Question

In Exercises 21–24, find the mean and median for each of the two samples, then compare the two sets of results.

Bank Queues Waiting times (in seconds) of customers at the Madison Savings Bank are recorded with two configurations: single customer line; individual customer lines. Carefully examine the data to determine whether there is a difference between the two data sets that is not apparent from a comparison of the measures of center. If so, what is it?

Single Line 390 396 402 408 426 438 444 462 462 462

Individual Lines 252 324 348 372 402 462 462 510 558 600

Short answer

The summarization of the two results:

Measure	Single line	Individual Line
Mean	429.0 seconds	429.0 seconds
Median	432.0 seconds	432.0 seconds

The variation in the datasets is not apparent from the measures of the center.

Step-by-step solution

Given information

The waiting times are recorded by the customers.

Single Line	390	396	402	408	426	438	444	462	462	462
Individual Line	252	324	348	372	402	462	462	510	558	600

Compute mean for each data set

The formula for the mean is:

$\overline{x}=\frac{\sum x}{n}$, where xrepresents the observations and ndenotes the count of the observations.

Substitute the values for a single line.

$$\begin{gathered} {\overline{x}}_S=\frac{390+396+402+...+462}{10} \\ =\frac{4290}{10} \\ \approx 429 \end{gathered}$$

Substitute the values for individual lines.

$$\begin{gathered} {\overline{x}}_I=\frac{252+324+348+...+600}{10} \\ =\frac{4290}{10} \\ =429 \end{gathered}$$

Thus, the mean values for single and individual lines are the same, i.e., 429.0 seconds.

Compute median for each set of measurements

To compute the median, find the middle values as per the total counts of the observations.

If the counts are even, the average of the middle values is the median.

Compute the median for the waiting lines of single lines.

The number of observations is10.

Arrange the observations in ascending order.

390
396
402
408
426
438
444
462
462
462

The middlemost observations are 426 and 438.

The median is given as:

$$\begin{gathered} M=\frac{426+438}{2} \\ =\frac{864}{2} \\ =432 \end{gathered}$$

Compute the median for individual measurements.

The number of observations is 10.

Arrange the observations in ascending order.

252
324
348
372
402
462
462
510
558
600

The middlemost observations are 402 and 462.

The median is given as:

$$\begin{gathered} M=\frac{402+462}{2} \\ =\frac{864}{2} \\ =432 \end{gathered}$$

Thus, the median values for single and individual lines are the same, 432.0 seconds.

Summarize the data and examine the difference between datasets

The summarization of the two results gives:

Measure	Single line	Individual Line
Mean	429.0 seconds	429.0 seconds
Median	432.0 seconds	432.0 seconds

The results are identical in both groups.

The waiting times for a single line have a minimum value of 390 and a maximum value of 462. On the other hand, the waiting times for individual lines vary between 252 and 600.4

The level of variation between the extreme values is large.

This characteristic of the data is not clear from the measure of the center. It can be established using the measures of variation.