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.
- 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:
\[ 29.16 + 5.76 + 1.96 + 1.96 + 0.16 + 0.16 + 0.36 + 2.56 + 6.76 + 43.56 = 92.32 \]
- Divide by the number of data points (N):
\[ \frac{92.32}{10} = 9.232 \]
- 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