Question

The average number of minutes Americans commute to work is 27.7 minutes. The average commute time in minutes for 48 cities are as follows.

Albuquerque	23.6	Jacksonville	26.5	Phoenix	28.6
Atlanta	28.6	Kansas City	23.7	Pittsburgh	25.3
Austin	24.9	Las Vegas	28.7	Portland	26.7
Baltimore	32.4	Little Rock	20.4	Providence	23.9
Boston	32.0	Los Angeles	32.5	Richmond	23.7
Charlotte	26.1	Louisville	21.7	Sacramento	26.1
Chicago	38.4	Memphis	24.1	Salt Lake City	20.5
Cincinnati	25.2	Miami	31.0	San Antonio	26.4
Cleveland	27.1	Milwaukee	25.1	San Diego	25.1
Columbus	23.7	Minneapolis	23.9	San Francisco	32.9
Dallas	28.8	Nashville	25.6	San Jose	28.8
Denver	28.4	New Orleans	32.0	Seattle	27.6
Detroit	29.6	New York	44.1	St. Louis	27.1
El Paso	24.7	Oklahoma City	22.3	Tucson	24.3
Fresno	23.3	Orlando	27.4	Tulsa	20.4
Indianapolis	25.1	Philadelphia	34.5	Washington, D.C.	33.1

(a)

What is the mean commute time (in minutes) for these 48 cities? (Round your answer to one decimal place.)

minutes

(b)

Compute the median commute time (in minutes).

minutes

(c)

Compute the mode(s) (in minutes). (Enter your answers as a comma-separated list.)

minutes

(d)

Compute the third quartile (in minutes).

minutes

Answer 1

a. Mean value is

b. The median is the middle number in a sorted list of numbers. So, to find the median, we need to place the numbers in value order and find the middle number.

Ordering the data from least to greatest, we get:

20.4   20.4   20.5   21.7   22.3   23.3   23.6   23.7   23.7   23.7   23.9   23.9   24.1   24.3   24.7   24.9   25.1   25.1   25.1   25.2   25.3   25.6   26.1   26.1   26.4   26.5   26.7   27.1   27.1   27.4   27.6   28.4   28.6   28.6   28.7   28.8   28.8   29.6   31   32   32   32.4   32.5   32.9   33.1   34.5   38.4   44.1   

As you can see, we do not have just one middle number but we have a pair of middle numbers, so the median is the average of these two numbers:

Median=

c. The mode of a set of data is the value in the set that occurs most often.

Ordering the data from least to greatest, we get:

20.4   20.4   20.5   21.7   22.3   23.3   23.6   23.7   23.7   23.7   23.9   23.9   24.1   24.3   24.7   24.9   25.1   25.1   25.1   25.2   25.3   25.6   26.1   26.1   26.4   26.5   26.7   27.1   27.1   27.4   27.6   28.4   28.6   28.6   28.7   28.8   28.8   29.6   31   32   32   32.4   32.5   32.9   33.1   34.5   38.4   44.1   

Since both 23.7 and 25.1 occur 3 times, the modes are 23.7 and 25.1 ( this data set is bimodal).

So mode is 23.7,25.1

d. The third quartile (or upper quartile or 75th percentile) is the median of the upper half of the numbers. So, to find the third quartile, we need to place the numbers in value order and find the upper half.

20.4   20.4   20.5   21.7   22.3   23.3   23.6   23.7   23.7   23.7   23.9   23.9   24.1   24.3   24.7   24.9   25.1   25.1   25.1   25.2   25.3   25.6   26.1   26.1   26.4   26.5   26.7   27.1   27.1   27.4   27.6   28.4   28.6   28.6   28.7   28.8   28.8   29.6   31   32   32   32.4   32.5   32.9   33.1   34.5   38.4   44.1   

So, the upper half is

26.4   26.5   26.7   27.1   27.1   27.4   27.6   28.4   28.6   28.6   28.7   28.8   28.8   29.6   31   32   32   32.4   32.5   32.9   33.1   34.5   38.4   44.1   

The median of these numbers is 28.8.