Questions: Describe the process of calculating a standard deviation. Give a simple example of its calculation (such as calculating the standard deviation of the numbers 2, 3, 4, 4, and 6 ). What is the standard deviation if all of the sample values are the same? Fill in the blanks to complete the process of calculating a standard deviation. Compute the mean of the data set. Then find the deviation from the mean for every data value by subtracting the mean from the data value Find the square of all the deviations from the mean, and then sum them together.

Solution Steps

To calculate the standard deviation, follow these steps:

1. Compute the mean (average) of the data set.
2. Find the deviation of each data value from the mean by subtracting the mean from each data value.
3. Square each of these deviations.
4. Sum all the squared deviations.
5. Divide this sum by the number of data values (for population standard deviation) or by the number of data values minus one (for sample standard deviation).
6. Take the square root of the result from step 5 to get the standard deviation.

For the given example, we will calculate the standard deviation of the numbers 2, 3, 4, 4, and 6.

If all sample values are the same, the standard deviation is 0 because there is no variation in the data.

The mean (\(\mu\)) of the data set \([2, 3, 4, 4, 6]\) is calculated as follows: \[ \mu = \frac{2 + 3 + 4 + 4 + 6}{5} = \frac{19}{5} = 3.8 \]

The deviations from the mean for each data value are: \[ \begin{align_} 2 - 3.8 &= -1.8 \\ 3 - 3.8 &= -0.8 \\ 4 - 3.8 &= 0.2 \\ 4 - 3.8 &= 0.2 \\ 6 - 3.8 &= 2.2 \\ \end{align_} \] So, the deviations are \([-1.8, -0.8, 0.2, 0.2, 2.2]\).

The squared deviations are: \[ \begin{align_} (-1.8)^2 &= 3.24 \\ (-0.8)^2 &= 0.64 \\ (0.2)^2 &= 0.04 \\ (0.2)^2 &= 0.04 \\ (2.2)^2 &= 4.84 \\ \end{align_} \] So, the squared deviations are \([3.24, 0.64, 0.04, 0.04, 4.84]\).

The sum of the squared deviations is: \[ 3.24 + 0.64 + 0.04 + 0.04 + 4.84 = 8.8 \]

For a sample, the variance (\(s^2\)) is calculated by dividing the sum of squared deviations by \(n-1\) (where \(n\) is the number of data values): \[ s^2 = \frac{8.8}{5-1} = \frac{8.8}{4} = 2.2 \]

The standard deviation (\(s\)) is the square root of the variance: \[ s = \sqrt{2.2} \approx 1.4832 \]

If all sample values are the same, such as \([4, 4, 4, 4, 4]\):

• The mean is \(4\).
• The deviations from the mean are all \(0\).
• The squared deviations are all \(0\).
• The sum of squared deviations is \(0\).
• The variance is \(0\).
• The standard deviation is \(0\).

Final Answer