Say that you're a high school teacher and that you're marking a recent test you gave your students. You're not too sure if you made it too easy or too difficult. Let's analyze the results using our 5 number summary calculator to find out, shall we?
The test was out of 50 points, and the results are:
32, 21, 38, 12, 44, 42, 37, 36, 21, 9, 40, 33, 22, 25, 27, 29, 30, 48, 19, 17, 30, 22, 45, 42.
There are twenty-four tests, and at first glance, it's very difficult to see if, generally speaking, it went well or not. Fortunately, the 5-number summary calculator will give us some insight into the answer, so let's see what we get.
All we need to do is input the entries one by one. Observe that when we open the 5-number summary calculator, we see only eight fields where we can input numbers. However, once we fill these in, new ones will appear, and, all in all, the calculator allows up to thirty values.
We write our numbers in the fields marked as #1 up to #24. Observe how a partial answer is already shown when we input the second value and how it changes with every number we give. Also, note how our entries are not ordered from smallest to largest. The calculator does the ordering for us and even gives us the tidied sequence under the variable fields.
Once we input the last number, we'll see the five-number summary of our statistics problem. Before we analyze it, let's grab a piece of paper and see how to find the 5-number summary ourselves.
First of all, we need to order our numbers from smallest to largest. Oh, bother... That's already some overtime in comparison to using the Omni Calculator. Anyway, here it is:
9, 12, 17, 19, 21, 21, 22, 22, 25, 27, 29, 30, 30, 32, 33, 36, 37, 38, 40, 42, 42, 44, 45, 48.
Just as we described in the above section, we begin by finding the minimum and the maximum:
minimum = 9,
maximum = 48.
Well, no one got zero, so that's a good thing. But no one got the maximum, either. Anyway, let's leave the analysis for later and get back to the other three values from the five-number summary.
Now, we calculate the median of our dataset. Since we have 24 entries (which is an even number), we'll need to find the average of two middle entries. Since 24 / 2 = 12, they'll be the 12th and the 13th. We look back at our ordered sequence, count through the values, and see that they are 30 and 30. Well, that makes the calculation a piece of cake:
median = (30 + 30) / 2 = 30.
Lastly, we need the first and third quartiles. We know that they are, respectively, the medians of the first and the second half of the entries, which, in our case, are the first 12 and the last 12 values. Again, 12 is an even number, which means that here we'll also have to find the average of two numbers.
Since 12 / 2 = 6, the first quartile will be the mean of the 6th and the 7th number, i.e., of 21 and 22. Similarly, the third quartile will need the 18th and the 19th (since 12 + 6 = 18), which are 38 and 40. This gives
1st_quartile = (21 + 22) / 2 = 21.5,
3rd_quartile = (38 + 40) / 2 = 39.
So what is the 5-number summary of our sequence? It's given by
- Minimum:
9; - First quartile:
21.5; - Median:
30; - Third quartile:
39; and - Maximum:
48.
This seems like quite a good distribution if you ask us. Let's also check how it looks on the box and whisker plot:
Certainly, most students scored slightly above the middle value of 25 points. This suggests that the students didn't just flip a coin before choosing the answer. Also, the median is at 30, so more than half of them passed.
This data analysis will be useful when we have to prepare the next test. But that will be the last one this school year, so maybe we should make it slightly easier? After all, most probably, they're already thinking about how they'll go to the beach once it's all over. And who are we to blame them? We could use a nap ourselves...
