Notes on Averages, Mean, Median, Mode and Their Applications

Chapter 3: Averages

Average

  • Definition: The average is a statistical concept that summarizes a group of numbers into one single number that represents a typical or representative value of the data set.

Arithmetic Mean

  • Example: A student's scores on five exams for the semester are given as follows:

    • Scores: 90, 83, 85, 72, 75

    • -

  • Mathematical Notation:

    • Mean of a Population: (\mu)

    • Sum of values: (\Sigma x)

    • Size of the population: (N)

Calculation of Arithmetic Mean

  • Given Scores:

    • 90, 83, 85, 72, 75

  • Calculation:

    • Total Sum, (\Sigma x = 90 + 83 + 85 + 72 + 75 = 405)

    • Number of Scores, (N = 5)

    • Mean Calculation:

      • (\text{Mean} = \frac{\Sigma x}{N} = \frac{405}{5} = 81)

Mean of a Sample

  • The formula for calculating the mean of a sample is similarly structured, defined as:

    • (x = \frac{S_x}{n})

    • Where, (S_x = 405) and (n = 5)

    • Therefore, the calculated sample mean is (x = 81).

Median

  • Definition: The median is defined as a positional average. It represents the value that lies in the middle position of a data set when arranged in ascending order.

  • Procedure:

    • Data must be organized from the smallest value to the largest value to find the median.

Example Calculation of Median

  • Given Scores:

    • 90, 83, 85, 72, 75

  • Arrange Values:

    • Ordered: 72, 75, 83, 85, 90

  • Since there is an odd number of data points, the median is the middle value:

    • Median = 83

Median - Even Number of Items

  • Example: With scores 90, 83, 85, 72, 75, and an additional score of 92:

    • Ordered: 72, 75, 83, 85, 90, 92

    • Since there are now an even number of values (6), the median is calculated as:

      • (\text{Median} = \frac{83 + 85}{2} = 84)

Mode

  • Definition: The mode is defined as the value of the variable that occurs most frequently in a data set.

Example of Mode Calculation

  • Data Set: Color of Car vs. Number of Cars:

    • Red: 12

    • Green: 15

    • Blue: 14

    • Brown: 20

    • Yellow: 12

  • Mode: The color that occurs the most frequently is Brown with 20 cars.

Level of Measurement

  • Categories & Averages Used:

    • Nominal: Mode

    • Ordinal: Median, Mode

    • Interval: Mean, Median, Mode

    • Ratio: Mean, Median, Mode

Averages – Special Cases

  • Weighted Mean: A weighted mean is utilized when certain values in a data set should have a higher contribution towards the calculated mean than others.

Example of Weighted Mean Calculation

  • Problem Context: Calculate the mean unemployment rate in a three-state area with the data given:

    • State: Pennsylvania (8%), New Jersey (10%), Delaware (3%)

Weighted Average Formula

  • Mathematical Representation:

    • (\text{Weighted Average} = \frac{\Sigma \omega x}{\Sigma \omega})

Example Calculation of Weighted Average

  • State Data:

    • Pennsylvania: Unemployment Rate 8% with Population 12 → (w \cdot x = 8 \cdot 12 = 96)

    • New Jersey: Unemployment Rate 10% with Population 8 → (w \cdot x = 10 \cdot 8 = 80)

    • Delaware: Unemployment Rate 3% with Population 1 → (w \cdot x = 3 \cdot 1 = 3)

    • Total Population ((\Sigma w)) = 21

    • Mean Calculation:

    • Total Weighted Sum = 96 + 80 + 3 = 179.

    • Mean Unemployment Rate

      • (\mu = \frac{179}{21} ≈ 8.52%)