Lesson 4.B.1 - Sampling Distribution for Two Means
Key Question: Has social media significantly impacted mental health?
Content: Sampling Distribution for Two Means & Conditions
Alignment: CED Topic 4.6
Video
Course Resources
Resources for teaching our AP® Statistics curriculum.
- Lesson Flow - timing and flow of class, using our lesson materials
- Pacing Guide - pacing our units, with daily or block schedules
- CED Alignment Guide - aligning our lessons to the AP® Statistics Course and Exam Description
Teaching Resources
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- Discussion Norms - our model discussion norms for the classroom
- Letter to Parents - letter to share with parents about our nonpartisan approach
- Teaching Math on Civic Topics - tips for teaching math lessons that cover civic topics
Lesson Notes
Lesson-specific insights from the creators of this lesson.
In this lesson, we return to a context covered previously in the course: Does social media harm teenagers’ mental health? Specifically, we take another look at an experiment conducted by researchers at Iowa State University, in which students were randomly assigned to use social media as normal or to limit their usage to 30 minutes per day. After some time passed, participants in both groups took psychological evaluations, and researchers compared the results. Previously, we explored the design of this experiment. In this lesson, we take a closer look at the results. In particular, students investigate whether the difference in average depression levels between the groups was large enough to suggest that it didn’t occur by chance alone.
- Calculate the sampling distribution for the difference between two population means
- Use the sampling distribution for the difference between two population means to evaluate claims
- Check the conditions for sampling two means and describe the purpose of each condition
Before proceeding: Familiarize yourself with the lesson materials linked above (e.g. handout, handout key, slides, video). Then, for additional background and teaching tips from the lesson creators, check out the sections below.
- We’ve found that many teenagers tend to be curious and open to having discussions about mental health and social media. However, that openness quickly fades if they feel that they’re being “given a lecturing” about social media. Starting the lesson with the examples (in the lesson slides and video) of past concerns about radio, television, and video games provides a helpful “we know some concerns have been overblown in the past” framing to the lesson. Then, presenting the more recent statistics about mental health declines feels less like a lecture, and more like the presentation of a compelling question: Are these trends just associated with the spread of social media, or are they causally connected?
- Many students are frequent social media users. Bringing that expertise into the room can feel empowering, and we find that students are often excited to dive into this lesson and share their own experiences. At the same time, other students may not feel as comfortable discussing their own experience with social media, especially as it relates to mental health. We recommend keeping this lesson’s discussions at the statistical level, while also allowing students to share their own experiences according to their own level of comfort.
- It’s helpful to focus on the fact that the sampling distribution in this lesson is a distribution of differences in sample means. This can help students distinguish the values in the sampling distribution from individual depression scale scores or individual sample means.
First, download this lesson's Handout Key and read through its Discussion Question section. Then, check out our model discussion norms and the additional background notes below.
- If students are stuck, here’s a helpful framing question to get them started: “Is a 5.8 unit difference surprising? Or could it happen by chance? How can you tell?” This will help students identify that they have to use the sampling distribution to judge whether a difference is unusual or not.
- If students need support understanding the concept of the sampling distribution, ask them to imagine repeatedly shuffling participants between two groups. Imagine no treatment is being applied to either group. For each shuffle, they find the difference in the mean depression scale scores between the two groups. These differences compose the sampling distribution, and they provide a reference for judging what we could expect from random assignment alone.
- The opening of the lesson mentions increased rates of major depressive episodes, anxiety disorders, and emergency department visits for self-harm among young people. It’s possible that the first two indicators could be attributed to more frequent detection and reporting of mental health disorders, rather than true increases in the underlying prevalence of these disorders. However, the same can’t be said about the third metric, since standards for reporting among emergency departments have not changed over the time periods mentioned in the lesson.
- Students may wonder: In the Iowa State social media experiment, why did researchers only ask participants to limit their social media use, rather than fully controlling it? In the full paper, the researchers state, “One consistent finding has been that complete abstinence from social media may not be sustainable for the average user. A less strict approach is to limit social media use by monitoring. Monitoring limited usage, as opposed to abstinence, may be more sustainable and practical.” So, researchers wanted to know if a more sustainable approach that required less coercion – merely telling folks to reduce their social media time – would still produce an effect. And it did.
- Students interested in exploring this topic further can check out the increasing body of experimental research on social media use and mental health, including the following articles: Brailovskaia, Swarlik, Grethe, Schillack, & Margraf (2023), Davis & Goldfield (2025), Graham, Mason, Riordan, Winter, & Scarf (2021), Hunt, All, Burns, & Li (2021), Kleemans, Daalmans, Carbaat, & Anschütz (2018), Lambert, Barnstable, Minter, Cooper, & McEwan (2022), Thai, Davis, Mahboob, Perry, Adams, & Gold (2023), and Yuen, Koterba, Stasio, et al. (2019).
- Emphasize the distinction between the difference in population means, μ1 − μ2, and the difference in sample means, x̄1 − x̄2. The population difference has a fixed value, although we typically do not know what that value is. The sample difference is a statistic that can vary across different samples or different random assignments.
- Check out the optional video here for more information about how the formula for the standard deviation of the sampling distribution is derived. For the AP Exam, students need to be able to recognize the formula and use it. They don’t need to know its derivation. However, for students who are curious about where the formula comes from, this resource is helpful and provides a window into more advanced statistics coursework.
- Technically, because this lesson describes an experiment with random assignment, the distribution used for inference is a randomization distribution rather than a sampling distribution. Randomization distributions describe the results of repeatedly reassigning treatments, whereas sampling distributions describe the results of repeatedly drawing random samples. For these procedures, however, the mathematical results are equivalent, so AP Statistics does not formally distinguish between the two.
Student Supports
Lesson-specific resources to support all learners.
- As students learn two sample procedures, they will need to distinguish them from the matched pairs procedures introduced in Lessons 4.A.3 and 4.A.5. Reinforce that two groups do not necessarily mean two independent samples: if observations across the groups have a meaningful pairing, such as two measurements from the same person or from identical twins, a matched pairs procedure is appropriate; otherwise, the groups should be treated as independent samples and a two-sample procedure is appropriate.
- Students may lose track of the direction of a difference when working with two groups. Encourage them to identify what Group 1 minus Group 2 means in context before calculating. A quick contextual statement such as “a positive difference means that Group 1 has the higher mean” can help prevent sign errors and incorrect interpretations later in the analysis.
- If students need support understanding the sampling distribution, ask them to imagine repeatedly shuffling participants between two groups. Imagine no treatment is being applied to either group. For each shuffle, they find the difference in the mean depression scale scores between the two groups. These differences compose the sampling distribution, and they provide a reference for judging what we could expect from random assignment alone.
- Vocabulary used in the context of the lesson may include words that are unfamiliar or have several meanings. In particular, the following mathematical terms may need clarification or a definition provided:
- Sampling distribution
- Independent samples
- Matched pairs
- In addition, the following contextual terms may need clarification or a definition provided:
- Depression scale
- Social media usage
- Mental health
- Treatment group
- Introducing the following language and framing can help students describe the difference between matched pairs procedures and two-sample procedures: “Matched pairs procedures involve finding the mean of many differences. Two-sample procedures involve finding the difference between two means.”