Question
The 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
Average of all Occupations: 50,620
1.Calculate the mean of the annual salaries of the occupations in the data. SHOW YOUR WORK/STEPS.
2.Calculate the median of the annual salaries of the occupations in the data. SHOW YOUR WORK/STEPS.
3.Determine the mode(s) of the annual salaries of the occupations in the data. SHOW YOUR WORK/STEPS.
4.How spread out are the salaries? (Find the range.) SHOW YOUR WORK/STEPS.
5.Calculate the interquartile range (IQR) of the occupation salaries. SHOW YOUR WORK/STEPS.
6.Calculate the variance of the occupation salaries.SHOW YOUR WORK/STEPS.
7.Calculate the standard deviation of the occupation salaries.SHOW YOUR WORK/STEPS.
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
Average of all Occupations: 50,620
1.Calculate the mean of the annual salaries of the occupations in the data. SHOW YOUR WORK/STEPS.
2.Calculate the median of the annual salaries of the occupations in the data. SHOW YOUR WORK/STEPS.
3.Determine the mode(s) of the annual salaries of the occupations in the data. SHOW YOUR WORK/STEPS.
4.How spread out are the salaries? (Find the range.) SHOW YOUR WORK/STEPS.
5.Calculate the interquartile range (IQR) of the occupation salaries. SHOW YOUR WORK/STEPS.
6.Calculate the variance of the occupation salaries.SHOW YOUR WORK/STEPS.
7.Calculate the standard deviation of the occupation salaries.SHOW YOUR WORK/STEPS.
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Answered by
GPT-4o mini
Let's calculate the statistical measures for the given annual salaries of occupations.
### Given Data:
1. Forensic Science Technician: 61,220
2. Police and Sheriff's Patrol: 61,050
3. Probation Officer: 56,630
4. Paralegal: 53,910
5. Fire Fighter: 51,930
6. Community and Social Service Specialists: 42,090
7. Correctional Officer/Jailer: 47,600
8. 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:
1. Find Q1 (1st quartile). This is the median of the first half of the data.
2. 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:
1. Mean: 51,278.75
2. Median: 52,920
3. Mode: None
4. Range: 24,520
5. Interquartile Range (IQR): 13,995
6. Variance: 67,153,157.9375
7. Standard Deviation: ~8,193.80
### Given Data:
1. Forensic Science Technician: 61,220
2. Police and Sheriff's Patrol: 61,050
3. Probation Officer: 56,630
4. Paralegal: 53,910
5. Fire Fighter: 51,930
6. Community and Social Service Specialists: 42,090
7. Correctional Officer/Jailer: 47,600
8. 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:
1. Find Q1 (1st quartile). This is the median of the first half of the data.
2. 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:
1. Mean: 51,278.75
2. Median: 52,920
3. Mode: None
4. Range: 24,520
5. Interquartile Range (IQR): 13,995
6. Variance: 67,153,157.9375
7. Standard Deviation: ~8,193.80
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