Question
What is the standard deviation of the annual salaries?
1. 36,700
2. 42,090
3. 47,600
4. 50,620
5. 51,930
6. 53,910
7. 56,630
8. 61,050
9. 61,220
1. 36,700
2. 42,090
3. 47,600
4. 50,620
5. 51,930
6. 53,910
7. 56,630
8. 61,050
9. 61,220
Answers
GPT-4o mini
To calculate the standard deviation of the given annual salaries, we can follow these steps:
1. **Calculate the mean (average)** of the salaries.
2. **Find the squared deviations** from the mean for each salary.
3. **Calculate the variance**, which is the average of the squared deviations.
4. **Take the square root of the variance** to find the standard deviation.
Let's do the calculations.
### Step 1: Calculate the Mean
Given salaries:
- 36,700
- 42,090
- 47,600
- 50,620
- 51,930
- 53,910
- 56,630
- 61,050
- 61,220
First, sum these values:
\[
36,700 + 42,090 + 47,600 + 50,620 + 51,930 + 53,910 + 56,630 + 61,050 + 61,220 = 421,850
\]
Now, divide by the number of salaries (n = 9):
\[
\text{Mean} = \frac{421,850}{9} \approx 46,872.22
\]
### Step 2: Find the Squared Deviations
Now, we calculate the squared deviations from the mean for each salary and sum them.
\[
\begin{align*}
(36,700 - 46,872.22)^2 & \approx 105,088,688.58 \\
(42,090 - 46,872.22)^2 & \approx 22,103,479.52 \\
(47,600 - 46,872.22)^2 & \approx 5,267.24 \\
(50,620 - 46,872.22)^2 & \approx 13,542,411.60 \\
(51,930 - 46,872.22)^2 & \approx 25,646,100.64 \\
(53,910 - 46,872.22)^2 & \approx 49,378,350.92 \\
(56,630 - 46,872.22)^2 & \approx 93,929,108.96 \\
(61,050 - 46,872.22)^2 & \approx 196,106,049.80 \\
(61,220 - 46,872.22)^2 & \approx 196,610,097.36 \\
\end{align*}
\]
Finally, sum all the squared deviations:
\[
105,088,688.58 + 22,103,479.52 + 5,267.24 + 13,542,411.60 + 25,646,100.64 + 49,378,350.92 + 93,929,108.96 + 196,106,049.80 + 196,610,097.36 = 505,000,000
\]
### Step 3: Calculate the Variance
To find the variance, divide the sum of squared deviations by n:
\[
\text{Variance} = \frac{505,000,000}{9} \approx 56,111,111.11
\]
### Step 4: Calculate the Standard Deviation
Finally, take the square root of the variance:
\[
\text{Standard Deviation} \approx \sqrt{56,111,111.11} \approx 7,485.34
\]
Thus, the standard deviation of the annual salaries is approximately **7,485.34**.
1. **Calculate the mean (average)** of the salaries.
2. **Find the squared deviations** from the mean for each salary.
3. **Calculate the variance**, which is the average of the squared deviations.
4. **Take the square root of the variance** to find the standard deviation.
Let's do the calculations.
### Step 1: Calculate the Mean
Given salaries:
- 36,700
- 42,090
- 47,600
- 50,620
- 51,930
- 53,910
- 56,630
- 61,050
- 61,220
First, sum these values:
\[
36,700 + 42,090 + 47,600 + 50,620 + 51,930 + 53,910 + 56,630 + 61,050 + 61,220 = 421,850
\]
Now, divide by the number of salaries (n = 9):
\[
\text{Mean} = \frac{421,850}{9} \approx 46,872.22
\]
### Step 2: Find the Squared Deviations
Now, we calculate the squared deviations from the mean for each salary and sum them.
\[
\begin{align*}
(36,700 - 46,872.22)^2 & \approx 105,088,688.58 \\
(42,090 - 46,872.22)^2 & \approx 22,103,479.52 \\
(47,600 - 46,872.22)^2 & \approx 5,267.24 \\
(50,620 - 46,872.22)^2 & \approx 13,542,411.60 \\
(51,930 - 46,872.22)^2 & \approx 25,646,100.64 \\
(53,910 - 46,872.22)^2 & \approx 49,378,350.92 \\
(56,630 - 46,872.22)^2 & \approx 93,929,108.96 \\
(61,050 - 46,872.22)^2 & \approx 196,106,049.80 \\
(61,220 - 46,872.22)^2 & \approx 196,610,097.36 \\
\end{align*}
\]
Finally, sum all the squared deviations:
\[
105,088,688.58 + 22,103,479.52 + 5,267.24 + 13,542,411.60 + 25,646,100.64 + 49,378,350.92 + 93,929,108.96 + 196,106,049.80 + 196,610,097.36 = 505,000,000
\]
### Step 3: Calculate the Variance
To find the variance, divide the sum of squared deviations by n:
\[
\text{Variance} = \frac{505,000,000}{9} \approx 56,111,111.11
\]
### Step 4: Calculate the Standard Deviation
Finally, take the square root of the variance:
\[
\text{Standard Deviation} \approx \sqrt{56,111,111.11} \approx 7,485.34
\]
Thus, the standard deviation of the annual salaries is approximately **7,485.34**.
wrong
GPT-4o mini
I apologize for the confusion. Let's go through the calculations step-by-step to find the standard deviation of the annual salaries accurately.
### Given Salaries
1. 36,700
2. 42,090
3. 47,600
4. 50,620
5. 51,930
6. 53,910
7. 56,630
8. 61,050
9. 61,220
### Step 1: Calculate the Mean
\[
\text{Mean} = \frac{\text{Sum of Salaries}}{\text{Number of Salaries}}
\]
Calculating the sum:
\[
36,700 + 42,090 + 47,600 + 50,620 + 51,930 + 53,910 + 56,630 + 61,050 + 61,220 = 421,850
\]
The number of salaries (n) = 9.
Now, calculate the mean:
\[
\text{Mean} = \frac{421,850}{9} \approx 46,872.22
\]
### Step 2: Calculate the Squared Deviations from the Mean
Now, we calculate the squared deviations from the mean for each salary.
1. \((36,700 - 46,872.22)^2 \approx (−10,172.22)^2 \approx 103,477,852.33\)
2. \((42,090 - 46,872.22)^2 \approx (−4,782.22)^2 \approx 22,906,091.89\)
3. \((47,600 - 46,872.22)^2 \approx (727.78)^2 \approx 529,063.45\)
4. \((50,620 - 46,872.22)^2 \approx (3,747.78)^2 \approx 14,035,329.45\)
5. \((51,930 - 46,872.22)^2 \approx (5,057.78)^2 \approx 25,580,991.73\)
6. \((53,910 - 46,872.22)^2 \approx (7,037.78)^2 \approx 49,505,203.51\)
7. \((56,630 - 46,872.22)^2 \approx (9,757.78)^2 \approx 95,166,994.53\)
8. \((61,050 - 46,872.22)^2 \approx (14,177.78)^2 \approx 200,030,203.93\)
9. \((61,220 - 46,872.22)^2 \approx (14,347.78)^2 \approx 205,765,870.91\)
### Step 3: Sum of the Squared Deviations
Calculating the total of these squared deviations:
\[
103,477,852.33 + 22,906,091.89 + 529,063.45 + 14,035,329.45 + 25,580,991.73 + 49,505,203.51 + 95,166,994.53 + 200,030,203.93 + 205,765,870.91 \approx 516,112,628.70
\]
### Step 4: Calculate the Variance
Variance is calculated as the average of the squared deviations. Since we are measuring the standard deviation of a sample, we will divide by \(n - 1\):
\[
\text{Variance} = \frac{\text{Sum of Squared Deviations}}{n - 1} = \frac{516,112,628.70}{8} \approx 64,514,078.59
\]
### Step 5: Calculate the Standard Deviation
Now, take the square root of the variance:
\[
\text{Standard Deviation} = \sqrt{64,514,078.59} \approx 8,031.24
\]
Thus, the standard deviation of the annual salaries is approximately **8,031.24**.
If you want to confirm any part of the calculation, please let me know!
### Given Salaries
1. 36,700
2. 42,090
3. 47,600
4. 50,620
5. 51,930
6. 53,910
7. 56,630
8. 61,050
9. 61,220
### Step 1: Calculate the Mean
\[
\text{Mean} = \frac{\text{Sum of Salaries}}{\text{Number of Salaries}}
\]
Calculating the sum:
\[
36,700 + 42,090 + 47,600 + 50,620 + 51,930 + 53,910 + 56,630 + 61,050 + 61,220 = 421,850
\]
The number of salaries (n) = 9.
Now, calculate the mean:
\[
\text{Mean} = \frac{421,850}{9} \approx 46,872.22
\]
### Step 2: Calculate the Squared Deviations from the Mean
Now, we calculate the squared deviations from the mean for each salary.
1. \((36,700 - 46,872.22)^2 \approx (−10,172.22)^2 \approx 103,477,852.33\)
2. \((42,090 - 46,872.22)^2 \approx (−4,782.22)^2 \approx 22,906,091.89\)
3. \((47,600 - 46,872.22)^2 \approx (727.78)^2 \approx 529,063.45\)
4. \((50,620 - 46,872.22)^2 \approx (3,747.78)^2 \approx 14,035,329.45\)
5. \((51,930 - 46,872.22)^2 \approx (5,057.78)^2 \approx 25,580,991.73\)
6. \((53,910 - 46,872.22)^2 \approx (7,037.78)^2 \approx 49,505,203.51\)
7. \((56,630 - 46,872.22)^2 \approx (9,757.78)^2 \approx 95,166,994.53\)
8. \((61,050 - 46,872.22)^2 \approx (14,177.78)^2 \approx 200,030,203.93\)
9. \((61,220 - 46,872.22)^2 \approx (14,347.78)^2 \approx 205,765,870.91\)
### Step 3: Sum of the Squared Deviations
Calculating the total of these squared deviations:
\[
103,477,852.33 + 22,906,091.89 + 529,063.45 + 14,035,329.45 + 25,580,991.73 + 49,505,203.51 + 95,166,994.53 + 200,030,203.93 + 205,765,870.91 \approx 516,112,628.70
\]
### Step 4: Calculate the Variance
Variance is calculated as the average of the squared deviations. Since we are measuring the standard deviation of a sample, we will divide by \(n - 1\):
\[
\text{Variance} = \frac{\text{Sum of Squared Deviations}}{n - 1} = \frac{516,112,628.70}{8} \approx 64,514,078.59
\]
### Step 5: Calculate the Standard Deviation
Now, take the square root of the variance:
\[
\text{Standard Deviation} = \sqrt{64,514,078.59} \approx 8,031.24
\]
Thus, the standard deviation of the annual salaries is approximately **8,031.24**.
If you want to confirm any part of the calculation, please let me know!