The 2010 roster of the Pittsburg Steelers professional footb

The 2010 roster of the Pittsburg Steelers professional football team included 7 defensive lineman and 9 offensive lineman. The weights in pounds of the defensive lineman were 305 325 305 300 285 280 298 and the weights of the offensive lineman were 338 324 325 304 344 315 304 319 318 Make a stemplot of the weights of the defensive lineman and find the five-number summary. Make a stemplot of the weights of the offensive lineman and find the five-number summary. Does either group contain one or more clear outliers Which group of players tends to be heavier. The Department of energy provides fuel economy ratings for all cars and light trucks sold in the United States. Here are the estimated miles per gallon for city driving for the 129 cars classified as midsize in 2010, arranged in increasing order:^6

Solution

We are given that the 2010 roster of the Pittsburgh Steelers professional football team included 7 defensive lineman and 9 offensive lineman.

The data for weights in pounds of the defensive linemen were,

305 325 305 300 285 280 298

and the data for weights in pounds of the offensive linemen were,

338 324 325 304 344 315 304 319 318

Stem-and-Leaf Plot: a display where the data is organized by place value, very useful if you need to find the mean, median, or mode of the data set.

Steps to make a stem plot :

i) Arrange data in increasing order.

i) Writing the data in numerical order may help to organize the data, but is NOT a required step. Ordering can be done later.

ii) Separate each number into a stem and a leaf. Since these are two digit numbers, the tens digit is the stem and the units digit is the leaf.

iii) Group the numbers with the same stems. List the stems in numerical order. (If your leaf values are not in increasing order, order them now.)

A stem-and-leaf plot shows the shape and distribution of data.

Stemplot of the weights of the defensive lineman are,

Now we have to find five number summary using this plot.

Five number summary are minimum, Q1(first quartile), Q2(second quartile or median), Q3(third quartile) and maximum.

Here total observations (n) = 7

n is odd,

therefore, median = (n+1)/2 ^th observation

median = (7 + 1) / 2^th observation

median = 4^th observation

median = 300

There are 3 values above (larger than) and below (smaller than) the median.

First quartile (Q1) is,

Q1 is the median of first three values.

First three values are 280 285 and 298.

Q1 = 285

Similarly Q3 (third quartile) is median of last three values.

Last three values are 305 305 and 325.

Q3 = 305

Minimum of the data is 280 and maximum of the data is 325.

Five number summary for  weights of the defensive lineman is,

Stemplot of the weights of the offensive lineman are,

Now we have to find five number summary using this plot.

Here total observations (n) = 9

n is odd,

therefore, median = (n+1)/2 ^th observation

median = (9 + 1) / 2^th observation

median = 5^th observation

median = 319

There are 4 values above (larger than) and below (smaller than) the median.

First quartile (Q1) is,

Q1 is the median of first four values.

First four values are 304 304 315 318

Here n is even that is 4.

So median of 4 values is [ (n/2)^th+(n/2 + 1) ]^th ] / 2

Q1 (median of 4 values) = [(4/2)^th + (4/2+1)^th ] / 2

Q1 = ( 2nd observation + 3rd observation ) / 2

Q1 = (304 + 315) / 2 = 309.5

Similarly Q3 (third quartile) is median of last four values.

Last four values are 324 325 338 344

Q3 = (325 + 338) / 2 = 331.5

Minimum of the data is 304 and maximum of the data is 344.

Five number summary for  weights of the offensive lineman is,

stem	leaf
28	0 5
29	8
30	0   5 5
31
32	5