1. **Problem:** Find the mean, median, and mode of the staff salaries given the table. 2. **Step 1: Calculate the mean salary.** - Mean is the total sum of all salaries divided by the total number of staff. - Total staff = $1+1+4+3+3+6+3+6+30+15+6 = 78$ - Total salary = $(1\times60000) + (1\times50000) + (4\times30000) + (3\times20000) + (3\times18000) + (6\times13000) + (3\times11500) + (6\times10000) + (30\times9000) + (15\times8000) + (6\times7000)$ - Calculate total salary: $$60000 + 50000 + 120000 + 60000 + 54000 + 78000 + 34500 + 60000 + 270000 + 120000 + 42000 = 889600$$ - Mean salary = $$\frac{889600}{78} \approx 11405.13$$ 3. **Step 2: Find the median salary.** - Median is the middle value when all salaries are arranged in order. - List salaries with frequencies: - 7000 (6), 8000 (15), 9000 (30), 10000 (6), 11500 (3), 13000 (6), 18000 (3), 20000 (3), 30000 (4), 50000 (1), 60000 (1) - Total staff = 78, median position = $\frac{78+1}{2} = 39.5$ (average of 39th and 40th values) - Counting up to 39th and 40th: - 7000: 1-6 - 8000: 7-21 - 9000: 22-51 - Both 39th and 40th salaries fall in the 9000 range. - Median salary = 9000 4. **Step 3: Find the mode salary.** - Mode is the salary with the highest frequency. - Frequencies: 9000 (30) is highest. - Mode salary = 9000 5. **Problem 2:** Raise salaries of 21 clerks (sales and general) to 10,000 and find new mean, median, mode. 6. **Step 1: Identify clerks and their salaries before raise:** - Sales clerks: 15 at 8000 - General clerks: 6 at 7000 - Total clerks = 21 7. **Step 2: Calculate new total salary after raise:** - Original clerks salary = $(15\times8000) + (6\times7000) = 120000 + 42000 = 162000$ - New clerks salary = $21 \times 10000 = 210000$ - Increase in salary = $210000 - 162000 = 48000$ - New total salary = $889600 + 48000 = 937600$ 8. **Step 3: Calculate new mean:** - Mean = $$\frac{937600}{78} \approx 12020.51$$ 9. **Step 4: Find new median:** - New salaries sorted: - 7000 (0), 8000 (0), 9000 (30), 10000 (6+21=27), 11500 (3), 13000 (6), 18000 (3), 20000 (3), 30000 (4), 50000 (1), 60000 (1) - Median position = 39.5 - Counting: - 9000: 1-30 - 10000: 31-57 - Median salary = 10000 10. **Step 5: Find new mode:** - Frequencies: - 9000 (30), 10000 (27) - Mode remains 9000 (highest frequency) 11. **Problem 3:** If all salaries increase by the same amount, which central tendency measures change? 12. **Step 1: Effect on mean:** - Mean increases by the same amount added. 13. **Step 2: Effect on median:** - Median increases by the same amount. 14. **Step 3: Effect on mode:** - Mode increases by the same amount. 15. **Justification:** - Adding a constant shifts all data points equally, so all measures increase by that constant. 16. **Problem 4:** If one or two clerks' salaries increase, which measure definitely changes? 17. **Answer:** - Mean definitely changes because total sum changes. - Median and mode may or may not change depending on values. 18. **Problem 5:** Foremen's salary increase to raise mean by 5000. 19. **Step 1: Current mean = 11405.13, new mean = 16405.13** - Total staff = 78 - Total salary needed = $16405.13 \times 78 = 1279600$ 20. **Step 2: Current total salary = 889600** - Increase needed = $1279600 - 889600 = 390000$ 21. **Step 3: Only foremen's salary changes, number of foremen = 6** - Increase per foreman = $\frac{390000}{6} = 65000$ 22. **Step 4: Current foreman salary = 13000** - New foreman salary = $13000 + 65000 = 78000$ 23. **Problem 6:** Lay off one foreman and two workmen, effect on mean salary? 24. **Step 1: Calculate total salary and staff before layoff:** - Total salary = 889600 - Staff = 78 25. **Step 2: Salaries of laid off staff:** - Foreman salary = 13000 - Workman salary = 9000 - Total laid off salary = $13000 + 2 \times 9000 = 31000$ 26. **Step 3: New total salary = $889600 - 31000 = 858600$** - New staff = $78 - 3 = 75$ 27. **Step 4: New mean = $\frac{858600}{75} = 11448$** 28. **Step 5: Compare new mean to old mean:** - Old mean = 11405.13 - New mean = 11448 - Mean salary increases 29. **Justification:** - Removing lower salary employees (workmen and foreman) increases average salary. **Final answers:** - Original mean = 11405.13, median = 9000, mode = 9000 - New mean after raise = 12020.51, median = 10000, mode = 9000 - All measures increase by same amount if all salaries increase equally - Mean definitely changes if one or two clerks' salaries increase - Foreman's new salary to raise mean by 5000 = 78000 - Mean salary increases if one foreman and two workmen laid off