Comparision between two random samples

Question

I have the following simple random samples which denote the scores obtained by some students over $10$:

Random sample A: 3.5 4.5 4.813 5.065 5.252 5.438 5.547 5.586 5.937 6.025 6.025 6.345 6.375 6.438 6.563 6.688 6.75 6.875 7.188 7.5 7.625 8 8.125 8.565 8.938

Random sample B: 3.313 3.69 4.625 4.688 4.875 4.985 5.063 5.375 5.436 5.5 5.563 5.579 5.792 5.948 5.955 6.035 6.268 6.344 6.603 6.814 6.826 7.001 7.014 8.035 8.221

I've been told to compare both samples, and started analyzing the box-plot, including median, quartiles, also the median, extreme values as maximum and minimum of both samples. I would like to interprete also the deviation, but our professor hasn't give us examples of data interpretation nor methods in order to treat the info. I would appreciate some help. This is an statistical summary:

• Count $A:25 \quad B:25$
• Sample mean $A:5.822 \quad B:6.387$
• Standard Deviation $A:1.171 \quad B:1.306$
• Coefficient of variation $A:20.119\% \quad B:20.442\%$
• Minimum $A:3.313 \quad B:3.500$
• Maximum $A:8.221 \quad B:8.938$
• Rank $A:4.908 \quad B:5.438$
• Median $A:5.792 \quad B:6.375$
• First quartile $A:5.063 \quad B:5.547$
• Third quartile $A:6,603 \quad B:7.188$
• Interquartile range $A:1.540 \quad B:1.641$

and the box-plot is:

Basically, what I've said is that the worst score is in group $A$, and I've given the difference between that score and the worst score in group $B$. The same for the best scores in both groups. I've also said that this by far provides useful information, and that mean gives us a $0.565$ points difference between the group $B$ and $A$. The median and the quartile division in the box-plot makes us infer that, as group, the group $B$ is better prepared than the group $A$. However, roughly, I see that the box diagrams of both samples have the same appearence, just displaced in score axis. Also I've tried to point out a small asymmetry in the case of the quartiles and median in group $A$. What do you recommend me to analyze now? Is what I've said ok? Thanks in advance.

Answer 1

Your analysis is comprehensive enough, just for the sake of completeness/ confirmation consider the following. When you comment on a boxplot there are two points that you should discuss

1. Location: Indeed due to the fact that the median, quartiles, max/min of group B are slightly higher than these of group A you can infer that group B is better prepared than group A (however, the difference is not very big).
2. Variability: Both boxplots indicate the same variability within each group. That can be seen from the length of the boxplots. You also mention this point "... both have the same appearence...", but you should (in my opinion) add the word variability.

Further points that can be made are

1. Outliers or extreme values: There are no such observations.
2. Skewness: You do also discuss this point. However, I do not agree that there is a slight asymmetry (this is too stringent). I rather believe that both boxplots indicate completely symmetrical distributions that can be adequately discribed by the normal distribution.

In sum, your approach is ok, just consider one-two additions with more proper vocabulary, and for me the correction that there is a slight asymmetry (I would write that they are symmetrical).