DS

Chapter 3 Part 1 Numerically Summarizing Data

Chapter 3: Numerically Summarizing Data Overview Focus: Understanding statistics to make informed decisions using data. Key characteristics of data distributions: shape, center, spread. Outliers are unusual data values that warrant analysis.

3.1 Measures of Central Tendency

Learning Objectives

  • Calculate arithmetic mean from raw data.

  • Determine the median from raw data.

  • Understand resistant statistics.

  • Calculate mode from raw data.

Measures of Central Tendency

Meaning: A measure that numerically describes the average or typical data value. Types:

  • Mean: Often referred to as average in media but can refer to median or mode.

  • Median: Value in the middle of a data set, resistant to outliers.

  • Mode: The most frequently occurring value in the data set.

Objective 1: Arithmetic Mean

DefinitionArithmetic Mean (μ): Total sum of values divided by number of observations.

Population vs. Sample Mean

  • Population Mean (μ): Uses all individuals, parameter.

  • Sample Mean (x̄): Uses sample data, statistic.

Formulas

  • Population Mean: μ = Σxi / N

  • Sample Mean: x̄ = Σxi / n

Example - Population MeanGiven: Scores of 10 students = 70, 80, 85, 90, 75, 95, 100, 60, 80, 85. Calculate: Add scores (70 + 80 + 85 + 90 + 75 + 95 + 100 + 60 + 80 + 85 = 790), divide by 10. Result: Mean = 790 / 10 = 79.

Objective 2: Median

DefinitionMedian (M): Middle value in an ordered data set.

Steps to Find Median

  1. Arrange data in ascending order.

  2. Count observations (n).

  3. Identify middle value:

    • Odd n: direct middle value.

    • Even n: average of two middle values.

Example - Median with Odd nData: 179, 201, 215, 217, 235, 242, 255, 260, 284.n = 9 (median is 217 seconds).

Example - Median with Even nOrdered Data: 62, 68, 76, 77, 79, 80, 82, 84, 87, 94; n = 10.Median = (77 + 79) / 2 = 78.

Objective 3: Statistical Resistance

Definition: A statistic is resistant if extreme values do not affect its value substantially. Example: Median is resistant; mean is not.

Objective 4: Mode

DefinitionMode: Most frequent observation in a data set. Can have no mode, one mode, or more than one mode (bimodal/multimodal).

Examples

  • For Quantitative Data: Example of O-ring failures, mode = 0 (most frequent).

  • For Exam Scores: No mode since all values are unique.

Summary of Measures of Central Tendency

Measure

Computation

When to Use

Mean

Population mean: Σxi / N; Sample mean: Σxi / n

Symmetric distribution with quantitative data.

Median

Arrange data, find middle value (M)

Skewed distribution, quantitative data.

Mode

Most frequent observation.

Qualitative data or information on most common values.

Example Applications

  • Mean Example: Basketball game ticket sales, find mean ticket price.(Example ticket prices: $20, $25, $30; Mean = ($20 + $25 + $30) / 3 = $25.)

  • Median Example: Assess typical phone call duration by finding median duration. (Example durations: 2 mins, 3 mins, 4 mins, 5 mins; Median = 3.5 mins.)

  • Mode Example: Identify common injury locations from rehabilitation data.(Example injury data: ankle, knee, ankle, elbow; Mode = ankle.)