Measures of central tendency and spread are essential tools for understanding data distribution. Quartiles, percentiles, and the median help us grasp where data points fall, while the interquartile range (IQR) reveals data spread and identifies outliers.

These measures paint a clear picture of dataset characteristics. By interpreting quartiles, percentiles, median, and IQR, we can compare individual data points to the overall set, assess central tendencies, and understand data variability and shape.

Measures of Central Tendency and Spread

Quartiles and percentiles calculation

• Quartiles divide an ordered dataset into four equal parts
  • First quartile (Q1) represents the 25th percentile, meaning 25% of data falls below this value
  • Second quartile (Q2) represents the 50th percentile or median, with 50% of data below this value
  • Third quartile (Q3) represents the 75th percentile, indicating 75% of data lies below this value
• Percentiles represent the percentage of data below a certain value
  • To calculate the $k$th percentile:
    1. Arrange the data in ascending order (smallest to largest)
    2. Calculate the rank using the formula: $rank = \frac{k}{100}(n+1)$, where $n$ is the total number of data points
    3. If the rank is an integer, the $k$th percentile corresponds to the data value at that rank (e.g., if rank is 5, the $k$th percentile is the 5th data point)
    4. If the rank is not an integer, interpolate between the two nearest data values (e.g., if rank is 7.5, the $k$th percentile is the average of the 7th and 8th data points)
• The five-number summary (minimum, Q1, median, Q3, maximum) provides a concise description of the data's distribution

Median as central tendency measure

• The median represents the middle value in an ordered dataset
  • For an odd number of values, the median is the exact middle value (e.g., in {1, 2, 3, 4, 5}, the median is 3)
  • For an even number of values, the median is the average of the two middle values (e.g., in {1, 2, 3, 4}, the median is (2 + 3) / 2 = 2.5)
• The median is less sensitive to extreme values or outliers compared to the mean, making it a robust measure of central tendency
• The median better represents the typical value for skewed distributions (e.g., income data, where a few high earners can pull the mean upward)
• Other measures of central tendency include the mean (arithmetic average) and mode (most frequent value)

Interquartile range for outlier identification

• The interquartile range (IQR) measures the spread of the middle 50% of data, calculated as the difference between the third quartile (Q3) and the first quartile (Q1)
  • $IQR = Q3 - Q1$
• Potential outliers are identified using the following criteria:
  • Lower outliers: Data values less than $Q1 - 1.5 \times IQR$ (e.g., if Q1 is 10 and IQR is 5, lower outliers are values below 10 - 1.5 × 5 = 2.5)
  • Upper outliers: Data values greater than $Q3 + 1.5 \times IQR$ (e.g., if Q3 is 20 and IQR is 5, upper outliers are values above 20 + 1.5 × 5 = 27.5)
• The IQR is resistant to extreme values, making it a robust measure of spread unaffected by outliers

Additional measures of spread

• Standard deviation measures the average distance of data points from the mean
• Variance is the square of the standard deviation, providing another measure of data dispersion

Using Measures of Location and Spread

Interpret quartiles and percentiles meaning

• Quartiles and percentiles provide insights into the distribution of data
  • A 25th percentile (Q1) value of 50 means 25% of data falls below 50 (e.g., in test scores, 25% of students scored below 50)
  • A 50th percentile (Q2 or median) value of 75 means 50% of data falls below 75 (e.g., half the students scored below 75)
  • A 75th percentile (Q3) value of 90 means 75% of data falls below 90 (e.g., 75% of students scored below 90)
• Quartiles and percentiles allow comparison of individual data points to the overall dataset (e.g., a student scoring in the 90th percentile performed better than 90% of their peers)

Median and IQR describe dataset characteristics

• The median indicates the central tendency of the dataset
  • A high median suggests the data values are generally higher (e.g., a median income of $100,000 indicates a wealthy population)
  • A low median suggests the data values are generally lower (e.g., a median age of 25 indicates a young population)
• The IQR represents the spread and variability of the dataset
  • A large IQR indicates greater spread in the data (e.g., an IQR of 20 years for age data suggests a wide range of ages)
  • A small IQR indicates data is more concentrated around the median (e.g., an IQR of 2 points for test scores suggests most scores are close to the median)
• The median and IQR together characterize the dataset's shape, including skewness (asymmetry) and potential outliers (e.g., a low median with a large upper IQR suggests right-skewness and possible high-end outliers)
• A box plot visually represents the five-number summary, making it easy to identify the median, quartiles, and potential outliers