of the upper half.
5. Calculate the IQR by subtracting Q1 from Q3.
6. For percentiles, use the percentile formula to find the desired value.

Example:

Data set: 7, 15, 36, 39, 40, 41

Step 1: Ordered data: 7, 15, 36, 39, 40, 41

Step 2: Median (Q2) = (36 + 39)/2 = 37.5

Step 3: Lower half: 7, 15, 36 → Q1 = 15

Step 4: Upper half: 39, 40, 41 → Q3 = 40

Step 5: IQR = 40 - 15 = 25

Step 6: To find the 90th percentile, Position = 0.9 * (6 + 1) = 6.3

The 90th percentile is between the 6th value (41) and the 7th value (not present), so it is approximately 41.

Advanced Concepts

In-Depth Theoretical Explanations

The measures of median, quartiles, percentiles, and interquartile range are crucial for understanding data distribution without being affected by outliers. Unlike the mean, which can be skewed by extreme values, the median provides a better central location for skewed distributions.

Mathematical Derivation of Quartiles:

For a data set with an odd number of observations, the quartiles are determined by the median of the lower and upper halves. For an even number, they are the medians of the divided data.

The calculation of percentiles can be approached using linear interpolation, especially when the desired percentile falls between two data points.

Proof of IQR Robustness:

The IQR is less sensitive to outliers compared to the range. Since it only considers the middle 50% of the data, extreme values do not influence it, making it a reliable measure of variability.

Complex Problem-Solving

Problem 1: A data set consists of the following ages of participants in a workshop: 22, 27, 29, 31, 35, 38, 40, 42, 45, 48, 50, 52. Calculate the median, Q1, Q3, IQR, and the 85th percentile.

Solution:

Ordered data: 22, 27, 29, 31, 35, 38, 40, 42, 45, 48, 50, 52

N = 12 (even number)

Median (Q2) = (35 + 38)/2 = 36.5

Lower half: 22, 27, 29, 31, 35, 38 → Q1 = (29 + 31)/2 = 30

Upper half: 38, 40, 42, 45, 48, 50, 52 → Q3 = (42 + 45)/2 = 43.5

IQR = 43.5 - 30 = 13.5

85th percentile position = 0.85 * (12 + 1) = 11.05

85th percentile ≈ 48 + 0.05*(50 - 48) = 48 + 0.1 = 48.1

Interdisciplinary Connections

These statistical measures are not confined to mathematics but are extensively used in various fields:

• Economics: Median income provides insights into the economic status of a population without being skewed by extremely high or low incomes.
• Medicine: Percentiles are used to assess growth charts in pediatrics, indicating a child's growth relative to peers.
• Environmental Science: IQR is utilized in analyzing pollutant levels to understand variations and detect anomalies.
• Education: Median scores help in evaluating student performance distributions, aiding in curriculum adjustments.

Advanced Techniques in Calculations

When dealing with large data sets or grouped data, the calculation of medians, quartiles, and percentiles requires formulas to estimate their values accurately.

Grouped Data: For data presented in frequency tables, the median, quartiles, and percentiles can be found using interpolation within the appropriate class intervals.

Software Applications: Statistical software and programming languages like R and Python provide functions to calculate these measures efficiently, especially for extensive datasets.

Handling Skewed Data Distributions

In skewed distributions, the median provides a better central location than the mean. The distance between Q1, Q2 (median), and Q3 can indicate the direction and degree of skewness.

Positive Skew: Q3 - Q2 > Q2 - Q1

Negative Skew: Q2 - Q1 > Q3 - Q2

Understanding skewness is essential in fields like finance for risk assessment and in quality control for process improvement.

Comparison Table

Summary and Key Takeaways