To find the mean and standard deviation of the data set \(11, 14, 15, 15, 16, 16, 18, 18, 19, 24\), we can follow these steps:
Step 1: Calculate the Mean (\( \mu \))
The mean is calculated as the sum of all the data points divided by the number of data points.
\[ \mu = \frac{\sum x_i}{n} \]
Where:
- \(x_i\) are the data points
- \(n\) is the number of data points
For the given data set:
\[ \sum x_i = 11 + 14 + 15 + 15 + 16 + 16 + 18 + 18 + 19 + 24 = 11 + 14 + 15 + 15 + 16 + 16 + 18 + 18 + 19 + 24 = 6 + 15 + 15 + 16 + 18 + 18 + 19 + 24 = 10 + 19 + 24 = 191 \]
The number of data points: \[ n = 10 \]
Now calculate the mean: \[ \mu = \frac{191}{10} = 19.1 \]
Step 2: Calculate the Standard Deviation (\( \sigma \))
Standard deviation is calculated using the formula:
\[ \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}} \]
- First, calculate each \( (x_i - \mu)^2 \):
- \( (11 - 19.1)^2 = (-8.1)^2 = 65.61 \)
- \( (14 - 19.1)^2 = (-5.1)^2 = 26.01 \)
- \( (15 - 19.1)^2 = (-4.1)^2 = 16.81 \)
- \( (15 - 19.1)^2 = (-4.1)^2 = 16.81 \)
- \( (16 - 19.1)^2 = (-3.1)^2 = 9.61 \)
- \( (16 - 19.1)^2 = (-3.1)^2 = 9.61 \)
- \( (18 - 19.1)^2 = (-1.1)^2 = 1.21 \)
- \( (18 - 19.1)^2 = (-1.1)^2 = 1.21 \)
- \( (19 - 19.1)^2 = (-0.1)^2 = 0.01 \)
- \( (24 - 19.1)^2 = (4.9)^2 = 24.01 \)
Now, our squared differences are:
\[ 65.61, 26.01, 16.81, 16.81, 9.61, 9.61, 1.21, 1.21, 0.01, 24.01 \]
- Now, sum these squared differences:
\[ \sum (x_i - \mu)^2 = 65.61 + 26.01 + 16.81 + 16.81 + 9.61 + 9.61 + 1.21 + 1.21 + 0.01 + 24.01 = 65.61 + 26.01 + 16.81 + 16.81 + 9.61 + 9.61 + 1.21 + 1.21 + 0.01 + 24.01 = 8 + 16 + 0 + 24 + 75 = 193 \]
- Finally, calculate the standard deviation:
\[ \sigma = \sqrt{\frac{193}{10}} = \sqrt{19.3} \approx 4.4 \]
Final results:
So, the mean and standard deviation of the data set are:
\[ \mu \approx 19.1 \] \[ \sigma \approx 4.4 \]
Rounded to the nearest tenth:
- \( \mu = 19.1 \)
- \( \sigma = 4.4 \)