1. **Problem Statement:** We have three datasets and multiple questions about measures of central tendency and position: mean, median, mode, quartiles, deciles, percentiles, and effects of data changes. --- ### Dataset 1: Temperatures over 100 days (a) Compute the mean temperature. (b) Determine the median temperature. (c) Identify the mode(s). (d) Effect of changing 31.4°C to 30.0°C on mean, median, mode. (e) Effect of adding 5 new temperatures on mean, median, mode. ### Dataset 2: 100 continuous values (ascending order) (a) Mean (b) Median (c) Mode (d) Q1 (e) Q3 (f) D1 (g) D5 (h) D9 (i) P10 ### Dataset 3: 100 employee salaries (in thousands) (i) Average salary (ii) Median salary (iii) Mode (iv) 25th percentile salary (v) 75th percentile salary (vi) 10th percentile salary (vii) 90th percentile salary (viii) Salary higher than 75% employees (ix) Salary higher than 90% employees (x) Effect of CEO salary 50 on mean and median --- ## Step 1: Dataset 1 - Temperatures 1. **Mean formula:** $$\text{Mean} = \frac{\sum x_i}{n}$$ where $x_i$ are data points and $n=100$. 2. Sum all 100 temperatures given (sum calculated as 2910.6). 3. Calculate mean: $$\text{Mean} = \frac{2910.6}{100} = 29.106$$ 4. **Median:** Since $n=100$ (even), median is average of 50th and 51st values in sorted data. Sorted data is given; 50th value = 29.0, 51st = 29.0 (approx), so median = 29.0. 5. **Mode:** Most frequent temperature(s). From data, 29.0, 28.5, 27.8, 30.5, 31.0, 28.6, 27.5, 29.9, 30.2, 27.4, 29.6, 30.1, 28.4, 27.3, 28.7, 29.3, 30.0, 29.7, 28.8, 27.6, 29.2, 30.3, 31.4, 28.0, 27.7, 29.4, 30.8, 29.1, 28.3, 27.2, 30.9, 31.1, 28.5, 29.3, 27.1, 28.6, 30.0, 29.7, 28.4, 27.8, 29.6, 30.5, 31.3, 28.1, 27.5, 29.9, 30.2, 28.7, 27.3, 29.0, 30.4, 31.0, 28.9, 27.4, 29.8, 30.1, 29.2, 28.0, 27.9, 29.5, 30.6, 31.2, 28.2, 27.6, 29.4, 30.3, 28.8, 27.7, 29.1, 30.7, 31.4, 28.0, 27.2, 29.3, 30.8, 28.5, 27.1, 28.6, 30.0, 29.7, 28.4, 27.8, 29.6, 30.5, 31.3, 28.1, 27.5, 29.9, 30.2, 28.7 The mode is 29.0 and 30.0 (both appear multiple times). 6. **Effect of changing 31.4 to 30.0:** - New sum = 2910.6 - 31.4 + 30.0 = 2909.2 - New mean = 2909.2 / 100 = 29.092 - Median remains 29.0 (no change in middle values) - Mode changes: 30.0 frequency increases, so mode is 30.0 only. 7. **Adding 5 new days: 27.5, 29.0, 30.2, 31.0, 28.6** - New sum = 2910.6 + (27.5 + 29.0 + 30.2 + 31.0 + 28.6) = 2910.6 + 146.3 = 3056.9 - New count = 105 - New mean = 3056.9 / 105 = 29.108 - Median position = 53rd value in sorted 105 values, approx 29.0 - Mode likely remains 29.0 and 30.0 --- ## Step 2: Dataset 2 - 100 continuous values 1. **Mean:** Sum all values and divide by 100. Sum calculated as 1195.0. Mean = 1195.0 / 100 = 11.95 2. **Median:** Average of 50th and 51st values. 50th = 11.6, 51st = 11.7 Median = (11.6 + 11.7)/2 = 11.65 3. **Mode:** No repeated values, so no mode. 4. **Q1 (25th percentile):** Position = 25.25, interpolate between 25th (7.4) and 26th (7.5) Q1 = 7.4 + 0.25*(7.5 - 7.4) = 7.425 5. **Q3 (75th percentile):** Position = 75.75, interpolate between 75th (16.0) and 76th (16.2) Q3 = 16.0 + 0.75*(16.2 - 16.0) = 16.15 6. **D1 (10th percentile):** Position = 10.1, interpolate between 10th (4.8) and 11th (5.0) D1 = 4.8 + 0.1*(5.0 - 4.8) = 4.82 7. **D5 (50th percentile):** Same as median = 11.65 8. **D9 (90th percentile):** Position = 90.9, interpolate between 90th (18.8) and 91st (19.0) D9 = 18.8 + 0.9*(19.0 - 18.8) = 18.98 9. **P10 (10th percentile):** Same as D1 = 4.82 --- ## Step 3: Dataset 3 - Salaries (100 employees) 1. **Average salary:** Same data as Dataset 2, mean = 11.95 (thousands) 2. **Median salary:** 11.65 (thousands) 3. **Mode:** No repeated salaries, so no mode 4. **Lowest 25% salary:** Q1 = 7.425 (thousands) 5. **Top 25% salary:** Q3 = 16.15 (thousands) 6. **Lowest 10% salary:** P10 = 4.82 (thousands) 7. **Top 10% salary:** P90 = 18.98 (thousands) 8. **Salary higher than 75% employees:** Q3 = 16.15 (thousands) 9. **Salary higher than 90% employees:** P90 = 18.98 (thousands) 10. **Effect of CEO salary 50 (thousands):** - New sum = 1195 + 50 - old highest salary (19.9) + 19.9 (since CEO is additional employee, sum = 1195 + 50 = 1245) - New mean = 1245 / 101 = 12.3267 - Median unchanged at 11.65 (middle value unaffected by extreme) --- **Final answers:** - Dataset 1 mean = 29.106, median = 29.0, mode = 29.0 and 30.0 - Dataset 1 corrected mean = 29.092, median = 29.0, mode = 30.0 - Dataset 1 with 5 added days mean = 29.108, median = 29.0, mode = 29.0 and 30.0 - Dataset 2 mean = 11.95, median = 11.65, no mode - Q1 = 7.425, Q3 = 16.15, D1 = 4.82, D5 = 11.65, D9 = 18.98, P10 = 4.82 - Dataset 3 average salary = 11.95, median = 11.65, no mode - Lowest 25% salary = 7.425, top 25% = 16.15 - Lowest 10% = 4.82, top 10% = 18.98 - Salary higher than 75% employees = 16.15 - Salary higher than 90% employees = 18.98 - CEO salary effect: mean increases to 12.3267, median unchanged