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

Answer 1

wrong

Answer 2

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**.

Answer 3

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!