Relative Standing Solution GuideTEKS S(4)(E) Standard Worked Solution: Relative Standing Topic Measures of Relative Position (z-scores) 5 Question Prompt Summary Statistics Comparison Determine which has a higher relative position: Condition i: A score of 75 on a statistics test with a mean of 60 and a standard deviation of 10. Condition ii: A score of 36 on an accounting test with a mean of 30 and a variance of 16. A. i only B. ii only C. Neither D. Cannot be determined Step-by-Step Mathematical Analysis Relative Position Formula: \(z = \frac{x - \mu}{\sigma}\) Scenario i: Statistics Test Given \(\sigma\) Given values: \(x = 75\), \(\mu = 60\), \(\sigma = 10\) \[z_i = \frac{x - \mu}{\sigma} = \frac{75 - 60}{10} = \frac{15}{10} = \mathbf{1.5}\] The statistics score lies exactly 1.5 standard deviations above the mean. Scenario ii: Accounting Test Trap: Variance Given! Given values: \(x = 36\), \(\mu = 30\), \(\sigma^2 = 16\) Find standard deviation: \(\sigma = \sqrt{16} = 4\) \[z_{ii} = \frac{x - \mu}{\sigma} = \frac{36 - 30}{4} = \frac{6}{4} = \mathbf{1.5}\] The accounting score also lies exactly 1.5 standard deviations above the mean. C Correct Answer Option C: Neither has a higher relative position Comparison: \(z_i = 1.5 = z_{ii}\) Identical standard score What is Relative Position? Raw scores (75 vs 36) cannot be directly compared across different distributions. A z-score standardizes them into standard deviation units from the mean, enabling an apples-to-apples comparison. The Distractor Trap (Option A) Students who forget to convert variance to standard deviation divide by 16: \(z = \frac{36 - 30}{16} = 0.375\) (Incorrect!) This causes them to incorrectly pick A (i only). Core Rule: Always verify whether given dispersion is variance (\(\sigma^2\)) or standard deviation (\(\sigma\)). Percentile Equivalent: \(P_{93.3}\)
Relative Standing Review WorksheetName: Date: Period: TEKS S(4)(E) Test Prep Relative Standing Review Standard Formula \(z = \frac{x - \mu}{\sigma}\) where \(\sigma = \sqrt{\text{Variance}}\) 1 Determine which has a higher relative position. Show all calculation steps. 4 pts Condition i: A score of 82 on a psychology test with a mean of 70 and a standard deviation of 8. Condition ii: A score of 48 on an economics test with a mean of 40 and a variance of 25. Work for Condition i (\(z_i\)): Work for Condition ii (\(z_{ii}\)): A i only B ii only C Neither D Cannot be determined 2 Determine which has a higher relative position. Show all calculation steps. 4 pts Condition i: A score of 91 on a biology test with a mean of 79 and a standard deviation of 6. Condition ii: A score of 74 on a chemistry test with a mean of 62 and a variance of 36. Work for Condition i (\(z_i\)): Work for Condition ii (\(z_{ii}\)): A i only B ii only C Neither D Cannot be determined 3 Error Analysis & Conceptual Justification 2 pts A classmate claims that in Question 2, Condition i has a higher relative standing because "a standard deviation of 6 is much smaller than a variance of 36, so the score of 91 must be more exceptional." Identify the statistical error in your classmate's reasoning. Remember: z-scores measure how many standard deviations an observation falls above or below the mean. Total: 10 Points
Relative Standing Review KeyTeacher Answer Key & Rubric Relative Standing Review Key Curriculum TEKS S(4)(E) Assessment 1 Psychology vs. Economics Correct: B (ii only) 4 pts Condition i (Psychology): \(z_i = \frac{82 - 70}{8} = \frac{12}{8} = \mathbf{1.50}\) 1.50 standard deviations above mean Condition ii (Economics): \(\sigma = \sqrt{25} = 5 \implies z_{ii} = \frac{48 - 40}{5} = \frac{8}{5} = \mathbf{1.60}\) 1.60 standard deviations above mean Comparison: Because \(z_{ii} (1.60) > z_i (1.50)\), Condition ii has a higher relative standing. Common error: Dividing by 25 yields \(z = 0.32\), wrongly picking A. 2 Biology vs. Chemistry Correct: C (Neither) 4 pts Condition i (Biology): \(z_i = \frac{91 - 79}{6} = \frac{12}{6} = \mathbf{2.00}\) 2.00 standard deviations above mean Condition ii (Chemistry): \(\sigma = \sqrt{36} = 6 \implies z_{ii} = \frac{74 - 62}{6} = \frac{12}{6} = \mathbf{2.00}\) 2.00 standard deviations above mean Comparison: Both standard scores are identical (\(z_i = z_{ii} = 2.00\)). Neither student has a higher relative standing; both fall in the 97.7th percentile. 3 Error Analysis & Conceptual Justification 2 pts Exemplar Student Response: "The classmate erroneously compared standard deviation directly to variance without standardizing the units. Variance is in squared units (\(\sigma^2 = 36\)), so the standard deviation of Test ii is \(\sigma = \sqrt{36} = 6\). Both tests have the exact same standard deviation (\(6\)), and both students scored exactly \(2\) standard deviations above their respective means (\(z = 2.0\)). Neither score is more exceptional." TEKS S(4)(E) Scoring Rubric Breakdown Calculation (4 pts each): 1 pt for finding \(\sigma = \sqrt{\text{Var}}\), 1 pt per correct \(z\)-score, 1 pt for final choice. Explanation (2 pts): 1 pt for stating variance must be square rooted; 1 pt for noting identical spread/z-scores. Common Misconception: Check for students who divide by variance directly without taking the square root first. Relative Standing Review Answer Key TEKS S(4)(E) Mastery
