Midterm Exam Review Guide: Frequency Distributions, Sums of Squares, Standard Deviation, Z-Scores, and T-Scores

Midterm Exam Logistics and Course Schedule

• ** File Navigation and Submission Procedure **:

  • The midterm review sheet is accessible on Canvas under Modules -> Midterm Exam Review via the left-side link.
  • Students complete the assignment in Google Sheets and must export/download the completed review or exam as an Excel file (.xlsx format) prior to submitting it to Canvas.

• ** Exam Structure **:

  • The exam directly mirrors the Google Sheets review file and contains three distinct tabs at the bottom of the workbook:
    • ** Tab 1 **: Frequency distributions (Frequency, Relative Frequency, Percentage, Cumulative Frequency).
    • ** Tab 2 **: Standard deviation and sums of squares (Definitional Formula, Raw Score Formula, and built-in spreadsheet functions).
    • ** Tab 3 **: Descriptive transformations ($z$-scores and $T$-scores).

• ** Time Restrictions and Availability Window **:

  • The exam is available between $8:00\,\text{AM}$ and $11:59\,\text{PM}$.
  • Once initiated, students are allotted $50\,\text{minutes}$ to complete the exam.
  • ** Hard Cutoff Exception **: The exam closes strictly at $11:59\,\text{PM}$. If a student begins the exam at $11:30\,\text{PM}$, the system will enforce the final cutoff, leaving only $29\,\text{minutes}$ of working time.

• ** Permitted Resources and Academic Integrity **:

  • The exam is open-note and open-slides; PowerPoint lecture slides may be referenced.
  • Students must complete the work entirely independently.
  • Consulting external resources (such as Google or AI programs) or communicating with peers is strictly prohibited.
  • Utilizing AI programs to generate exam responses is formally classified as plagiarism under the university academic handbook.

• ** Upcoming Schedule **:

  • ** Next Week **: No in-person lab session; students complete the midterm exam independently online.
  • ** Following Week **: No in-person lab session due to the Torrey Conference.
  • In-person lab meetings resume in three weeks.

Frequency Distributions and Spreadsheet Formulas

• ** Frequency ($f$) **:

  • Represents the total count of observations or participants receiving a specific score within a dataset (e.g., $6$ participants receiving a specific test score).

• ** Relative Frequency **:

  • ** Definition **: The proportion of times a specific event or score occurs relative to the total number of observations ($n$).
  • ** Mathematical Formula **:         $\text{Relative Frequency} = \frac{f}{n}$
  • ** Spreadsheet Syntax **: Requires explicit cell selection divided by the dataset sum. The formula must reference specific cell identifiers (e.g., =C3/SUM(...)).

• ** Percentage **:

  • ** Definition **: The relative frequency converted to a standard scale out of $100$.
  • ** Mathematical Formula **:         $\text{Percentage} = \text{Relative Frequency} \times 100$
  • ** Spreadsheet Syntax **: Involves multiplying the calculated relative frequency cell by $100$ using the asterisk operator (e.g., =C3*100).

• ** Cumulative Frequency **:

  • ** Definition **: The running total of frequencies accumulated row-by-row across the distribution table.
  • ** First Entry **: The cumulative frequency of the initial row is equal to the individual frequency of that row (e.g., $6$).
  • ** Subsequent Entries **: Equal to the sum of the previous row's cumulative frequency and the current row's individual frequency (e.g., $6 + 5 = 11$).
  • ** Spreadsheet Implementation **: Can be calculated manually (=6+5) or via reference chaining (= [Previous Cumulative Frequency Cell] + [Current Frequency Cell]). Dragging the chained formula down the column expedites computation.
  • ** Verification Rule **: The final entry in the cumulative frequency column must equal the sum total ($n$) of all observed frequencies.

Sums of Squares, Variance, and Standard Deviation

• ** Definitional Formula Method **:

  • ** Mathematical Formula for Sum of Squares ($SS$) **:         $SS = \sum (x - \bar{x})^2$
  • ** Step 1: Calculate Mean ($\bar{x}$) **:
    • Spreadsheet syntax: =AVERAGE(range) (e.g., =AVERAGE(B6:B11)).
  • ** Step 2: Calculate Deviation Scores ($x - \bar{x}$) **:
    • Subtract the mean from each raw score $x$.
    • Spreadsheet syntax: =B6 - $B$12.
    • Absolute cell locking: Dollar signs ($) must lock the mean cell row and column so the reference remains static when dragged downward.
  • ** Step 3: Calculate Squared Deviation Scores ($(x - \bar{x})^2$) **:
    • Square each deviation score.
    • Spreadsheet syntax: = cell^2 using the caret (^) exponent operator.
  • ** Step 4: Calculate Sum of Squares ($SS$) **:
    • Sum all squared deviation scores.
    • Spreadsheet syntax: =SUM(range) across the squared deviations column.
  • ** Step 5: Calculate Sample Variance ($s^2$) **:
    • Divide the calculated $SS$ by the degrees of freedom ($n - 1$).
    • For a sample size of $n = 6$ subjects, the denominator is $6 - 1 = 5$:             $s^2 = \frac{SS}{n - 1}$
    • Spreadsheet syntax: = SS_cell / 5.
  • ** Step 6: Calculate Sample Standard Deviation ($s$) **:
    • Take the square root of the sample variance:             $s = \sqrt{s^2}$
    • Spreadsheet syntax: =SQRT(variance_cell).

• ** Raw Score Formula Method **:

  • ** Mathematical Formula for Sum of Squares ($SS$) **:         $SS = \sum x^2 - \frac{(\sum x)^2}{n}$
  • ** Step 1: Compute Squared Raw Scores ($x^2$) **:
    • Square each individual raw score $x$.\n * Spreadsheet syntax: `= cell^2` and drag down.\n * ** Step 2: Compute Sum of Squared Raw Scores ($\sum x^2$) **:\n * Sum the squared raw score column.\n * Spreadsheet syntax: `=SUM(x^2_range)`.\n * ** Step 3: Compute Sum of Raw Scores ($\sum x$) **:\n * Sum the original raw score column.\n * Spreadsheet syntax: `=SUM(x_range)`.\n * ** Step 4: Compute Square of Total Raw Sum ((\sum x)^2$) **:\n * Square the value obtained in Step 3.\n * Spreadsheet syntax: `= (sum_x_cell)^2`.\n * ** Step 5: Compute Correction Factor (\frac{(\sum x)^2}{n}) **:\n * Divide the squared raw sum by the total sample size n$(for$n = 6$subjects, divide by$6$, NOT$n - 1).\n * Spreadsheet syntax: `= (sum_x_squared_cell) / 6`.\n * ** Step 6: Subtract Correction Factor to Obtain SS **:\n * Subtract Step 5 result from Step 2 result (`\sum x^2`).\n * Spreadsheet syntax: `= sum_x2_cell - correction_factor_cell`.\n * *Check*: This value must identically match the SS calculated using the definitional formula.\n * ** Step 7: Calculate Sample Variance (s^2) **:\n * Divide SS$by$n - 1$($5$for$n = 6).\n * ** Step 8: Calculate Sample Standard Deviation (s) **:\n * Spreadsheet syntax: `=SQRT(variance_cell)`.\n\n* ** Built-In Spreadsheet Function Method **:\n * ** Function Syntax **: `=STDEV(range)`\n * Highlight raw score data cells directly inside the parentheses (e.g., `=STDEV(B42:B47)` or `=STDEV(B2:B31)`).\n * Press Enter to yield standard deviation directly; use this to cross-verify manual formula calculations.\n\n# Standardization: Z-Scores and T-Scores\n\n* ** Z-Score Transformation **:\n * ** Mathematical Formula **:\n        z = \frac{x - \bar{x}}{s}\n * ** Dataset Parameters (Review Example) **:\n * Mean score (\bar{x}$):$59.4480\n * Standard deviation (s$):$18.30\n * ** Spreadsheet Formula Syntax **:\n * `=(x_cell - $mean_cell) / $sd_cell`\n * ** Operational Requirements **:\n * Parentheses are mandatory around the numerator (x - \bar{x}) to enforce correct mathematical order of operations.\n * Dollar sign absolute reference locking (`$`) must be applied to the mean and standard deviation cell identifiers so the reference cells remain fixed when dragging the formula down standard test score columns.\n\n* ** T-Score Transformation **:\n * ** Mathematical Formula **:\n        T = 50 + 10 \times z\n * ** Spreadsheet Formula Syntax **:\n * `=50 + 10 * z_cell`\n * ** Operational Requirements **:\n * An explicit asterisk operator (`*`) is required for multiplication between the scale factor 10$and the referenced$z-score cell.\n\n# Questions and Discussion\n\n* ** Student Clarification on Raw Score Calculation **:\n * ** Prompt **: A student inquired about which specific cells were selected to derive the sum of squares (SS) in the raw score formula table.
  • ** Response **: It was demonstrated that cell C32 (representing \sum x^2$$) was selected, from which the adjusted value (\sum x)^2 / n was subtracted.

• ** Student Clarification on Standard Deviation Function **:

  • ** Prompt **: A student requested a demonstration of the standard deviation function syntax due to screen size visibility issues.
  • ** Response **: Syntax was shown as =STDEV() in uppercase, highlighting the score range B2:B31 inside parentheses and pressing Enter to compute.

• ** Exam Grading Criteria Clarification **:

  • ** Formula Evaluation **: Full credit on relative frequency and percentage sections requires formula syntax linking corresponding cells. Simple numerical entry without dynamic cell references will not satisfy formula submission checks.
  • ** Cumulative Frequency Flexibility **: Manual addition (=6+5) and dynamic formula references (= previous_cell + current_cell) are both accepted for full credit.