Q. 3.22 - Elementary Statistics

Question

Consider a data set with m observations. If the data are sample data, you compute the sample standard deviation, s, whereas if the data are population data, you compute the population standard deviation, $\sigma$.

Part (a): Derive a mathematical formula that gives $\sigma$ in terms of s when both are computed for the same data set.

Part (b): Refer to the three data sets in Exercise 3.219. Verify that your formula in part (a) works for each of the three data sets.

Part (c): Suppose that a data set consists of 15 observations. You compute the sample standard deviation of the data and obtain $s=38.6$. Then you realize that the data are actually population data and that you should have obtained the population standard deviation instead. Use your formula from part (a) to obtain $\sigma$.

Step-by-Step Solution

Answer

Part (a): A mathematical formula that gives $\sigma$ in terms of s when both are computed for the same data set is $\sigma =s\sqrt{\frac{m-1}{m}}$.

Part (b): Use the formula, to find the value of standard deviation for the each of the three data sets using the sample standard deviation values, to verify that the formula in part (a) works.

Part (c): Using the formula of part (a) is 37.29.

Part (a) Step 1. Given information.

Consider the given question,

Total sample size is 15.

The standard deviation, $\mathrm{s}=38.6$.

Part (a) Step 2. Write the sample standard deviation.

Consider a data set with m observations. If the data are sample data, you compute the sample standard deviation, s whereas if the data are population data, you compute the population standard deviation $\sigma$.

Assume the m observation to be denoted as $x_1,x_2,x_3,...x_m$.

For a variable x and a sample of size m from a population, the sample standard deviation is given below,

$s=\sqrt{\frac{\Sigma {\left(x_i-\overline{\overline{x}}\right)}^2}{m-1}}$

The standard deviation of a finite population is obtained in a similar, but slightly different way.

For a variable x, the standard deviation of all possible observations for the entire population is called the population standard deviation of the variable x..

For a finite population, the formula is given below,

$\sigma =\sqrt{\frac{\Sigma {\left(x_i-\mu \right)}^2}{m}}$

Where, $\sigma =$population standard deviation.

The values of $\overline{x},\sigma$ are identical.

Part (a) Step 3. Write the ratio of these standard deviations.

Writing the ratio of these standard deviations, we get the relation between the sample and population standard deviation,

$$\begin{gathered} \frac{\sigma }{s}=\frac{\sqrt{{\displaystyle \frac{\Sigma {\left(x_i-\mu \right)}^2}{m}}}}{\sqrt{{\displaystyle \frac{\Sigma {\left(x_i-\overline{x}\right)}^2}{m-1}}}} \\ \frac{{\sigma }^2}{s^2}=\frac{{\displaystyle \frac{\Sigma {\left(x_i-\overline{x}\right)}^2}{m}}}{{\displaystyle \frac{{\displaystyle \Sigma {\left(x_i-\overline{x}\right)}^2}}{{\displaystyle m-1}}}} \end{gathered}$$

As $\overline{x},\mu$ are identical.

$$\begin{gathered} \frac{{\sigma }^2}{s^2}=\frac{m-1}{m} \\ m{\sigma }^2=\left(m-1\right)s^2 \end{gathered}$$

Therefore, the mathematical formula for $\sigma$ in terms of s when both computed on both data set, we get,

$$\begin{gathered} {\sigma }^2=\frac{m-1}{m}{s}^2 \\ \sigma =s\sqrt{\frac{m-1}{m}} \end{gathered}$$

Part (b) Step 1. Write the descriptive statistics for the given data.

The descriptive statistics for the given data set is given below,

Part (b) Step 2. Write the population standard deviation of data set.

The sample standard deviation of data set 1 is $s_1=2.16$ and sample size, m is 4.

Population standard deviation of data set,

$$\begin{gathered} {\sigma }_1=\sqrt{\frac{m-1}{m}}\times s_1 \\ =\sqrt{\frac{4-1}{4}}\times 2.16 \\ =1.87 \end{gathered}$$

The sample standard deviation of data set 2 is $s_2=2.04$ and sample size, m is 7.

Population standard deviation of data set,

$$\begin{gathered} {\sigma }_2=\sqrt{\frac{m-1}{m}}\times s_2 \\ =\sqrt{\frac{7-1}{7}}\times 2.04 \\ =1.88 \end{gathered}$$

The sample standard deviation of data set 3 is $s_3=2.01$ and sample size, m is 10.

$$\begin{gathered} {\sigma }_3=\sqrt{\frac{m-1}{m}}\times s_3 \\ =\sqrt{\frac{10-1}{10}}\times 2.01 \\ =1.91 \end{gathered}$$

Part (c) Step 1. Obtain the standard deviation.

Suppose that a data set consists of 15 observations. Compute the sample standard deviation of the data and obtain $s=38.6$.

Then we can say that the data are actually population data. Use the formula from part (a),

Population standard deviation in terms of sample standard deviation,

$$\begin{gathered} \sigma =\sqrt{\frac{n-1}{n}}\times s \\ =38.6\times \sqrt{\frac{14}{15}} \\ =38.6\times 0.9661 \\ =37.29 \end{gathered}$$