Q. 3.210 - Elementary Statistics

Question

Dallas Mavericks. From the ESPN website, in the Dallas Mavericks Roster, we obtained the following ages, in years, for the players on that basketball team for the $2013-2014$ season.

Obtain the following parameters for these ages. Use the appropriate mathematical notation for the parameters to express your answers.

a. Mean

b. Standard deviation

c. Median

d. Mode

e. IQR

Step-by-Step Solution

Answer

(a) The Mean is, $\mu =27.28$

(b) The Standard deviation is, $\sigma =5.1745$

(c) The median is, $\eta =25$

(d) The mode is, 

           Mode$=23, 25$

(e) The $IQR=8.5$

Part (a) Step 1: Given information

We are given that, 

The data set is, $$\begin{gathered} x_i=25, 24, 31, 36, 23, 32, 23, 25, 27, 30, 28, 38, 23, 20, 21, 35, 24, 25, \\ 23, 35, 25 \end{gathered}$$

No. of values are,

   $N=21$

Part (a) Step 2: Simplify

We know that,

The mean is given as,

$$\begin{gathered} \mu =\frac{\sum _{}{x}_i}{N}=\frac{25+24+31+36+....+25+23+35+25}{21} \\ =27.2857 \end{gathered}$$

Part (b) Step 1: Given information

We need to find out the standard deviation. 

Part (b) Step 2: Simplify

We know that the standard deviation is the square root of variance which is given as, 

AGE	DEVIATION	SQUARED DEVIATION
$25$	$-2.2857$	$5.2244$
$24$	$-3.22857$	$10.7958$
$31$	$3.7143$	$13.796$
$36$	$8.7143$	$75.939$
$23$	$-4.2857$	$18.3672$
$32$	$4.7143$	$22.2246$
$23$	$-4.2857$	$18.3672$
$25$	$-2.2857$	$5.2244$
$27$	$-0.2857$	$0.0816$

$30$	$2.7143$	$7.3674$
$28$	$0.7143$	$0.5102$
$38$	$10.7143$	$114.7962$
$23$	$-4.2857$	$18.3672$
$20$	$-7.2857$	$53.0814$
$21$	$-6.2857$	$39.51$
$35$	$7.7143$	$59.5104$
$24$	$-3.2857$	$10.7958$
$25$	$-2.2857$	$5.2244$
$23$	$-4.2857$	$18.3672$

$-35$

$7.7143$

$59.5104$

$25$

$-2.2857$

$5.2244$

TOTAL

$562.2857$

The standard deviation  is, 

    $\sigma =\sqrt{\frac{Total}{N}}=\sqrt{\frac{562.2857}{21}}=5.1745$

Part (c) Step 1: Given information

We need to find out the median of the data 

Part (c) Step 2: Simplify

Data in the ordered list is, 

$$\begin{gathered} 20, 21, 23, 23, 23, 23, 24, 24, 25, 25, 25, 25, 27, 28, 30, 31, 32, 35, 35, \\ 36, 38 \end{gathered}$$

The median of the sorted list is the middle value of the list,

Therefore, median is, $\eta =25$

Part(d) Step 1: Given information

We need to find out the mode of given data. 

Part(d) Step 2: Simplify

We know that mode is the most occurring value in the data set.

Therefore, mode is, $23, 25$

Part (e) Step 1: Given information

We need to find the value of IQR 

Part (e) Step 2: Simplify

As we know, 

The first quartile is the median of data below the median,

Therefore, $Q_1=\frac{23+23}{2}=23$

And the third quartile is the median of data above median,

Therefore, $Q_3=\frac{31+32}{2}=31.5$

Hence, the interquartile range is, 

       $IQR=Q_3-Q_1=31.5-23=8.5$