Psych Stats Lab Comprehensive Study Guide: Standard Scores, Z-Scores, T-Scores, and Google Sheets Applications

Administrative Announcements & Course Schedule

• Attendance Roster:

  • Kylie
  • Madeline
  • Naomi
  • Samantha
  • Abby Fraley
  • Second Abby

• Office Hours & Departmental Tutoring:

  • Psychology department website features a dedicated page for staff tutoring and TA office hours.
  • Students can access additional SAS TAs for support if standard office hours conflict with their schedules.
  • TAs provide support for homework assignments, exam preparation, and general course concepts.
  • Office hour schedules for TAs (including Lily) are published on both the department webpage and the course syllabus.

• Course Calendar & Milestones:

  • Current Session: Final lab session utilizing Google Sheets.
  • Next Session (Exam Review Day):
  • Held during standard lab time.
  • Comprehensive review covering frequencies, standard deviations, and standard scores (Z-scores and T-scores).
  • Includes review time for content questions relating to Dr. Ltie's main lecture class.
  • All review materials and exercise files presented in class will be uploaded directly to Canvas.
  • Exam Day:
  • Date: October 1 (two weeks from current lab date).
  • Format: Online via Canvas; no in-person lab meeting.
  • Availability Window: Exam opens at 8:00 AM and closes at 11:59 PM.
  • Time Limit: 50 minutes once started.
  • Permitted Resources: Open notes and open PowerPoint slides.
  • Academic Honesty Policy: Independent work only. Use of search engines (e.g., Google) or AI tools (e.g., ChatGPT) is strictly prohibited. Collaboration permitted on homework assignments is explicitly forbidden on exams.

• Google Sheets Exam Submission Protocol:

  • The exam Google Sheets document contains 3 distinct tabs at the bottom of the workbook.
  • Students must navigate through all 3 tabs, complete every exercise table, and answer all embedded questions.
  • Upon completion, download the Google Sheets file as an Excel format (.xlsx).
  • Upload the downloaded Excel file to the Canvas exam assignment page and press submit.

Standard Deviation Review

• Definition: Standard deviation ($s$ or $\text{SD}$) is a single numerical metric defining the typical distance between data points in a distribution and their mean.

• Interpretation: Higher standard deviation values indicate greater variability and a wider spread of individual data points around the center point (mean).

• Calculation Process (Three-Step Procedure):

  1. Calculate the Sum of Squares ($SS$) using either the definitional or computational formula.
  2. Compute the Variance ($s^2$) by dividing the Sum of Squares by the appropriate degrees of freedom ($n - 1$ for samples):      $\text{Variance} = \frac{SS}{n - 1}$
  3. Take the square root of the variance to determine the Standard Deviation ($s$):      $s = \sqrt{\text{Variance}}$

Standard Scores: Z-Scores

• Definition & Function:

  • Standard scores serve as the foundation for inferential statistical tests (such as T-tests in JASP).
  • Raw scores (e.g., SAT, ACT, GRE) lack inherent contextual meaning without a standardized frame of reference.
  • A Z-score quantifies the exact distance between a specific raw data point ($x$) and the distribution mean ($\mu$ or $\bar{x}$) in units of standard deviation ($\sigma$ or $s$).
  • While standard deviation evaluates average dispersion across an entire dataset, a Z-score evaluates the exact relative position of a single individual score.

• Z-Score Formula:$Z = \frac{x - \bar{x}}{s}$   Where:

  • $x$ = specific raw score
  • $\bar{x}$ = sample mean
  • $s$ = sample standard deviation

• Concrete Example (SAT Score Evaluation):

  • Given Data:
  • Tim's SAT score ($x$) = $1340$
  • Mean SAT score ($\bar{x}$) = $1060$
  • Standard Deviation ($s$) = $217$
  • Calculation:     $Z = \frac{1340 - 1060}{217} = \frac{280}{217} = 1.29$
  • Interpretation: Tim's SAT score is $1.29$ standard deviations above the mean.

• Rules of Thumb for Z-Score Interpretation:

  • $\pm 1$ Standard Deviation: Moderately above or moderately below the mean.
  • $\pm 2$ Standard Deviations: Significantly above or significantly below the mean.
  • $\pm 3$ Standard Deviations: Extremely above or extremely below the mean (indicates rare events that seldom occur in a normal distribution).

Standard Scores: T-Scores

• Definition & Purpose:

  • Z-scores frequently produce negative numbers and small decimals, making practical interpretation complex.
  • A T-score is a transformed standard score set to a fixed distribution mean of $50$ and a fixed standard deviation of $10$.
  • T-scores convert scores into positive whole numbers for ease of interpretation and directly prepare data for T-distributions and T-tests in JASP.

• T-Score Formula:$T = 50 + 10 \times Z$   Where:

  • $50$ = target distribution mean
  • $10$ = target distribution standard deviation
  • $Z$ = calculated Z-score

• Concrete Example (IQ/Standard Score Transformation):

  • Given Data:
  • Calculated Z-score ($Z$) = $-0.47$
  • Calculation:     $T = 50 + 10 \times (-0.47) = 50 - 4.7 = 45.33$
  • Interpretation: A Z-score of $-0.47$ converts to a standardized T-score of $45.33$.

Google Sheets Implementation & Formulas

• Dataset Structure:

  • Module 3 lab dataset contains individual test scores located in column B (cells B2 through B51).

• Core Google Sheets Functions:

  • Mean: =AVERAGE(B2:B51)
  • Standard Deviation: =STDEV(B2:B51)

• Z-Score Formula Syntax:

  • Input in cell C2:     =(B2 - $B$52) / $B$53
  • Absolute Referencing ($</code>): Absolute dollar signs locked onto cell coordinates (e.g., <code>$B$52</code> for mean, <code>$B$53 for SD) ensure that dragging formulas down preserves references to summary statistic cells while dynamically updating the raw score row (B2, B3, B4).

• T-Score Formula Syntax:

  • Input syntax:     =50 + 10 * C2     Where C2 represents the cell containing the corresponding Z-score.

• Spreadsheet Automation & Troubleshooting:

  • Autofill Procedure: Hover over the bottom-right corner of a completed cell until the black fill handle (+) appears, then click and drag down across the column range to autofill values.
  • Syntax Verification: Double-click autofilled cells to ensure references point to correct rows.
  • Formula Warnings: Relative frequencies and Sums of Squares ($SS$) calculations in Google Sheets frequently corrupt when dragged; manual double-clicking to verify formula syntax is required.

Questions & Classroom Discussion

• Q: Will exam calendar dates and review materials be available on Canvas?

  • A: Yes. Course calendar dates are published in the syllabus on Canvas, and all exercise review materials covered during class will be posted on Canvas.

• Q: Doesn't converting a Z-score to a T-score completely change the value of Z, and how does that help interpretation?

  • A: Converting to a T-score rescales the data specifically to reference the T-distribution (mean $50$, SD $10$). While it changes raw numeric outputs to eliminate negative numbers and decimals, it preserves exact relative positioning and facilitates downstream procedures like T-tests in JASP.

• Troubleshooting Google Sheets Formatting Issues:

  • Problem: Formulas producing incorrect numbers or displaying raw code.
  • Solution: Check for missing opening equal signs (=), replace invalid equal signs inside expressions with proper addition operators (+), and ensure range boundaries explicitly highlight the entire data column (B2:B51). Remove stray calculation boxes below summary statistics.

• Discussion on Regional Fast Food Establishments:

  • Comparison between In-N-Out Burger and Culver's.
  • Culver's originates in Wisconsin; the closest branch to the local campus is located in Phoenix, Arizona.
  • Culinary hierarchy discussed places Culver's at top tier (#1), with In-N-Out ranked secondary (#2 or #3).