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:
- \( (61220 - 45638.89)^2 \approx 243080024.41 \)
- \( (61050 - 45638.89)^2 \approx 235623815.68 \)
- \( (56630 - 45638.89)^2 \approx 124267515.09 \)
- \( (53910 - 45638.89)^2 \approx 67696093.79 \)
- \( (51930 - 45638.89)^2 \approx 39383631.51 \)
- \( (42090 - 45638.89)^2 \approx 1263752.41 \)
- \( (47600 - 45638.89)^2 \approx 390359.61 \)
- \( (36700 - 45638.89)^2 \approx 7934712.73 \)
- \( (50620 - 45638.89)^2 \approx 2442872.01 \)
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.