Question

The data below shows the daily number of cars that passes through a roundabout for some randomly selected days.
43 21 9 33 15 33 49 18 28 46 25
40 32 58 20 25 24 25 62 38 20

1.The quartile deviation is
2. Calculate the Bowley skewness coefficient, correct to two decimal places

Answer 1

(1) The quartile deviation is 10.5.

(2) The Bowley's Skewness coefficient is 0.28.

Step-by-step explanation:

We are given the data below which shows the daily number of cars that passes through a roundabout for some randomly selected days.

Arranging the data in ascending order we get;

9, 15, 18, 20, 20, 21, 24, 25, 25, 25, 28, 32, 33, 33, 38, 40, 43, 46, 49, 58, 62.

 

Firstly, we will calculate the all three quartiles, i.e;

$Q_1$ = First or lower quartile

$Q_2$ = Median

$Q_3$ = Third or upper quartile

Also, here number of observations (n) = 21

Now, the first or lower quartile is calculated as;

                    $Q_1 = (\frac{n+1}{4})^{th} \text{ obs.}$

                    $Q_1 = (\frac{21+1}{4})^{th} \text{ obs.}$  

                     $Q_1 = (\frac{22}{4})^{th} \text{ obs.}$

                     $Q_1 = 5.5^{th} \text{ obs.}$

So,  $Q_1 = 5^{th} \text{ obs} + 0.5[6^{th} \text{ obs} - 5^{th} \text{ obs} ]$

       $Q_1 = 20+ 0.5[21 - 20 ]$

        $Q_1 = 20+ 0.5 = 20.5$

Similarly, the third or upper quartile is calculated as;

                    $Q_3= 3(\frac{n+1}{4})^{th} \text{ obs.}$

                    $Q_3 = 3(\frac{21+1}{4})^{th} \text{ obs.}$  

                     $Q_3 = (\frac{66}{4})^{th} \text{ obs.}$

                     $Q_3 = 16.5^{th} \text{ obs.}$

So,  $Q_3= 16^{th} \text{ obs} + 0.5[17^{th} \text{ obs} - 16^{th} \text{ obs} ]$

       $Q_3= 40+ 0.5[43 - 40 ]$

        $Q_3= 40+ 1.5 = 41.5$

Now, the middle quartile or median is calculated as;

               $Q_2 =(\frac{n+1}{2})^{th} \text{ obs.}$

               $Q_2 =(\frac{21+1}{2})^{th} \text{ obs.}$

               $Q_2 =(\frac{22}{2})^{th} \text{ obs.}$

                $Q_2 =11^{th} \text{ obs.}$ = 28

(1) The quartile deviation is given by;

              Quartile deviation  =  $\frac{Q_3-Q_1}{2}$

                                              =  $\frac{41.5-20.5}{2}$ = 10.5

(2) Bowley's Skewness coefficient formula is given by;

                             =  $\frac{Q_1-2Q_2+Q_3}{Q_3-Q_1}$

                             =  $\frac{20.5-2(28)+41.5}{41.5-20.5}$

                             =  $\frac{2}{7}$  =  0.28