Chapter 3.1: Measures of Center

Instructor: Melony Parkhurst
Textbook: Essential Statistics, Navidi & Monk
Course: STAT 1401
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Qualitative vs. Quantitative Data

• Data Types:

  • Qualitative Data (Categorical)

  • Quantitative Data

  • Distinction further classified into:

    • Ordinal or Nominal

    • Discrete or Continuous

Frequency and Relative Frequency Distributions

• Importance of establishing bins for creating a Frequency or Relative Frequency Distribution:

  • This allows for easier interpretation and organization of Quantitative Data.

• Example context: Age of Students Enrolled at KSU in 2015. Sample Data:

  • Ages: 14, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 21, 22, 24, 24, 25, 27, 27, 28, 30, 30, 32, 33, 33, 33, 38, 40, 40, 42, 45, 49, 50, 52, 62, 62, 85

Chapter Coverage

• Chapters 3.1 & 3.2 focus on:

  • Calculations of Quantitative Data

  • Communication of data without visual aids

Measures of Center

Mean (Arithmetic Mean)

• Definition:

  • The mean is the most commonly used measure of central tendency.

  • It is often thought of as the average.

  • The mean is influenced by extreme values in the dataset.

  • Important to determine situations where an alternative measure of center may be more appropriate.

• Mathematical Representation:

  • Represented as:

  • $\bar{x}$ (Greek capital letter Sigma)

  • To compute, add all numbers in a list:

  • Each number in the list denoted as $x_i$

    • Examples: $x<em>1, x</em>2, ext{ … }, x_n$

  • Calculation formula: $\bar{x} = rac{ ext{Sum of all } x_i}{n}$

• Note:

  • Cannot calculate a mean using Qualitative Data.

Example of Calculating a Sample Mean

• Data collected related to KSU students’ dual enrollment during the Fall semesters of 2007 through 2012.

• Determine sample size (n) and find the Mean.

Median

• Definition:

  • The median is the middle value that splits the data set in half.

  • ½ of the data values are below the median, and ½ are above.

• Calculation:

  • For an odd number of observations (n): The middle number is the median.

  • For an even number of observations (n): Take the average of the two middle numbers.

• Example Data:

  • Collected quantitative data on KSU students who were dual enrolled from Fall 2007 to 2015.

Resistant and Non-Resistant Statistics

• Definition:

  • A statistic is resistant if it is not influenced by extreme values.

• Example contexts:

  • Comparison of the Mean and Median regarding sensitivity to extreme values:

  • Resistant to Extreme Values: Median

  • Not Resistant to Extreme Values: Mean

Mode

• Definition:

  • The mode is the value that appears most frequently in the dataset.

  • There can be more than one mode, or no mode if all values appear with the same frequency.

• Example problems analyzing modes in given datasets:

  • Dataset 1: 25, 79, 108, 125, 150, 186, 196, 302

  • Dataset 2: 2, 2, 5, 8, 11, 11, 11, 17, 17, 26

  • Dataset 3: 5, 8, 11, 11, 14, 17, 17, 26, 32, 41, 41

  • Dataset 4: Blue, Hazel, Green, Brown, Hazel

Describing Distribution Shape

• Analysis of Mean and Median:

  • Mean accounts for all values and is influenced by extreme values.

  • Median serves as the middle point in a dataset.

• Distribution Shapes:

  • Left Skewed Distribution: Median > Mean

  • Right Skewed Distribution: Mean > Median

  • In symmetric distributions, Mean = Median.

One-Variable Statistics on the TI-84 PLUS

• The 1-Var Stats command provides several key statistical values:

  • $\bar{x}$ - The sample mean

  • $ext{Sum of x}$ - Total of all data values

  • $ext{Sum of } x^2$ - Total of squared data values

  • $s$ - Sample standard deviation

  • $<br>ho$ - Population standard deviation

  • $ext{minX}$ - The smallest data value

  • $Q_1$ - First quartile

  • $ext{Med}$ - The median

  • $Q_3$ - Third quartile

  • $ext{maxX}$ - The largest data value

• Example context:

  • A student’s exam scores: 78, 83, 92, 68, 85.

  • Steps to find Mean and Median using TI-84 PLUS:

  1. Press STAT and choose 1:Edit to enter data into L1.

  2. Press STAT and navigate to CALC menu.

  3. Select 1-Var Stats and specify L1.

  4. Obtain Mean and Median.

Mean of Grouped Data

• Approximation methods for calculating the mean for grouped data are discussed in the context of ALEKS (previously Connect Math).