Chapter 4: Summarizing Data Numerically – Measures of Center

Chapter 4: Summarizing Data Numerically

Section 4.1: Measures of Center

Measures of center are statistical values that describe the central tendency of a data set. The most common measures are the mean, median, and mode. These values help summarize and compare data distributions.

Median

The median is the value that divides a data set into two equal halves, representing the 50th percentile.

• Definition: The median of a set of data values is the value at the 50th percentile.

• Finding the Median:

  • Arrange the observations from smallest to largest.

  • If the number of observations (n) is odd, the median is the middle observation.

  • If n is even, the median is the average of the two middle observations.

• Symbols: The symbol $\tilde{x}$ stands for the median of a sample; the letter $M$ stands for the median of a population.

Example: Comparing the Medians of Two Distributions

Minimum wages (in dollars per hour) of the Pacific states and the Mountain states are shown below:

Pacific State	Minimum Wage (dollars per hour)	Mountain State	Minimum Wage (dollars per hour)
California	9.89	Arizona	11.00
Oregon	12.00	Colorado	11.10
Washington	10.10	Idaho	7.25
Alaska	11.25	Montana	8.50
Hawaii	12.00	Nevada	8.25
		New Mexico	7.50
		Utah	7.25
		Wyoming	5.15

• Pacific States: Ordered minimum wages: 9.89, 10.10, 11.25, 12.00, 12.00

  • Number of observations: 5 (odd)

  • Median: Middle value = $11.25$ per hour

  • Interpretation: About half of the minimum wages are less than or equal to $11.25$, and half are greater than or equal to $11.25$.

• Mountain States: Ordered minimum wages: 5.15, 7.25, 7.25, 7.50, 8.25, 8.50, 11.00, 11.10

  • Number of observations: 8 (even)

  • Median: Average of the two middle values (7.50 and 8.25): $\frac{7.50 + 8.25}{2} = 7.875$ per hour

  • Interpretation: Half of the minimum wages are less than $7.875$, and half are greater than $7.875$.

• Comparison: The median minimum wage in Pacific states ($11.25$) is higher than in Mountain states ($7.875$), indicating that minimum wages tend to be higher in Pacific states.

Summary: The median is a useful measure of center, especially when comparing distributions or when data sets contain outliers or are skewed.