explain each step---**Question Stem:**
3. Find the mean and median salary of 60 employees of a company from the following table
**Table Content:**
| Salary (thousand $) | # of employees |
|-----------------------|----------------|
| 50 | 10 |
| 65 | 40 |
| 70 | 20 |
| 60 | 5 |
| 100 | 13 |
| 90 | 15 |
| 90 | 30 |
| 60 | 12 |
| 40 | 7 |
**Handwritten Notes/Calculations:**
xi | fi | CF | -i
50 | 10 | 10 |
65 | 40 | 50 | 60
70 | 20 | 70 | 1400
60 | 5 | 75 | 3
100 | 13 | [unclear, appears like 1]
90 | 15 | [unclear, appears like 0]
90 | 30 | [unclear, appears like 1]
60 | 12 | [unclear, appears like 1]
40 | 7 | [unclear, appears like 1]
mean = sum(xi * fi) / sum(fi) = 11150 / 152 = 73.355.
N = 152
N/2 = 152 / 2 = 76 -> Even
76th & 77th mean
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We need to find the mean and median salary from the given data table. First, let's note an important discrepancy: the question mentions sixty employees, but the table data actually sums to one hundred fifty-two employees. We'll proceed with the actual data showing one hundred fifty-two employees total.
Now let's calculate the mean salary. The mean is the sum of all salary times frequency values divided by the total number of employees. We multiply each salary by its frequency: forty times seven equals two hundred eighty, fifty times ten equals five hundred, and so on. The sum of all products is eleven thousand one hundred fifty. Dividing by one hundred fifty-two employees gives us a mean of seventy-three point three six thousand dollars.
Now let's find the median. Since we have one hundred fifty-two employees, which is an even number, the median is the average of the seventy-sixth and seventy-seventh values when data is sorted. We calculate cumulative frequencies to locate these positions. The seventy-sixth value falls where cumulative frequency first reaches or exceeds seventy-six, which is in the seventy thousand dollar salary group. Both the seventy-sixth and seventy-seventh values are seventy thousand, so the median is seventy thousand dollars.
Here's a visual representation of our salary data. The histogram shows the frequency distribution of salaries. The blue dashed line represents the mean at seventy-three point three six thousand dollars, while the red dashed line shows the median at seventy thousand dollars. Notice that the mean is slightly higher than the median, which indicates the presence of higher salaries pulling the average upward.
To summarize our findings: The mean salary is seventy-three point three six thousand dollars, calculated by dividing the sum of all salary-frequency products by the total number of employees. The median salary is seventy thousand dollars, found as the average of the seventy-sixth and seventy-seventh values in our sorted data. Since the mean is greater than the median, this indicates a right-skewed distribution where higher salaries pull the average upward from the middle value.