📊AP Statistics Review
AP Stats Unit 1 Practice FRQ Prompt Answers & Feedback
The FRQ section of the AP Stats exam can be tricky. To help, review these student responses to a unit 1 practice prompt, with corresponding feedback from Fiveable teacher Jerry Kosoff!
On the AP Statistics Exam, Section II consists of 6 free-response questions: 5 shorter/multi-part questions in 65 minutes and 1 investigative task in 25 minutes. This page models the style of a standard free-response question focused on exploring data, not the investigative task.
AP Statistics free-response questions are scored with point-based rubrics; the comments below use informal feedback language to describe quality of reasoning, not official score labels.
Unit 1 FRQ Practice Prompt
James drives his car to his job using an identical route each day. He noticed that the time it takes him to get to work (“commute time”) is sometimes impacted by the time at which he leaves his house. For 50 consecutive trips to his job, James noted how long his commute time took (in minutes), as well as whether he left his house before or after 8:00AM. The histograms below show the distribution of commute times on the 24 days he left his house after 8:00AM as well as on the 26 days he left his house before 8:00AM.
a. Write a few sentences comparing the distribution of commute times for days when James leaves his house before 8:00AM vs the days when James leaves his house after 8:00AM.
b. Based on the histograms, which set of days has a larger mean commute time? Justify your response.
FRQ Practice Submission 1
(a) Both the distributions for James’ commute times for days leaving his house before and after 8:00 am are unimodal with a peak approximately at 14-15 minutes for the travel time after 8 am with it being skewed to the right and a peak at approximately 15-16 minutes before 8 am. There appears to be no unusual features for both travel time distributions. The range of travel time after 8 am appears to be greater than the travel time before 8 am. The median for travel time after 8 am appears to be less than travel time before 8 am.
(b) The set of days after 8 am seems to have a larger mean commute. This is due to the travel time distrubution after 8 am having a right skew which would pull the mean towards the right indicating a larger value. While the set of days before 8 am doesn’t have a right skew so the mean would be smaller compared to travel time after 8 am.
Teacher FRQ Feedback
In part (a), you make a clear comparison of shape, spread, and center, and your answer is in context. Your descriptions of shape are correct, your comparison for spread is correct, but your comparison for center is incorrect; the median time for both distributions is between 15-16 minutes. This response correctly addresses shape, spread, and context, but the center comparison is inaccurate, so it would likely lose credit for the comparison of center.
In part (b), you correctly link the shape of the distribution after 8:00 AM to the likely impact on the mean commute time, and explain how that is not true in the distribution before 8:00 AM. This response would likely earn full credit for this part because it gives the correct conclusion and justifies it with the effect of right skew on the mean. Nicely done!
FRQ Practice Submission 2
a) James’ commute time to his work on days where he left the house after 8AM is approximately skewed to the right with a median between 14 and 15 minutes. On the other hand, when James leaves his house before 8AM, the graph of his commute time is approximately normal with a median between 15 and 16 minutes. The range is larger when he leaves after 8AM as compared to when he leaves earlier, and there don’t seem to be any outliers/gaps in either of the histograms.
b) Based on the histograms, the James’ mean commute time when he leaves the house after 8AM is approximately more than when he leaves the house before 8AM. Since the histogram is skewed to the right, that also pulls the mean higher, resulting in a likely higher mean than that of the second graph.
Teacher FRQ Feedback
In part (a), you make a clear comparison between the spread of the two distributions (“the range is larger when he leaves after 8AM as compared to when he leaves earlier”), but you do not do the same for your description of the median time. Also, check the graphs again: the median for the “after 8:00 AM” distribution should also be between 15-16 minutes. This response gets shape, spread, and context mostly right, but it would likely lose credit for the center comparison.
In part (b), you make a correct claim about the mean commute time being larger when he leaves after 8AM vs. before 8AM, but your explanation could use a little more clarity. When you say “the histogram” and then “the second graph,” be careful to specify which histogram you mean, and explain why the other graph does not have the same feature. This would likely earn credit for the correct conclusion, but stronger communication would make the justification more secure.
Nicely done—there are a few minor tweaks that will get you to a stronger, full-credit-style answer in no time, so keep practicing!
FRQ Practice Submission 3
A. When James leaves his house after eight am he gets to work in a median time that is between 15 and 16 minutes which is the same median time for if he left before 8 am. Neither graph appears to have any outliers or unusual features. The shape of the graph for leaving after 8 am is skewed right with a peak that is approximately between 14 and 15 minutes while the graph, for before 8 am the graph is unimodal and approximately normal with a peak approximately between 15 and 16 minutes. The range of travel time for after 8 am is at some time between 9 and 10 minutes, which is greater than the range of travel time for before 8 am which is at some time between 6 and 7 minutes.
B. The graph for after 8 am has a greater mean commute time. Since in part A we concluded that both graphs have the same median commute time and the graph for traveltime after 8 am is skewed right we can conclude that the graph for after 8 am has a greater commute time. Since that graph is skewed right we know that the mean is greater than the median. Since there in part A we concluded that travel time is before 8 am is approximately normal we can assume the mean is about the same as the median, which means the mean of the after graph is greater than the before graph.
Teacher FRQ Feedback
In part (a), you make a clear comparison of center (“same median time”), a clear comparison of spread, a correct description of shapes, and your answer is in context. This response would likely earn full credit for the comparison because it addresses center, shape, spread, and unusual features appropriately.
In part (b), you give a clear explanation of why the right skew in the “after 8 AM” distribution would impact the mean commute time, and you tie in your work from part (a) to strengthen your argument. This response would likely earn full credit because it correctly compares the likely means using shape and median. You did a good job of giving a thorough answer without saying more than you needed.
FRQ Practice Submission 4
(a) The distribution of travel times after 8am appears to be skewed right while the distribution of travel times after 8am appears approximately normal. The median travel time after 8am is the same for the median travel times before 8am. The range of the travel times before 8am appears to be smaller than the range of travel times after 8am.
(b) The set of data that has a larger mean commute time is the histogram leveled travel time after 8am. This is because the histogram distribution is skewed right meaning the mean is being pulled to the right. In this case, the mean is larger than the median where as the histogram who’s travel time is before 8am would have a similar mean and median since it is approximately normal.
Teacher FRQ Feedback
You’ve done a good job of getting to the point in both parts (a) and (b) and saying exactly what you need to say without much more. Your answer in part (a) includes appropriate comparisons and the context of the problem, though it looks like there is a typo because you say “after 8am” twice when comparing the two distributions. Your answer in part (b) mentions the impact of shape on the mean/median in both histograms, which is important for communicating your answer. These responses would likely earn full credit if that labeling mistake in part (a) is only a wording slip and not a misunderstanding.
FRQ Practice Submission 5
a) After 8am, James’ commute times are unimodal and skewed to the right. His commute time ranges from 12 minutes to 22 minutes. On most days, the commute is about 14-15 minutes. Before 8am, Jame’ commute times are unimodal and relatively symmetrical. The times range from 12 minutes to 19 minutes. The peak commute time is between 15 and 16 minutes.
b) Based on the data, the commute times after 8am likely have a higher mean than before 8am. The graph is skewed right, causing the mean to increase. We can assume that the mean increases beyond that of times before 8am because its graph is not skewed.
Teacher FRQ Feedback
In part (a), you do a good job of describing the distributions of commute times, but you do not compare them. If the AP exam asks you to compare distributions, they’re looking for you to use explicit comparison terms, such as “greater than,” “less than,” or “about the same as.” Also, be sure to explicitly compare shape, center, and spread. You mention the shape and spread of each distribution, but you do not directly compare the spreads, and you do not give a measure of center for either distribution; “peak” is useful description, but it is not a measure of center. As it stands, this response would likely not earn much, if any, credit for part (a).
In part (b), you do a good job of clearly communicating your choice (after 8am) and why it’s your choice (“skewed right”), as well as why the other graph is not your choice (“its graph is not skewed”). Good explanation! This response would likely earn full credit for part (b).
FRQ Practice Submission 6
(a) the median of both the after 8:00am graph and the before 8:00am graph is between 15-16 minutes. the after 8:00 graph has greater variability, range, and higher standard deviation compared to before 8:00; after 8:00 graph is also skewed slightly right. before 8:00 is normal with a jutting peak from 15-16 mins. since before 8:00 graph is normally distributed, its mean will also be between 15-16. on the other hand the after 8:00 graph is skewed right, the mean will be pulled up, resulting in a higher mean than median. both graphs are unimodal.
(b) as discussed in (a), the set of days after 8:00 has a higher mean. while both medians are between 15-16 mins, this set is skewed right, thus dragging the mean upwards.
Teacher FRQ Feedback
In part (a), you do a good job of comparing the distributions in context. Your comparison of medians is correct, and your comparison of spread is correct. Quick note: you could have used “variability,” “range,” or “standard deviation” to make these comparisons; you did not need all three. Similarly, you only need one measure of center. Be careful with your description of shape, though: calling the before-8:00 AM graph “normal” is too strong. “Approximately normal” or “roughly symmetric” would be safer, since we cannot verify a true Normal model from a histogram alone. That wording issue could cost credit on a strict rubric.
In part (b), you do a good job of clearly explaining your choice and connecting it to the fact that skew impacts the mean. This response would likely earn full credit.
FRQ Practice Submission 7
a.) The histogram comparing the commute times for days when James leaves his house before 8:00 AM is unimodal vs the days when James leaves his house after 8:00 AM is slightly skewed right. There is a peak at 14-15 minutes after 8:00 AM and a peak at 15-16 minutes before 8:00 AM. The median for both distributions is between 15-16 minutes. There are no apparent gaps/outliers in the histograms. The range of travel times for after 8:00 AM is bigger at between 12 to 22 minutes, while the range of travel times for before 8:00 AM is between 12 to 19 minutes.
b.) Based on the histogram, the set of days after 8:00 AM travel time has a larger mean commute because of the slight skew to the right. The right skewness will increase the mean, while the travel time before 8:00 AM does not have a skew to right causing it to have a smaller mean.
Teacher FRQ Feedback
In part (a), you do a good job of making in-context comparisons, correctly comparing the median and range for the two distributions, and making comments on the lack of obvious outliers. Be careful with your shape comments; typically, “unimodal” by itself is not enough to fully describe shape because a distribution can be unimodal and symmetric or unimodal and skewed. “Bimodal” by itself is more specific, but for a unimodal graph you should also say whether it is approximately symmetric or skewed.
In part (b), you choose the correct histogram and clearly defend your choice with a reference to the mean being impacted by the skew of the histogram. Nice job!
FRQ Practice Submission 8
a. The distribution of James’s commute time took for after 8:00 am is unimodal and skewed to the right while the distribution of James’s commute time took for before 8:00 am is unimodal and fairly symmetric. The center for the histogram for travel time after 8:00 am is the same as travel time before 8:00 am. The distribution of commute time for after 8:00 am has a larger spread than before 8:00 am. There is no apparent outlier.
b. The distribution of commute time for after 8:00 am has a larger mean. Looking at the graphs, the distribution for commute time for after 8:00 am is skewed right meaning the mean is being pulled to the larger values while the distribution of commute time before 8 am is fairly symmetric. A distribution that is highly right-skewed is likely to have a substantially larger difference between the mean and median.
Given that the median of both distributions is roughly the same, so it makes sense that the highly right-skewed distribution (after 8:00 am) is the one with the bigger gap between the mean and median compared to a more symmetric and, therefore, the one with the higher mean commute time.
Teacher FRQ Feedback
In part (a), you make clear comparisons of center, shape, and spread in context of the scenario. One thing to be careful with is your use of the term “center” being the same. As you correctly argue in part (b), the median travel times are about the same, but the mean travel times are not likely the same because of the skewness of the “after 8:00 AM” graph. Therefore, your use of the generic term “center” could be too vague. It’s important to be specific in cases like that.
As mentioned, your answer to part (b) is very good—it thoroughly explains your choice and why the other choice is incorrect. Well done!
FRQ Practice Submission 9
a.) The distribution of commute times before 8:00 am is more symmetric with a mean of approximately 15 min. it is also less variable than the distribution of times after 8:00 am with Range(before)= 7 and Range(after)= 10. The distribution of commute times after 8:00 am is more right skewed with a median around 15. (using median instead of the mean to determine the center because of its resistance to skews).
b.) Due to the apparent right skew in the after 8:00 am distribution, the mean (being non-resistant to skew) will be larger than the mean from the before 8:00 am distribution that has a symmetric shape, leaving the mean at approximately 15 minutes.
Teacher FRQ Feedback
In part (a), you make a clear comparison of the shapes of the graphs and a clear comparison of the spread/variability. Sidenote: be careful talking about ranges on histograms. The ranges are approximately 7 and approximately 10 minutes, respectively, because the numbers on the x-axis for a histogram are endpoints of intervals. The longest commute time after 8:00 AM may not have been exactly 22 minutes; it could have been 21.6 minutes. Unfortunately, even though you name the center for each distribution, you do not explicitly compare them. When asked to compare distributions, you must say something like “less than,” “more than,” or “about the same as.” This response would likely lose credit for center because it only implies the comparison.
In part (b), you correctly cite the right skew as having an impact on the mean and choose the correct distribution, but there may be a typo or wording mix-up in your explanation. Be sure to clearly distinguish mean from median.
FRQ Practice Submission 10
a) The distribution of commute times for days when James leaves his house before 8:00 AM is approximately normal and strongly skewed right. The center (median) of the distribution is approximately between 15 and 16 minutes.The spread (range) of the distribution is 10 minutes. There are no gaps or outliers in the distribution. Comparatively, the distribution of commute times for days when James leaves his house after 8:00 Am is fairly symmetrical. The center (median) of the distribution is between 15 and 16 minutes. The spread (range) of the distribution is 7 minutes. Additionally, there are no gaps or outliers in the distribution.
b) Since the normal distribution of commute times for days when James leaves his house before 8:00 AM is strongly skewed right, the mean will be greater than the median. Since the normal distribution of commute times for days when James leaves his house after 8:00 Am is fairly symmetrical, the mean will equal the median. Thus, the days when James leaves his house before 8:00 AM will have a larger commute time.
Teacher FRQ Feedback
In reading your descriptions, I think you may have switched labels in part (a). It looks like the first thing you describe is the “after” graph—the one with the larger spread and right skew—but you called it the “before” graph. Also, while you give several details, you do not explicitly compare the centers and spreads. When asked to compare graphs, you have to “spell it out” with phrases like “greater than,” “less than,” or “about the same as.” Also, some things to be careful with: (1) it’s impossible for a distribution to be both “approximately normal” and “strongly skewed right,” and (2) when discussing range on a histogram, we should use language like “approximately 10 minutes” because we do not know the exact min and max values from the bins alone. The after-8:00-AM histogram is the one that is right-skewed with the larger spread, while the before-8:00-AM histogram is approximately symmetric. A distribution cannot be both approximately normal and strongly right-skewed.
In part (b), your conclusion is incorrect. The distribution for commute times after 8:00 AM is the one that is skewed right, so it is the distribution more likely to have the larger mean. A correct explanation would be: “Because the after-8:00-AM distribution has about the same median as the before-8:00-AM distribution but is skewed right, its mean is pulled toward the larger values, so the after-8:00-AM days likely have the larger mean commute time.”
FRQ Practice Submission 11
A. The distributions of commute times on the 24 days he left after 8 am and the 26 days he left before 8 am both do not contain any outliers, and both their medians are about the same (approximately 15-16 minutes). However, the shape of the distribution of commute times on the 24 days he left his house after 8 am is more skewed (specifically to the right) than the distribution of commute times on the 26 days he left his house before 8 am (which is more unimodal). Also, the spread of the distribution of commute times on the 24 days he left after 8 am is larger than the spread of the distribution of commute times on the 26 days he left before 8 am.
B. The distribution of commute times on the 24 days he left after 8 am has the larger mean commute times in minutes. The medians of both the distributions are about the same — approximately 15-16 minutes. But since the distribution of commute times on the 24 days he left after 8 am is skewed to the right, this means its mean is greater than its median. And since the distribution of commute times on the 26 days he left before 8 am is more unimodal, this means its mean equals its median. So, therefore the mean of the distribution of commute times on the 24 days he left after 8 am is larger than the mean of the distribution of commute times on the 26 days he left before 8 am.
Teacher FRQ Feedback
Your answer to part (a) is very thorough and addresses the scenario in context with comparisons of center, shape, spread, and outliers. One small thing: be careful in your shape comparisons. It’s good to describe the after-8am times as skewed, but “unimodal” by itself for the before-8am times is a little too general. You should go with something like “more symmetric” to make the comparison more precise.
In part (b), you give a very clear correct answer with reasons relating to the shape of the graph. I have the same note about “unimodal” vs. “symmetric.”
FRQ Practice Submission 12
a) Comparing the distribution of commute times for days when James leaves his house before 8:00AM vs the days when James leaves his house after 8:00AM we observe that both are unimodal however after 8 is right-skewed while before 8 is roughly symmetrical. Also after 8 has a median between 15-16 while before 8 also has a median between 15-16. After 8 has a larger spread with a range of about 10 minutes while before 8 has a range of only 7 minutes. Lastly, neither distribution has any outliers as all values fall within the inner/outer fences.
b) After 8 will have a smaller mean as the majority of data is to the left where the time would be smaller whereas before 8 is roughly symmetrical meaning that the mean won’t be pulled as far left as in the other distribution. THus before 8 will have a larger commute time.
Teacher FRQ Feedback
In part (a), you give a very thorough comparison, including shape, center, and spread. You also say “about 10 minutes” for the range after 8am, which is important on histograms because you can’t know the exact min and max values for sure.
In part (b), you pick the wrong distribution. Remember that the mean is impacted by skew and outliers. A right-skewed distribution tends to have the mean pulled toward the larger values in the right tail, so it will usually have a mean greater than its median. Therefore, the after-8am distribution is more likely to have the larger mean in this case.
FRQ Practice Submission 13
a) the distribution when James leaves his house before 8:00am has a roughly symmetric shape whereas the other distribution it’s skewed to the right. the distribution for the travel time before 8 has a range of 7 and the other one has a range of 10. they have roughly the same median at 15-16 minutes. there seem to be no outliers.
b) since the graph is skewed to the right it means that its mean will be greater than the mean of the distribution of travel time before 8. this is because the graph is skewed and since the mean is non resistant to outlier it will get pulled towards the tail but towards the tail in this situation there are larger values which would make the mean greater.
Teacher FRQ Feedback
In part (a), you provide appropriate descriptions of center, shape, and spread for each of the distributions, but you do not provide comparisons. Comparisons in AP Stats involve explicit comparison words like “more,” “less,” or “about the same as.”
In part (b), you give a clear description of how the mean is impacted by the skew in the after-8am distribution. Nice job.
FRQ Practice Submission 14
a) the distribution of travel time after 8 am is slightly skewed to t he right while the distribution of travel time before 8am is approximately normal. both do not have any potential outliers. the center is the same for both distributions at 15-16 and the spread is larger for the distribution of travel time after 8 than before 8.
b) the distribution of travel time for after 8 am has a larger mean as the median for both distributions is 15-16 but since the distribution for before 8 is approximately normal, the mean will be a similar value to the median but the distribution of after 8 is skewed to the right which means the mean will be pulled to the tail, in this case towards a higher value. Therefore the distribution of travel time for after 8am will have a higher mean.
Teacher FRQ Feedback
Nice responses! You have clear comparisons in part (a) of center, shape, and spread. You could strengthen part (a) by referring specifically to the median as your measure of center; the means will likely be different, as you correctly defend in part (b).
FRQ Practice Submission 15
a. On the graph of days when James leaves his house after 8 am the shape is fairly unimodal with a right skew whereas in the graph for before 8 am the shape is unimodal and moderately symmetric. The center for both graphs, according to the median appears to be between 15 and 16 minutes for his commute time. The spread for travel time after 8 am is greater than the spread for travel time before 8 am. Neither graph seems to show any apparent outliers.
b. The graph showing travel time after 8 am has a larger mean commute time because, while the median shows both graphs having a similar center, the mean is pulled in the direction of skew so since the after 8 am graph is skewed to the right, it pulls the mean up making it larger than the travel time before 8 am because that one is fairly symmetric.
Teacher FRQ Feedback
Both answers are very solid and written in a way that would likely earn points on a typical rubric. You provide clear comparisons in part (a) of center, shape, and spread, and provide a strong description in part (b) that is in context and mentions both distributions. Nice job!