Question

How many days in advance do travelers purchase their airline tickets? Below are data showing the
advance days for a sample of 13 passengers on United Airlines Flight 812 from Chicago to Los Angeles.

11, 7, 11, 4, 15, 14, 71
29, 8, 7, 16, 29, 249

(a) Calculate the mean, median, and mode.
(b) Which is the best measure of central tendency? Why?
(c) Base on your discussion in part (b), which is the best measure of variation? Determine its
value.

Answer 1

Below are data showing the
advance days for a sample of 13 passengers on United Airlines Flight 812 from Chicago to Los Angeles.

11, 7, 11, 4, 15, 14, 71, 29, 8, 7, 16, 29, 249

a) defin, n= total number of observations

● For non frequency type discrete data mean is given by, where, Xi = value of ith observation.

● For non frequency type discrete data median is the middle most value of the data set. For calculating median of a discrete data set at first we ordered the data in ascending order the If the total number of observations n is odd then the median is th observation and if total number of observations n is even then median is the any value between th observation and th observation , for simplicity we take the average of th observation and observation.

● For non frequency type discrete data mode is that value which occur most frequently that is which value has highest frequency is called mode of that data set.

So mean is given by ,  

Now we ordered the data in ascending order and we get,

4,7,7,8,11,11,14,15,16,29,29,71,249.

Here, total number of obs = n = 13 is odd. So the median is th =   observation. From the ordered data we clearly see that 7th observation is 14. i.e, median = 14

And from the data we clearly see that value 7, 11, 29 has frequency 2 and others value has frequency 1. So this data set has 3 mode and 7, 11, 29 this three value is the mode of this data set.

b) Form the data we cleary say that every value is in between 4 to 29 excluding two value which are 71 and 249 so this two value is called outliers.

Median is the best measure of central tendency for this data set.

Because:-

We know that median is the appropriate measure to control this type of situation because mean can be influenced by the outliers. And for this data set we get three mode. So from this discussion we say that median is the best measure of central tendency.

c) Some important measures of variation measure is Range, Standard deviation, Mean absolute deviation, Quartile Deviation.

Now range is given by, Range = Highest value -Lowest value

So if outlier present range is highly affected by outlier.

Standard deviation and mean absolute deviation also affected by outliers because this two measure is based on mean of the data set and if outliers present we know that mean is also affected.

Quartile deviation is useful measure of variation because it is less affected by outliers or a skewed data set.

So for our given data set Quartile deviation is the best measure of variation. Now Quartile deviation is given by the formula,

where, Q3 = 3rd quartile and Q1 = 1st quartile

Now, th observation and th observation.

So, th observation and,

th observation

Now for simplicity we take the average of 3rd and 4th observation from ordered data set as a Q1. From the ordered data set 3rd observation= 7 and 4th observation= 8. So Q1=(8+7)/2 = 7.5 and we take average of 10th and 11th observation as a Q3. From the ordered data 10th and 11th both observation is 29. So Q3= 29.

Now quartile deviation is given by,