Question

3. On the first week of classes, thirty-four students sat for a Math Diagnostic Test. Their scores (out of a maximum score of 30) arranged in ascending order were: 10, 12, 13, 13,15, 15, 16, 17, 18, 19, 21, 23, 23, 23, 23, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 27, 27, 28, 28, 29, 29,29,30,30 (a) [2 marks] Find the 5-number summary for these data. (b) [2 marks] Are there any outliers? Show your work on how you identify outliers. (c) [2 marks] Draw a boxplot of this distribution.

Answer 1

The data are as follows: 10, 12, 13, 13,15, 15, 16, 17, 18, 19, 21, 23, 23, 23, 23, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 27, 27, 28, 28, 29, 29,29,30,30.

The number of students (n) appeared in test are 34 (even) that is n=34

(a). The five number summary are: minimum, Q1, Median(Q2), Q3, Maximum.

So minimum=10 and maximum=30

Median=(n/2th term+(n/2+1)th term)/2= (17th term+18th term)/2

=(25+25)/2= 25

Q1= (n+1)/4th term=(34+1)/4th term = 8.75th term

Here 8th term is 17 and 9th term is 18.

so Q1= 0.75*18+(1-0.75)*17= 17.75

and Q3= 3(n+1)/4th term= 3*35/4= 26.25th term

Here 26th term is 27 and 27th term is 27.

So Q3= 0.25*27+(1-0.25)*27= 27

So the five number summary are 10, 17.75, 25, 27, 30.

(b). Outliers can be identified from the following formulas:

Q1-1.5(IQR)

Q3+1.5(IQR)

If any value in the data lie outside these values, then it is called outlier(s).

where IQR stands for Inter Quartile Range and is defined as; Q3-Q1= 27-17.75=9.25

Hence the outlier limits are

17.75-1.5*9.25=3.875

27+1.5*9.25=40.875.

In our case no value is lying outside (3.875, 40.875). Therefore, we have no outlier here.

(c). Draw a boxplot.

I have plotted here a boxplot using R software and attached the image of that plot. Here the line at bottom show the minimum value and just above it represents the first quartile(Q1), the darkest line is for median , and the line just above it, shows third quartile (Q3) and the last line represent the maximum value of the data. There is no other point beyond the box, that means there are no outliers in the data