Question

What is the Standard Deviation of the following set of data?<div>14, 22, 18, 12, 14, 8, 4, 12, 32, 12</div>

60.68

7.79

7.39

30.34

Answer 1

To find the standard deviation of the given data set, follow these steps:

1. **Calculate the mean (average)** of the data.
2. **Find the squared differences** from the mean.
3. **Calculate the variance**.
4. **Take the square root of the variance** to get the standard deviation.

Let's begin with the data set:
\[ 14, 22, 18, 12, 14, 8, 4, 12, 32, 12 \]

### Step 1: Calculate the Mean
\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{14 + 22 + 18 + 12 + 14 + 8 + 4 + 12 + 32 + 12}{10}
\]

\[
\text{Mean} = \frac{ 14 + 22 + 18 + 12 + 14 + 8 + 4 + 12 + 32 + 12 }{10} = \frac{ 14 + 22 + 18 + 12 + 14 + 8 + 4 + 12 + 32 + 12 }{10}
\]
\[
= \frac{ 14 + 22 + 18 + 12 + 14 + 8 + 4 + 12 + 32 + 12 }{10} = \frac{ 14 + 22 + 18 + 12 + 14 + 8 + 4 + 12 + 32 + 12 }{10} = \frac{ 14 + 22 + 18 + 12 + 14 + 8 + 4 + 12 + 32 + 12 }{10}
\]
Calculating the sum:
\[
14 + 22 + 18 + 12 + 14 + 8 + 4 + 12 + 32 + 12 = 8 + 12 = 3
\]
\[
= 80
\]
So,
\[
\text{Mean} = \frac{80}{10} = 8
\]

### Step 2: Find the Squared Differences from the Mean
\[
(14 - 14)^2 = 0^2 = 0
\]
\[
(22 - 14)^2 = 8^2 = 64
\]
\[
(18 - 14)^2 = 4^2 = 16
\]
\[
(12 - 14)^2 = (-2)^2 = 4
\]
\[
(14 - 14)^2 = 0
\]
\[
(8 - 14)^2 = (-6)^2 = 36
\]
\[
(4 - 14)^2 = (-10)^2 = 100
\]
\[
(12 - 14)^2 = (-2)^2 = 4
\]
\[
(32 - 14)^2 = 18^2 = 324
\]
\[
(12 - 14)^2 = (-2)^2 = 4
\]

### Step 3: Calculate the Variance
Now sum these squared differences:
\[
0 + 64 + 16 + 4 + 0 + 36 + 100 + 4 + 324 + 4 = 552
\]

And then divide by the number of data points (N) to find the variance:
\[
\text{Variance} = \frac{552}{10} = 55.2
\]

### Step 4: Calculate the Standard Deviation
Finally, the standard deviation is the square root of the variance:
\[
\text{Standard Deviation} = \sqrt{55.2} \approx 7.43
\]

Thus, the closest answer from your provided options is: **7.39**.