The planets of the solar system have the numbers of moons listed below in order from the sun. (Pluto is not included because it was uninvited from the solar system party in 2006.) Include appropriate units whenever relevant.

0 0 1 2 17 28 21 8

a. Find the mean.

b. Find the median.

c. Find the mode.

d. Find the range.

e. Find the standard deviation.

f. Find the variance.

g. Use the range rule of thumb to identify the values separating significant values from those that are not significant.

h. Based on the result from part (g), do any of the planets have a number of moons that is significantly low or significantly high? Why or why not?

i. What is the level of measurement of the data: nominal, ordinal, interval, or ratio?

j. Are the data discrete or continuous?

The number of moons of all the 8 planets is listed.

a.

The mean number of moons is computed below:

\(\begin{array}{c}{\rm{Mean}} = \frac{{0 + 0 + 1 + 2 + 17 + 28 + 21 + 8}}{8}\\ = \frac{{77}}{8}\\ = 9.6\end{array}\)

Thus, the meanis equal to 9.6 moons.

b.

The number of moons arranged in ascending order is shown below:

Since the number of items is even (8), the following formula is used to compute the median value:

\[\begin{array}{c}M{\rm{edian}} = \frac{{{{\left( {\frac{n}{2}} \right)}^{th}}{\rm{obs}} + {{\left( {\frac{n}{2} + 1} \right)}^{th}}}}{2}\\ = \frac{{{{\left( {\frac{{\rm{8}}}{{\rm{2}}}} \right)}^{{\rm{th}}}}{\rm{obs + }}{{\left( {\frac{{\rm{8}}}{{\rm{2}}}{\rm{ + 1}}} \right)}^{{\rm{th}}}}}}{{\rm{2}}}\\{\rm{ = }}\frac{{{{\left( 4 \right)}^{th}}{\rm{obs}} + {{\left( 5 \right)}^{th}}{\rm{obs}}}}{2}\end{array}\]

\[\begin{array}{c} = \frac{{2 + 8}}{2}\\ = \frac{{10}}{2}\\ = 5\end{array}\]

The median value is equal to 5.0 moons.

c.

The mode is the number that occurs most frequently.

It can be observed that only the number “0” occurs twice and all the rest of thenumbers occur only once.

Thus, the mode value is equal to 0 moons

d.

The range value is computed as shown below:

\(\begin{array}{c}{\rm{Range}} = {\rm{Maximium}}\;{\rm{Value}} - {\rm{Minimum}}\;{\rm{Value}}\\ = 28 - 0\\ = 28\end{array}\)

Thus, the value of the range is equal to 28.0 moons.

e.

The value of the standard deviation is computed as shown below:

\(\begin{array}{c}S.D. = \sqrt {\frac{{{{\sum {\left( {{x_i} - \bar x} \right)} }^2}}}{n}} \\ = \sqrt {\frac{{{{\left( {0 - 9.6} \right)}^2} + {{\left( {0 - 9.6} \right)}^2} + ..... + {{\left( {8 - 9.6} \right)}^2}}}{{8 - 1}}} \\ = 11.0\end{array}\)

Thus, the value of the standard deviation is equal to 11.0 moons.

f.

The value of the variance is computed below:

\(\begin{array}{c}{\rm{Variance}} = \frac{{{{\sum {\left( {{x_i} - \bar x} \right)} }^2}}}{n}\\ = \frac{{{{\left( {0 - 9.6} \right)}^2} + {{\left( {0 - 9.6} \right)}^2} + ..... + {{\left( {8 - 9.6} \right)}^2}}}{{8 - 1}}\\ = 120.3\end{array}\)

Thus, the value of the variance is equal to 120.3 moons squared.

g.

The range rule of thumb is used to identify the significant values.

Significantly high values are separated by the following value:

\(\begin{array}{c}Mean + 2S.D. = 9.625 + 2\left( {11.0} \right)\\ = 31.6\end{array}\)

Thus, values of the number of moons greater than or equal to 31.6 are considered significantly high values.

Significantly low values are separated by the following value:

\(\begin{array}{c}Mean - 2S.D. = 9.6 - 2\left( {11.0} \right)\\ = - 12.4\end{array}\)

Thus, values of the number of moons less than or equal to -12.4 are considered significantly low values.

And the values that are not significant will lie between -12.4 moons and 31.6 moons.

h.

Since none of the values is less than or equal to -12.4, none of the planets have a significantly low number of moons.

Also, none of the given values is greater than or equal to 31.6, none of the planets have a significantly high number of moons.

i.

The following levels of measurement are defined:

Nominal: Data are categorical and cannot be arranged in an order.

Ordinal: Data are categorical and the labels can be arranged in an order. The distance between each label is not known.

Interval: Data are numerical and can be arranged in a specific order. Also, the distance between each value can be computed, but their ratios cannot be computed.

Ratio: Data are numerical. The ratios of the values can be computed and the natural zero starting point of the data is meaningful (unlike the interval scale).

Here, the variable at hand is the number of moons of each planet.

Since the number of moons is quantitative, their ratios can be computed and the value of 0 moons represents no moons of that planet, therefore, the number of moons of each planet ismeasured on a ratio scale.

j.

Discrete data are data that cannot contain decimals and can only be counted.

Continuous data are data that can contain decimals and are measured on a scale.

Here, it is known that the number of moons cannot be in decimals.

Thus, the number of moons of each planet is a discrete data.