2026 AP Statistics FRQ #1 Walkthrough | Full Solution & Explanation for Free Response Question 1

Añadido: 2026-05-09

[00:00:00]What's up my sad stars? In this video, we're going to go over the 2026 AP Statistics FRQ question number one. Now, this question dealt with the weight of several goats and we had to analyze the data. Let's dive into the question right now.

[00:00:14]So, the question starts off by talking to us about a goat farmer who raises two breeds of goats, breed H and breed J. He wants to compare the weights of the two breeds and takes independent random samples of 14 goats from each breed.

[00:00:25]The weights in pounds of each goat is measured and the weights of the 14 brand H goats are recorded and we are given a nice list of all of the weights of the 14 brand H goats.

[00:00:35]Now, the first question is pretty simple. It says, "Use the data in the list to determine the five-number summary for the sample of breed H." Now, you got to remember what a five-number summary is. It's your min, your max, Q1, Q3, and your median. Now, you can actually plug all this data into Desmos, NumWorks, or your TI-84 calculator and it will calculate all of those values for you without doing any work and you don't have to show any work. All you can do is list them. But, it's actually pretty easy to do by hand as well. First off, there are 14 goats. Add one, 15, divide by two is seven and a half. That means that the median is located right between the seventh and the eighth goat.

[00:01:11]Now, they were very kind to put the data already in order for us. So, here is the seventh goat, here is the eighth goat.

[00:01:17]So, the 7.5 is going to be right in between, so it's right here is where that median's going to fall. It's between 62 and 66. So, add 62 and 66 and divide by two and we get a median of 64.

[00:01:28]Now, that puts seven goats below that median and seven is a nice odd number, which means there is a middle and that's Q1. The middle of the bottom seven values, 56 is Q1. The minimum is obviously 48. Then, the middle of the top seven goats is also a nice number because there are seven goats up there, which means there's a nice middle right there at 72 and that's going to be Q3 and then our max is 80. So, pretty simple to simply list the min is 48, Q1 is 56, the median is 64, Q3 is 72, and the max is 80. But again, you do not have to do all that work by hand, even though it didn't take that long. You're more than welcome to use a calculator to find it.

[00:02:08]Now, part B introduces to brand J goats.

[00:02:11]The distribution of the weights for brand J goats is displayed in a box plot. Use the five-number summary from part A and the box plot for breed J to compare the center and variability of the distributions. So, they want us to compare just the center and the spread for goats H and goats J.

[00:02:32]All right. Now, the very first thing I thought to do was, well, grab the five-number summary from the box plot.

[00:02:37]So, this is for breed J. So, I estimated that that min was around 48, Q1 I said a little bit above 55, so 56. The median looked right around 64, a little bit below 65. Q3 was right at 80, it looked like, and the max I estimated to be around 88 or 87, 89, somewhere around there, but I said 88. So, now that I have a good feel for the five-number summary for brand J, I could write a nice conclusion here about comparing them. Now, the first thing I noticed that the medians, the center, were pretty much the same, right around 64. But the spreads were it was a little bit different. So, here's what I wrote. The distributions of the weight brand H and brand J have similarities and differences. The center for both brand H and J are approximately the same, around 64 lb. So, I noticed that that was something they had in common.

[00:03:24]However, the variabilities differ. The range for brand H is 49 to 80, while the brand Excuse me, I keep saying brand.

[00:03:32]Breed J is a little more spread out, going from 48 to 88. So, it does have a higher max, so it does go a little bit higher for the range.

[00:03:40]And then the interquartile range for breed J is also more spread out, going from 56 to 80, while the while the middle 50% for brand H is only 56 to 72.

[00:03:53]And I got a couple typos in there. I'm very sorry about that. So again, what we notice is that the overall spread min to max is bigger for breed J, but also that middle 50% the interquartile range between Q1 and Q3 is also a little bit more spread out. Now realistically, the biggest thing is that right hand side.

[00:04:11]That right hand side is more spread out for J than it is for H. Other than that, the left hand side below the median are roughly the same.

[00:04:21]All right. Then we move into one more question, part C.

[00:04:25]A stem and leaf plot is another way to display the weights of goats. The distribution of weight for breed H goats is displayed in the given stem and leaf plot. So this was the data that was given to us way back in the very beginning. These are the weights for breed H goats. So consider the five number summary that we actually found back in part A and the stem and leaf plot. If a box plot were created using the five number summary that we found in part A, what characteristics of the shape of the distribution of the weights for breed H goats would be apparent from the stem and leaf plot but not from the box plot? And then they want us to explain why a box plot would not display that particular characteristic. So the first thing we want to talk about is shape, right? Shape is what you don't necessarily see in a box plot but you can see in a stem and leaf plot. So we see a bimodal shape here. We see these two kind of um peaks in the 50s and in the 70s and we're not going to see that in the box plot. So here is my write-up as to why that is. I said the characteristics that would be seen in the stem and leaf plot but not the box plot would be the bimodal shape with a peak in the 50s and in the 70s.

[00:05:34]Now, why? Why is that? Well, the bimodal shape will not be seen because a box plot shows how values are clustered within the intervals between the min, Q1, median, Q3, and the max. A stem and leaf plot on the other hand displays the individual data values, so you can actually see where the data are concentrated. If the data are bimodal, the stem and leaf plot will reveal two separate clusters or peaks, which we did see. The box plot cannot show the two peaks because different distributions can have the same five-number summary.

[00:06:03]The box plot compresses all the detailed information into a five values, so the presence of two modes is hidden.

[00:06:10]So, hopefully that makes sense in terms of first off, what it is that we were able to see in the stem and leaf plot, and then why that is. So, did you have to have word for word everything I had down here to get a perfect score?

[00:06:21]Absolutely not, but something along the idea that it's the shape that we're not going to be able to see, and the reason is why is because remember, a box plot is really meant to show us the four sections of the of the data, the bottom 25, the top 25%, and that middle 50%, and where they fall and how they're spread out. You know, a box plot's really more just show us the center and the spread, not showing us the shape.

[00:06:44]All right, that's it for question one. I didn't think it was overall too bad.

[00:06:46]Some kids might struggle a little bit here with part C, but hopefully you got some of it right that you're able to get a couple points.