Question

Flight Arrivals. Listed below are the arrival delay times (min) of randomly selected American Airlines flights that departed from JFK in New York bound for LAX in Los Angeles. Negative values correspond to flights that arrived early and ahead of the scheduled arrival time. Use these values for Exercises 1–4.

-30 -23 14 -21 -32 11 -23 28 103 -19 -5 -46

Statistics Find the mean, median, standard deviation, and range. Are the results statistics or parameters?

Short answer

Mean:-3.6minutes

Standard deviation: 39.9 minutes

Median: -20.0 minutes

Range: 149.0 minutes

The values obtained are statistics as they are computed from a sample.

Step-by-step solution

Given information

The arrival delay times (in minutes) of a sample of flights are provided.

Mean

The mean value of the arrival delay time is computed below.

\(\begin{array}{c}\bar x = \frac{{\left( { - 30} \right) + \left( { - 23} \right) + ...... + \left( { - 46} \right)}}{{12}}\\ = - 3.6\end{array}\).

Thus, the mean value is equal to -3.6 minutes.

Median

Arrange the data values in ascending order, as follows.

-46	-32	-30	-23	-23	-21
-19	-5	11	14	28	103

Asthe number of observations in the sample is an even number (12), the following formula is used to compute the median value:

\(\begin{array}{c}{\rm{Median}} = \frac{{{{\left( {\frac{n}{2}} \right)}^{th}}obs + {{\left( {\frac{n}{2} + 1} \right)}^{th}}obs}}{2}\\ = \frac{{{{\left( {\frac{{12}}{2}} \right)}^{th}}obs + {{\left( {\frac{{12}}{2} + 1} \right)}^{th}}obs}}{2}\\ = \frac{{{6^{th}}obs + {7^{th}}obs}}{2}\\ = \frac{{\left( { - 21} \right) + \left( { - 19} \right)}}{2}\\ = - 20.0\end{array}\).

Thus, the value of the median is equal to -20.0 minutes.

Step 4:Standard deviation

The standard deviation is computed as shown below.

\(\begin{array}{c}s = \sqrt {\frac{{\sum\limits_{i = 1}^n {{{({x_i} - \bar x)}^2}} }}{{n - 1}}} \\ = \sqrt {\frac{{{{\left( { - 30 - 3.6} \right)}^2} + {{\left( { - 23 - 3.6} \right)}^2} + .... + {{\left( { - 46 - 3.6} \right)}^2}}}{{12 - 1}}} \\ = 39.9\end{array}\).

Thus, the value of the standard deviation is equal to 39.9 minutes.

Range

The value of the range is computed as follows.

\(\begin{array}{c}{\rm{Range}} = {\rm{Maximum}}\;{\rm{Value}} - {\rm{Minimum}}\;{\rm{Value}}\\ = 103 - \left( { - 46} \right)\\ = 149.0\end{array}\)

Thus, the value of the range is equal to 149.0 minutes.

Statistics vs. parameter

A value is said to be a statistic is if it represents a sample and is computed from a sample.

A value is said to be a parameter if it represents a population and is computed from a population.

Here, the data values form a sample that is selected from the entire population of American Airlines flights from New York to LA.

Thus, the results/values computed from this sample are considered statistics and not parameters.