Chapter 3.CR, Problem 4CR

To determine

To show:

One example of a data set with large variation and one example with small variation.

Explanation of Solution

Given:

One data set contains large variation and one data set contains small variation.

Formula used:

The formula to calculate the mean is given as,

$\mu =\frac{\sum x_i}{N}\mathrm{...................}(1)$

The formula to calculate the standard variation is,

${\sigma }^2=\frac{\sum {\left(x_i-\mu \right)}^2}{N}\mathrm{.............}(2)$

Calculation:

Assume the data set of 6 observation as given below,

$1,8,27,10,39,5$

Substitute 1 for $x_1$, 8 for $x_2$, 27 for $x_3$, 10 for $x_4$ and 39 for $x_5$ and 5 for $x_6$ in equation (1),

$\begin{array}{rll}\mu & =& \frac{1+8+27+10+39+5}{6}\\ & =& \frac{90}{6}\\ & =& 15\end{array}$

Substitute 1 for $x_1$, 8 for $x_2$, 27 for $x_3$, 10 for $x_4$ and 39 for $x_5$, 5 for $x_6$ and 15 for $\mu$ in equation (2),

$\begin{array}{rll}{\sigma }^2& =& \frac{{\left(1-15\right)}^2+{\left(8-15\right)}^2+{\left(27-15\right)}^2+{\left(10-15\right)}^2+{\left(39-15\right)}^2+{\left(5-15\right)}^2}{6}\\ & =& \frac{{\left(-14\right)}^2+{\left(-7\right)}^2+{\left(12\right)}^2+{\left(-5\right)}^2+{\left(24\right)}^2+{\left(-10\right)}^2}{6}\\ & =& \frac{1090}{6}\\ & =& 181.66\end{array}$

Square root both sides,

$\sigma =13.47$

The standard deviation comes out to be $13.47$, so, the large variance represent that the data values are more spread out.

Now, assume the data set of 6 observation as given below,

$42,38,37,39,44,34$

Substitute 42 for $x_1$, 38 for $x_2$, 37 for $x_3$, 39 for $x_4$ and 44 for $x_5$ and 34 for $x_6$ in equation (1),

$\begin{array}{rll}\mu & =& \frac{42+38+37+39+44+34}{6}\\ & =& \frac{234}{6}\\ & =& 39\end{array}$

Substitute 42 for $x_1$, 38 for $x_2$, 37 for $x_3$, 39 for $x_4$ and 44 for $x_5$, 34 for $x_6$ and 39 for $\mu$ in equation (2),

$\begin{array}{rll}{\sigma }^2& =& \frac{{\left(42-39\right)}^2+{\left(38-39\right)}^2+{\left(37-39\right)}^2+{\left(39-39\right)}^2+{\left(44-39\right)}^2+{\left(34-39\right)}^2}{6}\\ & =& \frac{{\left(3\right)}^2+{\left(-1\right)}^2+{\left(-2\right)}^2+{\left(0\right)}^2+{\left(5\right)}^2+{\left(-5\right)}^2}{6}\\ & =& \frac{64}{6}\\ & =& 10.6\end{array}$

Square root both sides,

$\sigma =3.25$

The standard deviation comes out to be $3.25$, so, the small variance represent that the data values are close together.