ch 2

CENTRAL TENDENCY AND VARIABILITY

LEARNING OBJECTIVES

  • Create and interpret frequency distribution graphs and tables.
  • Identify when to use the mean, median, or mode when describing a distribution's central tendency.
  • Compute and interpret the mean, median, and mode.
  • Identify when to use the range or standard deviation when describing a distribution's variability.
  • Compute and interpret the standard deviation for a population or sample.
  • Construct a scientific conclusion based on central tendency and variability statistics.

INTRODUCTION

  • This chapter describes the use of graphs and measures of central tendency and variability in answering research questions.
  • You will learn about:
    • Three types of graphs:
    • Frequency bar graphs
    • Boxplots
    • Data plots
    • Three measures of central tendency:
    • Mean
    • Median
    • Mode
    • Two measures of variability:
    • Range
    • Standard deviation
  • Graphs, measures of central tendency, and variability illustrate the use of data in answering research questions.

EVIDENCE OF BENEFITS FROM HUMAN TOUCH

  • Research indicates that human touch benefits psychological and physical health:
    • Massage: Associated with reductions in anxiety, depression, and pain (Moyer et al., 2004).
    • Skin-to-skin contact: Can help infants gain weight (Boo & Jamli, 2007).
    • Touch: Can improve immune system function (Field, 2010).
  • Therapeutic Touch (TT): A controversial treatment involving no physical contact where practitioners claim to manipulate human energy fields (HEFs) for relaxation, pain reduction, and immune enhancement (therapeutictouch.org).

TESTING THERAPEUTIC TOUCH

  • Emily Rosa's Experiment: Investigated claims of TT practitioners by testing their ability to sense HEFs.
    • Procedure:
    • TT practitioners faced a divider not allowing them to see their own or Rosa's hands.
    • Rosa randomly placed her hand above one of their hands and practitioners guessed the hand location.
    • Each practitioner completed 10 trials.
  • Hypothesis: If practitioners can sense HEFs, they should be correct significantly more than chance (5 out of 10).
    • Expectations:
      • Close to 10 correct responses would support the hypothesis.
      • Around 5 correct responses would indicate guessing.

FREQUENCY DISTRIBUTION GRAPHS AND TABLES

Bar Graphs
  • Before analyzing data, it's essential to define supportive evidence: if practitioners could sense HEFs, they should score near 10 correct.
  • Figures 2.1a and 2.1b illustrate:
    • Figure 2.1a: Supportive evidence with results clustering near 10.
    • Figure 2.1b: Disconfirming evidence clustering around 5, indicative of chance performance.
  • Identifying supportive evidence before examining actual data minimizes biases.
DATA ANALYSIS
  • Collected data from 28 TT practitioners—scores of correct responses include:
    • 1, 2, 3 (×8), 4 (×5), 5 (×7), 6 (×2), 7 (×3), 8.
  • Presenting data as a list is unhelpful; multiple data analysis methods improve interpretation.
  • Figure 2.2 shows a frequency bar graph of the raw data.
  • A histogram reveals the 'center' around 5 and the variability of scores.

BOXPLOTS AND ADDITIONAL GRAPH TYPES

Boxplots
  • Figure 2.4: Boxplot summarizes data using percentiles.
    • 25th Percentile: Bottom of the box represents a score of 3, indicating 25% of scores are at or below this.
    • Median: Middle score, or 50th percentile (4) lies at the height of the line inside the box.
    • 75th Percentile: Top of the box indicates a score of 5, suggesting 75% of all scores are equal to or below this value.
    • Minimum and Maximum: Displayed by vertical lines extending from the box (1 min, 8 max).
Tables
  • Table 2.1: Provides a comprehensive frequency table displaying counts, percentages, and cumulative percentages.
    • Example: 25% scored 5, 28.6% scored 3; cumulative percentages help understand distribution thresholds.

MEASURES OF CENTRAL TENDENCY

Defining Central Tendency
  • Three central tendency measures:
    • Mean: Arithmetic average.
    • Median: 50th percentile score.
    • Mode: Most frequently occurring score.
  • Each measure is influenced by the distribution shape and scale of measurement (nominal, ordinal, interval, ratio).
Assessing each Measure
  • For nominal data: Use Mode (e.g., undergraduate majors, psychology as most common at 17).
  • For ordinal data: Use Mode or Median:
    • Example: Class ranks summarized by either measure; median class rank can be considered more reliable.
Interval or Ratio Data
  • Use Mean, Median, or Mode.
  • Example: Text messages sent in class—outlier significantly affects mean calculation.
  • Table 2.2 summarizes measures.

COMPUTING MEASURES OF CENTRAL TENDENCY

Application to TT Data
  • Computing Mean (4.39), Median (4), and Mode (3) for TT practitioners, identify the best summary: when the distribution is symmetrical, mean is preferred.

VARIABILITY: RANGE OR STANDARD DEVIATION

Defining Variability
  • Range: Difference between highest and lowest scores (8 - 1 = 7).
    • A poor measure due to its insensitivity to middle scores.
  • Standard Deviation: Represents typical distance of scores from the mean (σ).
Steps for Computing Standard Deviation
  1. Compute sum of squared deviation scores (SS).
  2. Compute variance (σ²).
  3. Compute standard deviation (σ).
  • Detailed steps and formulas for both population and sample standard deviations are outlined.
Conclusion Drawing
  • Review statistical evidence alongside findings from other studies. Claims of TT and HEFs by practitioners are questionable based on this data. Evidence signals that claims may lack support, as only 1 of 28 practitioners performed at an expected level of accuracy.
  • Citing other studies suggests TT may lead to no more than placebo effects.