Question

Find the mean and standard deviation for the following data set. Round to the nearest tenth.

11, 14, 15, 15, 16, 16, 17, 18, 19, 23



(1 point)
Responses

Mean= 16.4, Standard Deviation= 3.0
Mean= 16.4, Standard Deviation= 3.0

Mean= 16.6, Standard Deviation= 3.7
Mean= 16.6, Standard Deviation= 3.7

Mean= 16.6, Standard Deviation= 3.7
Mean= 16.6, Standard Deviation= 3.7

Mean= 16.4, Standard Deviation= 3.0

Answer 1

To find the mean and standard deviation of the given data set (11, 14, 15, 15, 16, 16, 17, 18, 19, 23), follow these steps:

### Step 1: Calculate the Mean
The mean (average) is calculated as follows:

\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}
\]

The sum of the data points is:

\[
11 + 14 + 15 + 15 + 16 + 16 + 17 + 18 + 19 + 23 = 11 + 14 + 15 + 15 + 16 + 16 + 17 + 18 + 19 + 23 = 144
\]

The number of data points is 10.

Now, calculate the mean:

\[
\text{Mean} = \frac{144}{10} = 14.4
\]

### Step 2: Calculate the Standard Deviation
The standard deviation (SD) is calculated using the formula:

\[
SD = \sqrt{\frac{\sum (x_i - \text{Mean})^2}{N}}
\]

Where \( x_i \) is each data point, \( \text{Mean} \) is the mean we calculated above, and \( N \) is the number of data points.

1. Calculate the squared deviations from the mean:

\[
\begin{align*}
(11 - 16.4)^2 & = (-5.4)^2 = 29.16 \\
(14 - 16.4)^2 & = (-2.4)^2 = 5.76 \\
(15 - 16.4)^2 & = (-1.4)^2 = 1.96 \\
(15 - 16.4)^2 & = (-1.4)^2 = 1.96 \\
(16 - 16.4)^2 & = (-0.4)^2 = 0.16 \\
(16 - 16.4)^2 & = (-0.4)^2 = 0.16 \\
(17 - 16.4)^2 & = (0.6)^2 = 0.36 \\
(18 - 16.4)^2 & = (1.6)^2 = 2.56 \\
(19 - 16.4)^2 & = (2.6)^2 = 6.76 \\
(23 - 16.4)^2 & = (6.6)^2 = 43.56 \\
\end{align*}
\]

Now sum these squared deviations:

\[
29.16 + 5.76 + 1.96 + 1.96 + 0.16 + 0.16 + 0.36 + 2.56 + 6.76 + 43.56 = 92.32
\]

2. Divide by the number of data points (N):

\[
\frac{92.32}{10} = 9.232
\]

3. Take the square root to find the SD:

\[
SD = \sqrt{9.232} \approx 3.0
\]

### Final Results
- Mean = 16.4
- Standard Deviation = 3.0

Therefore, the correct response is:

**Mean= 16.4, Standard Deviation= 3.0**