7.7 Paired data

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7. Quantitative data

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Paired data

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Inference with paired data

In the last two sections, we worked with confidence intervals and significance tests for the difference between two means using independent random samples. Now we’ll look at a different situation: a quantitative variable measured twice for the same individual (or for two very similar, closely matched individuals).

Definitions

Paired data: A setting in which a quantitative variable is measured twice on the same individual (or on two closely matched individuals). Instead of comparing two separate groups, you create one new variable — the difference between the paired measurements — and then do one-sample inference on the mean of those differences.

Why pairing matters: an SAT example

A classic example is testing the effectiveness of a new SAT preparation course. Suppose you collect unpaired data by taking one group of $50$ students who take the prep course and a completely different group of $50$ students who don’t, and then you compare the two group means. That comparison might be misleading because the two groups could differ in important ways before the course even begins. For example:

The prep-course group might include higher-achieving students to start with (or the opposite).
Any pre-existing differences between the groups would show up in the results, even if the course had no effect.

There’s also a practical concern: students and parents might view it as unfair to offer the course to only some students, raising ethical questions about the study design.

A paired design addresses both issues by measuring the same students twice. For example, you could:

Start with one group of all $100$ students.
Give everyone the SAT before the prep course.
Have everyone complete the SAT preparation course (the course is the “treatment”).
Give everyone the SAT after the course.
Compute each student’s score change (after − before) and analyze whether the mean change is significantly different from $0$ .

In practice, you’d also think about issues like practice effects or test familiarity when interpreting a before-and-after design.

The calculations for paired data are the same as in the previous section; the difference is the data that we should collect in order to get the most accurate results possible by controlling for differences between the two groups regardless of the applied treatment.

Other situations that call for paired data

Example 1: Medical trials

If researchers measure people’s blood pressure before and after they take a new medication, the researchers will have a better understanding of the specific impacts that the medication has on blood pressure. If they tested on two different groups (one group of people who has not taken the medication and one group of people who has), it would not account for the differences in blood pressure that the groups may have regardless of the medication. Certain people have higher blood pressure than others for a variety of reasons.

Example 2: Product testing

Consumers try both Brand A and Brand B of a product and rate each one on a scale of $1$ to $5$ . The reason in this case that it is best to use paired data rather than the non-paired method is that it’s possible that consumers consuming Brand A normally rank products lower in general or vice versa. Some people are quick to give any decent product a perfect $5$ -star rating while others very rarely give $5$ stars and would give only a $4$ -star rating for a product that exceeded their expectations but wasn’t their very favorite. In short, some people are less easily impressed than others. Also, people who have been on the receiving end of ratings understand that even a $4$ -star rating can be considered a “bad” rating to certain employers and would result in the worker being disciplined if they get too many bad ratings. People who have experienced this are more likely to give a $5$ -star rating almost no matter what as long as the product was not terrible. Using paired data means that the way people choose to rate products is controlled for.

Paired or independent? Ask yourself:

Are the two sets of measurements taken from the same pool, or are they from two separate and unrelated pools?

If they are from the same pool, the best option is most likely to use paired data.
If they are from two separate and unrelated pools, then using the non-paired approach makes much more sense in order to accurately compare and figure out any potential differences between the two groups.

Paired data and paired design

Two measurements from same individual or closely matched individuals
Analyze differences between paired measurements (new variable)
Use one-sample inference on mean of differences

Advantages of paired design

Controls for individual differences (reduces variability)
Increases power to detect true effects
Avoids confounding from pre-existing group differences

Examples of paired data use

Medical trials: before-and-after measurements on same person
Product testing: same consumer rates both products, controls for rating tendencies

Choosing paired vs. independent samples

Paired: measurements from same pool/individuals
Independent: measurements from two unrelated groups

Analysis approach

Calculate difference for each pair (e.g., after − before)
Apply one-sample procedures to the set of differences

Next | 8.1 Goodness of fit test

Paired data

Inference with paired data

Definitions

Paired data: A setting in which a quantitative variable is measured twice on the same individual (or on two closely matched individuals). Instead of comparing two separate groups, you create one new variable — the difference between the paired measurements — and then do one-sample inference on the mean of those differences.

Why pairing matters: an SAT example

The prep-course group might include higher-achieving students to start with (or the opposite).
Any pre-existing differences between the groups would show up in the results, even if the course had no effect.

There’s also a practical concern: students and parents might view it as unfair to offer the course to only some students, raising ethical questions about the study design.

A paired design addresses both issues by measuring the same students twice. For example, you could:

Start with one group of all $100$ students.
Give everyone the SAT before the prep course.
Have everyone complete the SAT preparation course (the course is the “treatment”).
Give everyone the SAT after the course.
Compute each student’s score change (after − before) and analyze whether the mean change is significantly different from $0$ .

In practice, you’d also think about issues like practice effects or test familiarity when interpreting a before-and-after design.

Other situations that call for paired data

Example 1: Medical trials

If researchers measure people’s blood pressure before and after they take a new medication, the researchers will have a better understanding of the specific impacts that the medication has on blood pressure. If they tested on two different groups (one group of people who has not taken the medication and one group of people who has), it would not account for the differences in blood pressure that the groups may have regardless of the medication. Certain people have higher blood pressure than others for a variety of reasons.

Example 2: Product testing

Consumers try both Brand A and Brand B of a product and rate each one on a scale of $1$ to $5$ . The reason in this case that it is best to use paired data rather than the non-paired method is that it’s possible that consumers consuming Brand A normally rank products lower in general or vice versa. Some people are quick to give any decent product a perfect $5$ -star rating while others very rarely give $5$ stars and would give only a $4$ -star rating for a product that exceeded their expectations but wasn’t their very favorite. In short, some people are less easily impressed than others. Also, people who have been on the receiving end of ratings understand that even a $4$ -star rating can be considered a “bad” rating to certain employers and would result in the worker being disciplined if they get too many bad ratings. People who have experienced this are more likely to give a $5$ -star rating almost no matter what as long as the product was not terrible. Using paired data means that the way people choose to rate products is controlled for.

Paired or independent? Ask yourself:

Are the two sets of measurements taken from the same pool, or are they from two separate and unrelated pools?

If they are from the same pool, the best option is most likely to use paired data.
If they are from two separate and unrelated pools, then using the non-paired approach makes much more sense in order to accurately compare and figure out any potential differences between the two groups.

Achievable AP Statistics

7. Quantitative data

Our AP Statistics course is currently in development and is a work-in-progress.

Paired data

4 min read

Font

Discuss

Feedback

Inference with paired data

Definitions

Paired data: A setting in which a quantitative variable is measured twice on the same individual (or on two closely matched individuals). Instead of comparing two separate groups, you create one new variable — the difference between the paired measurements — and then do one-sample inference on the mean of those differences.

Why pairing matters: an SAT example

The prep-course group might include higher-achieving students to start with (or the opposite).
Any pre-existing differences between the groups would show up in the results, even if the course had no effect.

There’s also a practical concern: students and parents might view it as unfair to offer the course to only some students, raising ethical questions about the study design.

A paired design addresses both issues by measuring the same students twice. For example, you could:

Start with one group of all $100$ students.
Give everyone the SAT before the prep course.
Have everyone complete the SAT preparation course (the course is the “treatment”).
Give everyone the SAT after the course.
Compute each student’s score change (after − before) and analyze whether the mean change is significantly different from $0$ .

In practice, you’d also think about issues like practice effects or test familiarity when interpreting a before-and-after design.

Other situations that call for paired data

Example 1: Medical trials

If researchers measure people’s blood pressure before and after they take a new medication, the researchers will have a better understanding of the specific impacts that the medication has on blood pressure. If they tested on two different groups (one group of people who has not taken the medication and one group of people who has), it would not account for the differences in blood pressure that the groups may have regardless of the medication. Certain people have higher blood pressure than others for a variety of reasons.

Example 2: Product testing

Consumers try both Brand A and Brand B of a product and rate each one on a scale of $1$ to $5$ . The reason in this case that it is best to use paired data rather than the non-paired method is that it’s possible that consumers consuming Brand A normally rank products lower in general or vice versa. Some people are quick to give any decent product a perfect $5$ -star rating while others very rarely give $5$ stars and would give only a $4$ -star rating for a product that exceeded their expectations but wasn’t their very favorite. In short, some people are less easily impressed than others. Also, people who have been on the receiving end of ratings understand that even a $4$ -star rating can be considered a “bad” rating to certain employers and would result in the worker being disciplined if they get too many bad ratings. People who have experienced this are more likely to give a $5$ -star rating almost no matter what as long as the product was not terrible. Using paired data means that the way people choose to rate products is controlled for.

Paired or independent? Ask yourself:

Are the two sets of measurements taken from the same pool, or are they from two separate and unrelated pools?

If they are from the same pool, the best option is most likely to use paired data.
If they are from two separate and unrelated pools, then using the non-paired approach makes much more sense in order to accurately compare and figure out any potential differences between the two groups.

Paired data and paired design

Two measurements from same individual or closely matched individuals
Analyze differences between paired measurements (new variable)
Use one-sample inference on mean of differences

Advantages of paired design

Controls for individual differences (reduces variability)
Increases power to detect true effects
Avoids confounding from pre-existing group differences

Examples of paired data use

Medical trials: before-and-after measurements on same person
Product testing: same consumer rates both products, controls for rating tendencies

Choosing paired vs. independent samples

Paired: measurements from same pool/individuals
Independent: measurements from two unrelated groups

Analysis approach

Calculate difference for each pair (e.g., after − before)
Apply one-sample procedures to the set of differences

Next | 8.1 Goodness of fit test

Paired data

Inference with paired data

Definitions

Paired data: A setting in which a quantitative variable is measured twice on the same individual (or on two closely matched individuals). Instead of comparing two separate groups, you create one new variable — the difference between the paired measurements — and then do one-sample inference on the mean of those differences.

Why pairing matters: an SAT example

The prep-course group might include higher-achieving students to start with (or the opposite).
Any pre-existing differences between the groups would show up in the results, even if the course had no effect.

There’s also a practical concern: students and parents might view it as unfair to offer the course to only some students, raising ethical questions about the study design.

A paired design addresses both issues by measuring the same students twice. For example, you could:

Start with one group of all $100$ students.
Give everyone the SAT before the prep course.
Have everyone complete the SAT preparation course (the course is the “treatment”).
Give everyone the SAT after the course.
Compute each student’s score change (after − before) and analyze whether the mean change is significantly different from $0$ .

In practice, you’d also think about issues like practice effects or test familiarity when interpreting a before-and-after design.

Other situations that call for paired data

Example 1: Medical trials

If researchers measure people’s blood pressure before and after they take a new medication, the researchers will have a better understanding of the specific impacts that the medication has on blood pressure. If they tested on two different groups (one group of people who has not taken the medication and one group of people who has), it would not account for the differences in blood pressure that the groups may have regardless of the medication. Certain people have higher blood pressure than others for a variety of reasons.

Example 2: Product testing

Consumers try both Brand A and Brand B of a product and rate each one on a scale of $1$ to $5$ . The reason in this case that it is best to use paired data rather than the non-paired method is that it’s possible that consumers consuming Brand A normally rank products lower in general or vice versa. Some people are quick to give any decent product a perfect $5$ -star rating while others very rarely give $5$ stars and would give only a $4$ -star rating for a product that exceeded their expectations but wasn’t their very favorite. In short, some people are less easily impressed than others. Also, people who have been on the receiving end of ratings understand that even a $4$ -star rating can be considered a “bad” rating to certain employers and would result in the worker being disciplined if they get too many bad ratings. People who have experienced this are more likely to give a $5$ -star rating almost no matter what as long as the product was not terrible. Using paired data means that the way people choose to rate products is controlled for.

Paired or independent? Ask yourself:

Are the two sets of measurements taken from the same pool, or are they from two separate and unrelated pools?

If they are from the same pool, the best option is most likely to use paired data.
If they are from two separate and unrelated pools, then using the non-paired approach makes much more sense in order to accurately compare and figure out any potential differences between the two groups.