Descriptive Statistics: Measures of Position

Overview

• Measures of position indicate where a data point sits in a distribution or dataset.
• Compare data points within the same or across different datasets.

Quartiles

• Quartiles divide ordered data into four parts.
  • Q2 (Median): Splits data into top 50% and bottom 50%.
  • Q1 (First Quartile): 25% below, 75% above.
  • Q3 (Third Quartile): 75% below, 25% above.
• Quartiles correspond to percentiles:
  • Q1 = 25th percentile
  • Q2 = 50th percentile (Median)
  • Q3 = 75th percentile

Finding Quartiles

1. Sort data in ascending order.
2. Find Q2 (Median):
   • Even number of data points: Average of two middle values.
   • Odd number of data points: Middle value.
3. Exclude the median if it's a data point and find Q1 and Q3:
   • Q1: Median of lower half.
   • Q3: Median of upper half.

Software and Quartile Calculation

• Different software (SPSS, Excel, Jamovi) may give different quartile values.
• SPSS, Excel, and Jamovi may differ in methods, especially for Q1 and Q3.

Interquartile Range (IQR)

• IQR = Q3 - Q1
• Represents the middle 50% of data.
• More robust to outliers compared to total range due to focus on central data.

Five Number Summary

• Components: Minimum, Q1, Q2 (Median), Q3, Maximum.
• Provides measures of central tendency and variability (IQR, Range).

Box Plots

• Visual representation of five number summary.
• Box: Q1 to Q3.
• Line inside Box: Q2 (Median).
• Whiskers: Extend to minimum and maximum, unless outliers are present.
• Useful for showing distribution shape, symmetry, and skewness.

Interpreting Box Plots

• Symmetrical Distribution: Whiskers are equal length, median centered.
• Positive Skew: Long whisker towards higher values, median towards lower end.
• Negative Skew: Long whisker towards lower values, median towards higher end.

Modified Box Plots

• Show outliers explicitly using symbols (e.g., asterisks, circles).
• Outliers:
  • Less than Q1 - 1.5 * IQR or more than Q3 + 1.5 * IQR.
  • Extreme Outliers: Less than Q1 - 3 * IQR or more than Q3 + 3 * IQR.
• Whiskers adjusted to nearest non-outlier value.

Example

• Example data ranging from 1 to 10 with analysis:
  • Q2 = 5.5,
  • Q1 = 3,
  • Q3 = 8.
• Demonstrates calculation of quartiles and visualization in box plots.

Important Notes

• Manual calculation methods may differ from software outputs.
• Follow specific instructions for manual calculations when required.
• Box plots preferred for skewed distributions due to robustness to outliers.