AP Statistics 1.9 Comparisons of the Distributions for One Quantitative Variable Study Notes - New Syllabus
AP Statistics 1.9 Comparing Quantitative Distributions and Z-Scores Study Notes – New Syllabus
AP Statistics 1.9 Comparing Quantitative Distributions and Z-Scores Study Notes – As per latest AP Statistics Syllabus.
LEARNING OBJECTIVES
- 1.9.A Compare multiple quantitative one-variable graphical representations.
- 1.9.B Compare multiple quantitative one-variable graphical representations of summary statistics.
- 1.9.C Justify a claim using multiple quantitative one-variable graphical representations.
- 1.9.D Calculate z-scores with population parameters.
- 1.9.E Compare z-scores as measures of relative position for distributions.
ESSENTIAL KNOWLEDGE:
- 1.9.A.1 Graphical representations of a quantitative variable can be used to compare important features between two or more distributions of the same quantitative variable. Histograms, back-to-back stem-and-leaf plots, and dotplots may be used to compare center, variability, shape, outliers, clusters, or gaps in two or more distributions. Boxplots may be used to compare center, variability, outliers, and skewness (or symmetry).
- 1.9.B.1 A comparison of graphical representations for two or more distributions can include any of the numerical summaries (e.g., mean, standard deviation, etc.).
- 1.9.C.1 Multiple quantitative one-variable graphical representations may reveal information that can be used to justify claims about the variable in context.
- 1.9.D.1 A standardized score measures the number of standard deviations a data value falls above or below the mean.
- 1.9.D.2 A z-score is calculated as
\( z=\dfrac{x_i-\mu}{\sigma} \)
where \(x_i\) is the data value, \(\mu\) is the population mean, and \(\sigma\) is the population standard deviation. A z-score measures how many standard deviations a data value is above (positive z-score) or below (negative z-score) the mean. When the population mean and standard deviation are unknown, the sample mean and standard deviation may be used to determine a z-score. - 1.9.E.1 z-scores may be used to compare relative positions of individual values within a distribution or between distributions.
1.9.A.1 Comparing Quantitative Distributions Using Graphical Representations
Graphical representations reveal important characteristics that help compare distributions.
Different types of graphs provide different information.
Histograms, dotplots, and stem-and-leaf plots are especially useful for examining:
- Center
- Spread
- Shape
- Outliers
- Clusters
- Gaps
Boxplots are especially useful for comparing:
- Center
- Spread
- Outliers
- Skewness or Symmetry
Comparing Center
The center represents the typical value of a distribution.
When comparing graphs:
- A distribution located farther to the right generally has a larger center.
- A distribution located farther to the left generally has a smaller center.
Measures often used:
- Mean
- Median
Comparing Variability (Spread)
Spread describes how dispersed the data values are.
- A distribution with a wider spread has greater variability.
- A distribution with a narrower spread has less variability.
Measures often used:
- Range
- IQR
- Standard Deviation
Comparing Shape
The shape of a distribution may be:
- Symmetric
- Skewed Right
- Skewed Left
- Uniform
- Unimodal
- Bimodal
Histograms and dotplots are particularly useful for identifying shape.
Comparing Outliers
Outliers appear as observations that are unusually far from the rest of the data.
Outliers may affect:
- Mean
- Range
- Standard Deviation
When comparing distributions, mention any unusual observations that appear.
Comparing Clusters and Gaps
Some distributions contain:
- Clusters (groups of observations)
- Gaps (intervals with no observations)
These features may indicate different subgroups within the data.
| Graph Type | Features Best Compared |
|---|---|
| Histogram | Center, Spread, Shape, Clusters, Gaps, Outliers |
| Dotplot | Center, Spread, Shape, Clusters, Gaps, Outliers |
| Back-to-Back Stem-and-Leaf Plot | Center, Spread, Shape, Outliers |
| Boxplot | Center, Spread, Outliers, Skewness |
AP Statistics Comparison Template
When comparing two quantitative distributions, use the following order:
Center → Spread → Shape → Outliers
This structure appears frequently on AP Free-Response Questions.
A strong comparison should describe:
- Which distribution has the larger center
- Which distribution has greater variability
- How the shapes differ
- Whether outliers are present
Example
Two boxplots summarize test scores from two classes.
Class A:
\( \mathrm{Median=85,\ IQR=10} \)
Class B:
\( \mathrm{Median=78,\ IQR=18} \)
No outliers are shown on either boxplot.
Compare the two distributions.
▶️ Answer / Explanation
Center:
Class A has the larger median. Since: \( \mathrm{85>78} \)
Class A generally achieved higher test scores.
Spread:
Class B has the larger IQR.
Since:
\( \mathrm{18>10} \)
Class B’s scores are more variable.
Shape:
The shape cannot be determined from the information provided.
A complete boxplot would be needed to evaluate skewness or symmetry.
Outliers:
No outliers are present in either distribution.
Overall, Class A performed better on average, while Class B showed greater variability in scores.
Example
Two histograms display the daily exercise times of two groups of adults.
Group A has a roughly symmetric distribution centered around:
\( \mathrm{45} \) minutes.
Group B has a right-skewed distribution centered around:
\( \mathrm{35} \) minutes with several large values above: \( \mathrm{90} \) minutes.
Compare the two distributions.
▶️ Answer / Explanation
Center:
Group A has the larger center because its typical exercise time is around: \( \mathrm{45} \) minutes compared with: \( \mathrm{35} \) minutes for Group B.
Spread:
Group B appears to have greater variability because its values extend much farther to the right.
Shape:
Group A is approximately symmetric.
Group B is skewed right.
Outliers:
The unusually large exercise times above:
\( \mathrm{90} \)
minutes may be potential outliers in Group B.
Therefore, Group A has a higher typical exercise time, while Group B is more variable and more strongly skewed.
1.9.B Compare Multiple Quantitative One-Variable Graphical Representations of Summary Statistics
Graphical representations and summary statistics work together to describe quantitative distributions.
While graphs provide a visual display of the data, numerical summaries provide precise measurements that help compare distributions.
When comparing two or more distributions, statisticians often use:
- Mean
- Median
- Range
- IQR
- Standard Deviation
- Five-Number Summary
These numerical summaries help support conclusions that may be suggested by graphical displays such as:
- Histograms
- Dotplots
- Stem-and-Leaf Plots
- Boxplots
A strong statistical comparison combines graphical evidence with numerical summaries.
1.9.B.1 Comparing Distributions Using Numerical Summaries
A comparison of two or more distributions may include any numerical summary statistic.
These statistics provide objective evidence about the distributions and help justify statistical conclusions.
Common numerical summaries include:
| Statistic | Used to Compare |
|---|---|
| Mean | Typical value (center) |
| Median | Typical value (center) |
| Range | Overall spread |
| IQR | Spread of middle 50% |
| Standard Deviation | Variability around the mean |
| Five-Number Summary | Center, spread, and position |
When comparing distributions, numerical summaries help answer questions such as:
- Which distribution has the higher center?
- Which distribution is more variable?
- Which distribution is more consistent?
- Are the distributions similar or different?
Using Mean and Standard Deviation
When distributions are approximately symmetric and do not contain strong outliers, comparisons often focus on:
- Mean
- Standard Deviation
A larger mean indicates a larger typical value.
A larger standard deviation indicates greater variability.
Using Median and IQR
When distributions are skewed or contain outliers, comparisons often focus on:
- Median
- IQR
These statistics are resistant and provide a better description of the distribution.
AP Statistics Comparison Strategy
When comparing multiple distributions:
- Compare the centers.
- Compare the spreads.
- Describe any differences in shape.
- Mention any outliers.
- Support conclusions using numerical summaries.
This follows the AP Statistics comparison framework:
Center → Spread → Shape → Outliers
| Feature | Possible Numerical Summary |
|---|---|
| Center | Mean or Median |
| Spread | Range, IQR, Standard Deviation |
| Shape | Mean-Median Relationship, Graphs |
| Outliers | IQR Rule, Standard Deviation Rule |
Example
Two classes took the same AP Statistics quiz.
| Statistic | Class A | Class B |
|---|---|---|
| Mean | 88 | 82 |
| Median | 89 | 81 |
| Standard Deviation | 5 | 12 |
Compare the performance of the two classes using the numerical summaries.
▶️ Answer / Explanation
Center:
Class A has a higher mean and median than Class B.
Therefore, students in Class A generally scored higher on the quiz.
Spread:
Class B has the larger standard deviation.
Since:
\( \mathrm{12>5} \)
Class B’s scores are more spread out and less consistent.
Conclusion:
Class A performed better overall and had more consistent scores.
Class B had lower scores on average and greater variability.
Example
Two boxplots summarize weekly exercise times.
Group A:
\( \mathrm{Median=240} \) minutes, \( \mathrm{IQR=60} \) minutes
Group B:
\( \mathrm{Median=180} \) minutes, \( \mathrm{IQR=90} \) minutes
Use the numerical summaries to compare the two groups.
▶️ Answer / Explanation
Group A has the larger median.
Therefore, Group A typically exercises more each week.
Group B has the larger IQR.
Therefore, Group B shows greater variability in exercise time.
The middle 50% of exercise times are more spread out in Group B than in Group A.
Overall, Group A exercises more consistently and for a greater amount of time.
1.9.C Justify a Claim Using Multiple Quantitative One-Variable Graphical Representations
Graphs are powerful tools because they allow statisticians to visualize and compare distributions.
When multiple graphical representations of the same quantitative variable are available, they can provide evidence to support statistical claims.
Rather than relying on opinions or assumptions, statisticians use information displayed in graphs to justify conclusions.
Common graphical representations used to support claims include:
- Histograms
- Dotplots
- Stem-and-Leaf Plots
- Back-to-Back Stem-and-Leaf Plots
- Boxplots
These graphs reveal important characteristics of distributions that can be used as statistical evidence.
1.9.C.1 Using Multiple Graphical Representations to Justify Claims
Multiple quantitative one-variable graphical representations may reveal information that supports claims about a variable in context.
A strong statistical claim should be supported by specific graphical evidence.
When examining graphs, statisticians commonly look for:
- Center
- Spread
- Shape
- Outliers
- Clusters
- Gaps
These characteristics provide evidence that can support conclusions about a distribution.
Claims About Center
Graphs can reveal which distribution has the larger typical value.
Evidence may include:
- Larger median on a boxplot
- Distribution shifted farther right on a histogram
- Larger mean or median from a graphical summary
Claims About Variability
Graphs can reveal which distribution is more spread out.
Evidence may include:
- Larger IQR on a boxplot
- Wider histogram
- Larger overall range
Claims About Shape
Graphs can reveal:
- Symmetry
- Skewness
- Unimodality
- Bimodality
- Uniformity
Shape often provides important evidence about how the data are distributed.
Claims About Outliers
Graphs may reveal unusually large or unusually small observations.
Outliers often appear:
- Beyond whiskers in boxplots
- Separated from the main cluster in dotplots
- Far from most observations in histograms
Outliers should always be mentioned when they affect the interpretation of a distribution.
| Feature | Possible Graphical Evidence |
|---|---|
| Center | Median position, location of distribution |
| Spread | IQR, range, overall width |
| Shape | Symmetry or skewness |
| Outliers | Isolated observations |
| Clusters | Concentrated groups of observations |
| Gaps | Intervals containing no observations |
AP Statistics Claim Structure
When justifying a claim using graphs:
- State the claim.
- Identify graphical evidence.
- Interpret the evidence in context.
A strong response always connects the graphical feature directly to the claim being made.
Avoid statements such as:
“The graph looks higher.”
Instead write:
“The boxplot for Group A has a larger median than Group B, indicating that Group A typically has larger values.”
Example
Two boxplots compare weekly study times for two groups of students.
Group A:
- Median = \( \mathrm{12} \) hours
- IQR = \( \mathrm{4} \) hours
Group B:
- Median = \( \mathrm{8} \) hours
- IQR = \( \mathrm{9} \) hours
Use the graphical information to justify a claim about the study habits of the two groups.
▶️ Answer / Explanation
The boxplots suggest that Group A typically studies more than Group B.
This claim is supported by the medians:
\( \mathrm{12>8} \)
hours.
The boxplots also show that Group B has greater variability because:
\( \mathrm{IQR=9} \)
is larger than:
\( \mathrm{IQR=4} \)
hours.
Therefore, Group A generally studies more, while Group B’s study times vary more from student to student.
Example
Two histograms display daily commute times for workers in two cities.
- City A has a distribution centered near: \( \mathrm{25} \) minutes and appears approximately symmetric.
- City B has a distribution centered near: \( \mathrm{40} \) minutes and is skewed right with several very long commute times.
Use the histograms to justify a claim comparing the commuting patterns of the two cities.
▶️ Answer / Explanation
The histograms suggest that workers in City B generally have longer commute times than workers in City A.
This claim is supported by the centers of the distributions.
City B is centered near:
\( \mathrm{40} \)
minutes, whereas City A is centered near:
\( \mathrm{25} \)
minutes.
The histogram for City B is also skewed right, indicating that some workers have unusually long commute times.
Therefore, City B tends to have longer and more variable commute times than City A.
1.9.D.1 Standardized Scores (z-Scores)
A standardized score, or z-score, measures the number of standard deviations a data value lies above or below the population mean.
A z-score converts an original data value into a standardized value.
The z-score formula is:
\( z=\frac{x-\mu}{\sigma} \)
where:
- \( z \) = z-score
- \( x \) = observed data value
- \( \mu \) = population mean
- \( \sigma \) = population standard deviation
Important AP Statistics Ideas
- A positive z-score means the observation is above the mean.
- A negative z-score means the observation is below the mean.
- The farther the z-score is from 0, the more unusual the observation is.
- Z-scores allow comparisons between different distributions.
For example:
- \( z=2 \) means the value is 2 standard deviations above the mean.
- \( z=-1.5 \) means the value is 1.5 standard deviations below the mean.
| Position Relative to Mean | Sign of z-Score |
|---|---|
| Above Mean | Positive |
| At Mean | Zero |
| Below Mean | Negative |
Example
A population of exam scores has:
- Population Mean: \( \mu=70 \)
- Population Standard Deviation: \( \sigma=10 \)
A student earns a score of:
\( x=85 \)
Calculate the student’s z-score.
▶️ Answer / Explanation
Step 1: Use the z-score formula.
\( z=\frac{x-\mu}{\sigma} \)
Step 2: Substitute the values.
\( z=\frac{85-70}{10} \)
\( z=\frac{15}{10} \)
\( z=1.5 \)
Therefore:
\( \mathrm{z=1.5} \)
Interpretation:
The student’s score is:
\( \mathrm{1.5} \)
standard deviations above the population mean.
Example
A population of adult heights has:
- Population Mean: \( \mu=170 \) cm
- Population Standard Deviation: \( \sigma=8 \) cm
A person has a height of:
\( x=158 \) cm
Calculate the z-score and interpret its meaning.
▶️ Answer / Explanation
Step 1: Apply the z-score formula.
\( z=\frac{x-\mu}{\sigma} \)
\( z=\frac{158-170}{8} \)
\( z=\frac{-12}{8} \)
\( z=-1.5 \)
Therefore:
\( \mathrm{z=-1.5} \)
Interpretation:
The person’s height is:
\( \mathrm{1.5} \)
standard deviations below the population mean.
The negative z-score indicates the observation lies below average.
1.9.D.2 Calculating and Interpreting z-Scores
A z-score is a standardized score that measures how many standard deviations a data value lies above or below the mean.
The z-score is calculated by subtracting the population mean from the data value and then dividing by the population standard deviation.
z-Score Formula
\( z=\frac{x_i-\mu}{\sigma} \)
where:
- \( x_i \) = data value (observation)
- \( \mu \) = population mean
- \( \sigma \) = population standard deviation
- \( z \) = standardized score
The z-score tells us how far a data value is from the mean in units of standard deviations.
Interpretation of z-Scores
- A positive z-score indicates the value is above the mean.
- A negative z-score indicates the value is below the mean.
- A z-score of 0 indicates the value is exactly equal to the mean.
| z-Score | Meaning |
|---|---|
| \( z=0 \) | At the mean |
| \( z=1 \) | 1 standard deviation above the mean |
| \( z=-1 \) | 1 standard deviation below the mean |
| \( z=2 \) | 2 standard deviations above the mean |
| \( z=-2 \) | 2 standard deviations below the mean |
Using Sample Statistics
Sometimes the population mean and population standard deviation are unknown.
In that situation, the sample mean and sample standard deviation may be used to calculate an approximate z-score.
Sample z-Score Formula
\( z=\frac{x_i-\bar{x}}{s} \)
where:
- \( x_i \) = data value
- \( \bar{x} \) = sample mean
- \( s \) = sample standard deviation
This formula standardizes observations relative to the sample rather than the entire population.
Important AP Statistics Idea
The sign of the z-score tells the direction from the mean:
- Positive → Above the mean
- Negative → Below the mean
The magnitude of the z-score tells how unusual the observation is:
- Small absolute value → Close to the mean
- Large absolute value → Far from the mean
Example
The weights of a population of dogs have:
- Population Mean: \( \mu=40 \) pounds
- Population Standard Deviation: \( \sigma=5 \) pounds
A dog weighs:
\( x=50 \)
pounds.
Calculate the z-score and interpret its meaning.
▶️ Answer / Explanation
Step 1: Use the z-score formula.
\( z=\frac{x-\mu}{\sigma} \)
Step 2: Substitute the values.
\( z=\frac{50-40}{5} \)
\( z=\frac{10}{5} \)
\( z=2 \)
Interpretation:
The dog’s weight is:
\( \mathrm{2} \)
standard deviations above the population mean.
Example
A sample of students has:
- Sample Mean: \( \bar{x}=72 \)
- Sample Standard Deviation: \( s=8 \)
One student scored: \( x=60 \) on the exam.
Calculate the student’s z-score using the sample statistics.
▶️ Answer / Explanation
Step 1: Use the sample z-score formula.
\( z=\frac{x-\bar{x}}{s} \)
Step 2: Substitute the values.
\( z=\frac{60-72}{8} \)
\( z=\frac{-12}{8} \)
\( z=-1.5 \)
Interpretation:
The student’s score is:
\( \mathrm{1.5} \)
standard deviations below the sample mean.
The negative sign indicates that the score falls below average.
1.9.E Compare z-Scores as Measures of Relative Position for Distributions
A raw data value by itself does not always indicate how unusual or impressive an observation is.
For example, a score of: \( \mathrm{85} \) might be outstanding in one class but average in another.
To make meaningful comparisons, statisticians use z-scores.
Because z-scores measure distance from the mean in standard deviation units, they provide a standardized way to compare observations.
This allows comparisons:
- Within the same distribution
- Between different distributions
1.9.E.1 Using z-Scores to Compare Relative Position
A z-score describes an observation’s relative position within a distribution.
The z-score formula is:
\( z=\frac{x-\mu}{\sigma} \)
or when population parameters are unknown:
\( z=\frac{x-\bar{x}}{s} \)
A z-score indicates how many standard deviations an observation lies above or below the mean.
Because all z-scores are measured on the same scale, they can be used to compare observations from completely different distributions.
Comparing Relative Position Within a Distribution
Within a single distribution:
- A larger z-score indicates a higher relative position.
- A smaller z-score indicates a lower relative position.
Example:
- Student A: \( z=2.0 \)
- Student B: \( z=1.2 \)
Student A performed better relative to the rest of the group because Student A’s score is farther above the mean.
Comparing Relative Position Between Different Distributions
Z-scores are especially useful when comparing observations from different distributions.
Even if the original scales are different, z-scores place observations on a common standardized scale. The observation with the larger z-score has the stronger relative performance.
| z-Score | Relative Position |
|---|---|
| \( z=0 \) | At the mean |
| \( z=1 \) | Above average |
| \( z=2 \) | Far above average |
| \( z=-1 \) | Below average |
| \( z=-2 \) | Far below average |
Important AP Statistics Idea
When comparing observations from different distributions:
The larger z-score corresponds to the stronger relative position.
The original data values should not be compared directly because the distributions may have different centers and spreads.
| Observation | Raw Value | z-Score | Better Relative Position? |
|---|---|---|---|
| A | Higher | Smaller | No |
| B | Lower | Larger | Yes |
Example
Two students take the same exam.
Student A has:
\( z=1.8 \)
Student B has:
\( z=0.9 \)
Which student performed better relative to the rest of the class?
▶️ Answer / Explanation
The larger z-score indicates the stronger relative performance.
Since:
\( \mathrm{1.8>0.9} \)
Student A’s score is farther above the class mean.
Therefore, Student A performed better relative to the rest of the class.
Example
Maria scored: \( \mathrm{88} \) on a mathematics exam.
The mathematics scores have:
- \( \mu=80 \)
- \( \sigma=4 \)
Her z-score is:
\( z=\frac{88-80}{4}=2 \)
James scored: \( \mathrm{92} \) on a science exam.
The science scores have:
- \( \mu=90 \)
- \( \sigma=2 \)
His z-score is: \( z=\frac{92-90}{2}=1 \)
Who performed better relative to their classmates?
▶️ Answer / Explanation
Compare the z-scores:
Maria:
\( z=2 \)
James:
\( z=1 \)
Since:
\( \mathrm{2>1} \)
Maria’s score is farther above the mean of her distribution.
Therefore, Maria performed better relative to her classmates, even though James had the larger raw score.
This example shows why z-scores are useful when comparing observations from different distributions.