Q 3.64. - Elementary Statistics

Question

Consider the data set 3,3,3,3,3,3.

Part (a) Guess the value of the sample standard deviation without calculating it. Explain your reasoning

Part (b) Use the defining formula to calculate the sample standard deviation.

Part (c) Complete the following statement and explain your reasoning. If all observation in a data set are equal, the sample standard deviation is

Part (d) Complete the following statement and explain your reasoning. If the sample standard deviation of a data set is 0 then... 

Step-by-Step Solution

Answer

Part (a)  The data set's standard deviation is 0.

Part (b) The data set's standard deviation is 0.

Part (c) The data set's standard deviation is 0.

Part (d) The data set's values are all the same.

Part (a) Step 1. Given information.

The data points are as follows:

3, 3, 3, 3, 3, 3

Part (a) Step 2. The given data's sample standard deviation.

A data set's range is given by range = max-min = 0.

Standard deviation is the measure of deviation in data. As there is no variation in the data because the values of all the data points are the same (one can verify that using range).

As a result, the data set's standard deviation is 0.

Part (b) Step 1. Given information.

The data points are as follows:

3, 3, 3, 3, 3, 3

Part (b) Step 2. The given data's sample standard deviation.

A data set's range is given by range = max-min.

The formula computes the standard deviation.

$s = \sqrt{\frac{\sum _{}{x_i}^2-{\left(\sum _{}{x}_i\right)}^2/n}{n-1}}$

Determine the mean of the given data set.

$$\begin{gathered} \overline{x}=\frac{3+3+3+3+3+3}{6} \\ \overline{x}=\frac{18}{6} \\ \overline{x}=3 \end{gathered}$$

$$\begin{gathered} s=\sqrt{\frac{{\left(3-3\right)}^2+{\left(3-3\right)}^2+{\left(3-3\right)}^2+{\left(3-3\right)}^2+{\left(3-3\right)}^2{\left(3-3\right)}^2}{6-1}} \\ s=0 \end{gathered}$$

As a result, the required value is 0.

Part (c) Step 1. Given information.

The data points are as follows:

3, 3, 3, 3, 3, 3

Part (c) Step 2. The given data's sample standard deviation.

A data set's range is given by range = max-min.

The formula computes the standard deviation.

$s = \sqrt{\frac{\sum _{}{x_i}^2-{\left(\sum _{}{x}_i\right)}^2/n}{n-1}}$

Standard deviation is zero if all observations in a data set are equal.

As a result, the data set's standard deviation is 0.

Part (d) Step 1. Given information.

The data points are as follows:

3, 3, 3, 3, 3, 3

Part (d) Step 2. The data set's values.

We can observe that the standard deviation is zero because all the observations are the same.

So, if a data set's sample standard deviation is zero, all observations in the data set are equal.