Question

Calculate the mean of the annual salaries of the occupations in the bar graph. SHOW YOUR WORK/STEPS.
61,220+61050+56630+53910+51930+42090+47600+36700+50620=410750/ 9=45638
Calculate the median of the annual salaries of the occupations in the bar graph. SHOW YOUR WORK/STEPS.
36700, 42090, 47600, 50620, 51930, 53910, 56630, 61050, 61220 the median is 51,930
Determine the mode(s) of the annual salaries of the occupations in the bar graph. SHOW YOUR WORK/STEPS.
How spread out are the salaries? (Find the range.) SHOW YOUR WORK/STEPS.
Calculate the interquartile range (IQR) of the occupation salaries. SHOW YOUR WORK/STEPS.
Calculate the variance of the occupation salaries.SHOW YOUR WORK/STEPS.
Calculate the standard deviation of the occupation salaries.SHOW YOUR WORK/STEPS.

Answer 1

To analyze the annual salaries of the occupations, let's go through each calculation step-by-step:

### Given Salaries:
- 61,220
- 61,050
- 56,630
- 53,910
- 51,930
- 42,090
- 47,600
- 36,700
- 50,620

### 1. Calculate the Mean
The mean is calculated by adding all the salaries and dividing by the number of salaries.

\[
\text{Mean} = \frac{\text{Sum of Salaries}}{\text{Number of Salaries}} = \frac{61,220 + 61,050 + 56,630 + 53,910 + 51,930 + 42,090 + 47,600 + 36,700 + 50,620}{9}
\]

Calculating the sum:

\[
61,220 + 61,050 + 56,630 + 53,910 + 51,930 + 42,090 + 47,600 + 36,700 + 50,620 = 410,750
\]

Now, dividing by the number of entries (9):

\[
\text{Mean} = \frac{410,750}{9} \approx 45,638.89
\]

### 2. Calculate the Median
To find the median, we first need to sort the salaries in ascending order:

\[
36,700, 42,090, 47,600, 50,620, 51,930, 53,910, 56,630, 61,050, 61,220
\]

Since there are 9 values (odd), the median is the middle value, which is the 5th value in the sorted list:

\[
\text{Median} = 51,930
\]

### 3. Determine the Mode
The mode is the value that appears most frequently in the data set. Looking at our salaries:

- 61,220
- 61,050
- 56,630
- 53,910
- 51,930
- 42,090
- 47,600
- 36,700
- 50,620

Each salary is unique; therefore, there is **no mode**.

### 4. Calculate the Range
The range is found by subtracting the smallest salary from the largest salary:

\[
\text{Range} = \text{Max} - \text{Min} = 61,220 - 36,700 = 24,520
\]

### 5. Calculate the Interquartile Range (IQR)
To find the IQR, we need to determine the first (Q1) and third quartiles (Q3).

**Sorted Salaries**:
\[
36,700, 42,090, 47,600, 50,620, 51,930, 53,910, 56,630, 61,050, 61,220
\]

- **Q1 (the median of the first half)**: (first 4 values; Q1 is the median of 36,700, 42,090, 47,600, 50,620)
- Q1 = (42,090 + 47,600) / 2 = 44,845

- **Q3 (the median of the second half)**: (last 4 values; Q3 is the median of 51,930, 53,910, 56,630, 61,050)
- Q3 = (53,910 + 56,630) / 2 = 55,270

Now, calculate the IQR:

\[
\text{IQR} = Q3 - Q1 = 55,270 - 44,845 = 10,425
\]

### 6. Calculate the Variance

First, calculate the deviations from the mean, square them, and then find the average of those squared deviations.

\[
\text{Mean} = 45,638.89
\]

Now, the variances:

- \((61220 - 45638.89)^2\)
- \((61050 - 45638.89)^2\)
- \((56630 - 45638.89)^2\)
- \((53910 - 45638.89)^2\)
- \((51930 - 45638.89)^2\)
- \((42090 - 45638.89)^2\)
- \((47600 - 45638.89)^2\)
- \((36700 - 45638.89)^2\)
- \((50620 - 45638.89)^2\)

Calculating squared differences:

1. \( (61220 - 45638.89)^2 \approx 243080024.41 \)
2. \( (61050 - 45638.89)^2 \approx 235623815.68 \)
3. \( (56630 - 45638.89)^2 \approx 124267515.09 \)
4. \( (53910 - 45638.89)^2 \approx 67696093.79 \)
5. \( (51930 - 45638.89)^2 \approx 39383631.51 \)
6. \( (42090 - 45638.89)^2 \approx 1263752.41 \)
7. \( (47600 - 45638.89)^2 \approx 390359.61 \)
8. \( (36700 - 45638.89)^2 \approx 7934712.73 \)
9. \( (50620 - 45638.89)^2 \approx 2442872.01 \)

Now, calculating the variance:

\[
\text{Variance} = \frac{\text{Sum of squared deviations}}{n - 1}
\]
where \( n = 9 \):
- Sum of squared deviations = (243080024.41 + 235623815.68 + 124267515.09 + 67696093.79 + 39383631.51 + 1263752.41 + 390359.61 + 7934712.73 + 2442872.01) \approx 408897812.54
- Variance = \( \frac{408897812.54}{8} \approx 51112226.57 \)

### 7. Calculate the Standard Deviation
The standard deviation (SD) is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{51112226.57} \approx 7140.60
\]

### Summary of Results
- Mean: \( \approx 45,638.89 \)
- Median: \( 51,930 \)
- Mode: \( \text{None} \)
- Range: \( 24,520 \)
- IQR: \( 10,425 \)
- Variance: \( 51,112,226.57 \)
- Standard Deviation: \( \approx 7,140.60 \)

This concludes the detailed calculations of the salaries.