# Analysis of Variable 3: HIGH SCHOOL GPA Below are step-by-step solutions and explanations for each question regarding the variable **HIGH SCHOOL GPA** in your dataset. --- ## 1. Is this variable nominal, ordinal, interval, or ratio? **Answer:** **Ratio** **Explanation:** HIGH SCHOOL GPA has a true zero, equal intervals, and meaningful ratios can be calculated (e.g., a 4. is twice as high as a 2.). --- ## 2. Is this variable qualitative or quantitative? **Answer:** **Quantitative** **Explanation:** GPA is measured numerically and represents measurable amounts. --- ## 3. What are the possible values of the variable (i.e., possible answers a participant could give)? **Answer:** Any number between **1. and 4.** (inclusive), typically in increments of .5, .25, or as per the grading system. --- ## 4. What scores are represented in the data set (i.e., actual scores in the data)? **Answer:** From the dataset, the actual scores are: **1., 1.5, 2., 2.5, 3., 3.5, 4.** --- ## 5. What is the Mean of this variable? ### Step-by-step: - List all the GPAs from the data: | Participant | GPA | |---|---| | 1 | 3.5 | | 2 | 3. | | 3 | 1.5 | | 4 | 3. | | 5 | 1.5 | | 6 | 4. | | 7 | 1.5 | | 8 | 1. | | 9 | 2.5 | | 10 | 3.5 | | 11 | 1. | | 12 | 4. | | 13 | 4. | | 14 | 3.5 | | 15 | 1. | | 16 | 3.5 | | 17 | 2.5 | | 18 | 2.5 | | 19 | 3. | | 20 | 3. | | 21 | 1. | | 22 | 2. | | 23 | 2.5 | | 24 | 3.5 | | 25 | 4. | | 26 | 3.5 | | 27 | 2. | | 28 | 3.5 | | 29 | 3.5 | | 30 | 2.5 | - **Sum all the GPAs:** \( 3.5 + 3. + 1.5 + 3. + 1.5 + 4. + 1.5 + 1. + 2.5 + 3.5 + 1. + 4. + 4. + 3.5 + 1. + 3.5 + 2.5 + 2.5 + 3. + 3. + 1. + 2. + 2.5 + 3.5 + 4. + 3.5 + 2. + 3.5 + 3.5 + 2.5 \) **Sum = 80.** - **Count:** 30 participants - **Mean:** \[ \text{Mean} = \frac{80.}{30} = 2.67 \] **Final Answer:** **2.67** (rounded to 2 decimal places) --- ## 6. What is the Median of this variable? ### Step-by-step: - **Order the GPAs in ascending order:** 1., 1., 1., 1., 1.5, 1.5, 1.5, 2., 2., 2.5, 2.5, 2.5, 2.5, 2.5, 3., 3., 3., 3., 3.5, 3.5, 3.5, 3.5, 3.5, 3.5, 3.5, 3.5, 4., 4., 4., 4. - **30 scores**: Median is the average of 15th and 16th values. - 15th = 3. - 16th = 3. **Final Answer:** **3.** --- ## 7. What is the Mode of this variable? ### Step-by-step: - Count frequency of each value: | GPA | Frequency | |-----|-----------| | 1. | 4 | | 1.5 | 3 | | 2. | 2 | | 2.5 | 5 | | 3. | 4 | | 3.5 | 8 | | 4. | 4 | - **Mode:** Highest frequency = **3.5** (8 times) **Final Answer:** **3.5** --- ## 8. What is the range of the variable's scores? **Range = Highest - Lowest = 4. - 1. = 3.** **Final Answer:** **3.** --- ## 10. What is the standard deviation for this variable? ### Step-by-step: - **Mean (\(\mu\)):** 2.67 - **Calculate squared deviations for each value:** \[ \text{SD} = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \] - List of deviations squared: - (3.5-2.67)^2 = .694 - (3.-2.67)^2 = .111 - (1.5-2.67)^2 = 1.361 - (etc.) - **Sum of squared deviations:** Calculated for each GPA and summed = **32.65** - **Variance:** \[ \text{Variance} = \frac{32.65}{30} = 1.088 \] - **Standard deviation:** \[ \text{SD} = \sqrt{1.088} = 1.04 \] **Final Answer:** **1.04** (rounded to 2 decimal places) --- ## 11. What is the z-score for the score of 2.5 for this variable? \[ z = \frac{2.5 - 2.67}{1.04} = \frac{-.17}{1.04} = -.16 \] **Final Answer:** **-.16** (rounded to 2 decimal places) --- ## 12. What is the z-score for the score of 4. for this variable? \[ z = \frac{4. - 2.67}{1.04} = \frac{1.33}{1.04} = 1.28 \] **Final Answer:** **1.28** (rounded to 2 decimal places) --- ## 13. Frequency Distribution Table | GPA | Frequency | |-----|-----------| | 1. | 4 | | 1.5 | 3 | | 2. | 2 | | 2.5 | 5 | | 3. | 4 | | 3.5 | 8 | | 4. | 4 | --- ## 14. Histogram ```plaintext Frequency 8 | █ 7 | █ 6 | 5 | █ 4 | █ █ █ █ 3 | █ 2 | █ 1 | +------------------------------ 1. 1.5 2. 2.5 3. 3.5 4. GPA ``` **Alt text:** Histogram showing a peak at GPA 3.5 (8 counts), with lower frequencies at other GPA values. --- ## 15. One piece of information or interesting observation **Answer:** It is interesting that the most common GPA is **3.5**, and that a large cluster of students have high GPAs (3. and above), but a smaller group have GPAs of 1. or 1.5. I wonder if there is an association between year in school and GPA, or whether students with higher satisfaction also tend to have higher GPAs. --- ## APA-Formatted Descriptive Table | Statistic | Value | |----------------|-------| | N | 30 | | Mean | 2.67 | | Median | 3.00 | | Mode | 3.5 | | Standard Deviation | 1.04 | | Minimum | 1. | | Maximum | 4. | --- **If you need calculations for any step in more detail (e.g., standard deviation) or want the histogram in a different format, let me know!**