Let's calculate the statistical measures for the given annual salaries of occupations.
Given Data:
- Forensic Science Technician: 61,220
- Police and Sheriff's Patrol: 61,050
- Probation Officer: 56,630
- Paralegal: 53,910
- Fire Fighter: 51,930
- Community and Social Service Specialists: 42,090
- Correctional Officer/Jailer: 47,600
- EMT and Paramedic: 36,700
1. Calculate the Mean
The mean is calculated by summing all the salaries and dividing by the number of salaries.
\[ \text{Mean} = \frac{\text{Sum of Salaries}}{\text{Number of Salaries}} \]
Sum of Salaries: \[ 61,220 + 61,050 + 56,630 + 53,910 + 51,930 + 42,090 + 47,600 + 36,700 = 410,230 \]
Number of Salaries: 8
\[ \text{Mean} = \frac{410,230}{8} = 51,278.75 \]
2. Calculate the Median
To find the median, first sort the salaries in ascending order:
- 36,700
- 42,090
- 47,600
- 51,930
- 53,910
- 56,630
- 61,050
- 61,220
Since there are 8 salaries (an even number), the median will be the average of the 4th and 5th numbers.
\[ \text{Median} = \frac{51,930 + 53,910}{2} = \frac{105,840}{2} = 52,920 \]
3. Determine the Mode
The mode is the value that appears most frequently.
All salaries in the data set are unique; thus, there is no mode.
\[ \text{Mode} = \text{None} \]
4. Calculate the Range
The range is the difference between the highest and lowest values.
\[ \text{Range} = \text{Highest Salary} - \text{Lowest Salary} \]
\[ \text{Range} = 61,220 - 36,700 = 24,520 \]
5. Calculate the Interquartile Range (IQR)
The interquartile range is calculated as follows:
- Find Q1 (1st quartile). This is the median of the first half of the data.
- Find Q3 (3rd quartile). This is the median of the second half of the data.
First Half:
- 36,700
- 42,090
- 47,600
- 51,930
Median of first half (Q1): (42,090 + 47,600) / 2 = 44,845
Second Half:
- 53,910
- 56,630
- 61,050
- 61,220
Median of second half (Q3): (56,630 + 61,050) / 2 = 58,840
Now calculate IQR: \[ IQR = Q3 - Q1 = 58,840 - 44,845 = 13,995 \]
6. Calculate the Variance
Variance is calculated using the formula:
\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} \]
Where \( \mu \) is the mean.
Calculating each deviation from the mean (51,278.75):
- (61,220 - 51,278.75)² = 97,093,728.0625
- (61,050 - 51,278.75)² = 93,773,728.0625
- (56,630 - 51,278.75)² = 28,456,828.5625
- (53,910 - 51,278.75)² = 6,401,778.5625
- (51,930 - 51,278.75)² = 428,078.0625
- (42,090 - 51,278.75)² = 83,596,242.5625
- (47,600 - 51,278.75)² = 13,614,901.5625
- (36,700 - 51,278.75)² = 214,750,578.0625
Sum of squared deviations: \[ 97,093,728.0625 + 93,773,728.0625 + 28,456,828.5625 + 6,401,778.5625 + 428,078.0625 + 83,596,242.5625 + 13,614,901.5625 + 214,750,578.0625 = 537,225,263.5 \]
Calculating the variance: \[ \sigma^2 = \frac{537,225,263.5}{8} = 67,153,157.9375 \]
7. Calculate the Standard Deviation
Standard deviation (σ) is the square root of the variance. \[ \sigma = \sqrt{\sigma^2} = \sqrt{67,153,157.9375} \approx 8,193.80 \]
Summary of Results:
- Mean: 51,278.75
- Median: 52,920
- Mode: None
- Range: 24,520
- Interquartile Range (IQR): 13,995
- Variance: 67,153,157.9375
- Standard Deviation: ~8,193.80