How unusual is a value in its own group?
You will be able to: Calculate and compare standardized scores without assuming a normal distribution.
How unusual is a value in its own group?
An 80-point result means something different on a test averaging 60 than on one averaging 85. The typical spread matters too.
A useful starting point: How can two groups be compared fairly? →
Words and symbols before equations
- Standardized score z
- Distance from the mean measured in standard deviations.
- μ
- Population mean.
- σ
- Population standard deviation, assumed positive here.
- Relative position
- Location compared with the distribution’s center and spread.
What this picture assumes
Population mean fixed at 60 points; standard deviation is positive. The plot is a number line, not a normal curve. No percentile or probability is implied.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- z=(80−60)/10=2. This is 2 population standard deviations above the mean. No percentile is implied.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Subtract the mean to measure direction and distance, then divide by the standard deviation to express the distance on a common scale: z=(x−μ)/σ.
If x=80, μ=60 and σ=10, then z=2: the value is two standard deviations above the mean. A negative z is below the mean; z=0 is at the mean.
A z-score can be calculated for a nonnormal distribution. It does not determine a percentile without further information about distribution shape. When population parameters are unknown, sample mean and s can describe standardized position in that sample.
A worked example, step by step
Score A is 84 in a population with μ=72, σ=6. Score B is 90 with μ=80, σ=10. Which is higher relative to its group?
- For A, subtract 72 from 84 to get 12.
- Divide by 6: z_A=2.
- For B, z_B=(90−80)/10=1.
- A is farther above its own mean, although B has the higher raw score. No percentile is determined from these facts alone.
A z-score has no measurement unit. It is not itself a percentage or a probability.
What if σ=0?
Compare with an explanation
All population values are identical, so division by σ is undefined; do not calculate a z-score.
Predict. Change one thing. Explain.
Change the score and then the population standard deviation. Explain why a fixed raw difference is less unusual when spread is larger.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
z=(80−60)/10=2. This is 2 population standard deviations above the mean. No percentile is implied.
Population mean fixed at 60 points; standard deviation is positive. The plot is a number line, not a normal curve. No percentile or probability is implied.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the data values, graph scales, summary statistics or study-design conditions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA measurement has μ=30 cm and σ=4 cm. Find z for 36 cm, then recover x for z=−0.5. Interpret both.
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Compare with the answer and four-point rubric
- 1 point: z=(36−30)/4=1.5.
- 1 point: 36 cm is 1.5 standard deviations above the mean.
- 1 point: x=30+(−0.5)×4=28 cm.
- 1 point: 28 cm is half a standard deviation below the mean; neither z specifies a percentile without shape information.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What does a negative z mean?
Below the reference mean.
RECALL 2Are z-scores measured in centimeters?
No; the measurement units cancel.
RECALL 3Can z be used on nonnormal data?
Yes, but normal percentile rules do not automatically apply.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How unusual is a value in its own group?
- z=(x−μ)/σ for σ>0.
- x=μ+zσ.
- A percentile requires information about distribution shape.
Remember: A z-score has no measurement unit. It is not itself a percentage or a probability.
Conditions: Population mean fixed at 60 points; standard deviation is positive. The plot is a number line, not a normal curve. No percentile or probability is implied.
Refresh Kid · AP Statistics Unit 1 · Objectives 1.9.D, 1.9.E · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 1.9, objectives 1.9.D, 1.9.E. Framework effective Fall 2026, checked September 17, 2026. Unit 1 includes one-variable data and data collection; it is part of the revised five-unit course.
Examples and datasets are synthetic, independently authored teaching material. Quartile calculations state a median-of-halves convention. Outlier screens identify values to investigate, not data to discard. Random selection and random assignment have different inferential roles. These lessons introduce design and descriptive reasoning; formal inference comes in later units.
The Organic Chemistry Tutor companion title and destination were checked; the full video was not reviewed. Khan Academy’s destination was checked, but its lesson content was not fully readable by the research tool. OpenStax provides optional reference reading. No provider scripts, questions or graphics were copied. Refresh Kid is not affiliated with these providers.
GitHub’s 3D website collection informed optional spatial inspection. Our original sampling model uses self-hosted Three.js with its MIT license. It shows labeled units in four groups; camera rotation does not change the sampling procedure. Quantitative graphs remain 2D to avoid perspective distortion. Complete labeled diagrams, selected IDs and explanations remain available without 3D.
Independent teacher review and observation of students remain pending. Technical checks do not certify statistical accuracy, accessibility or learning effectiveness. This is a review edition.
Released AP Statistics questions and scoring guides are optional. Older exams use the earlier framework, so check alignment before selecting parts. All practice on this page is original, not official AP material.
Learn → Explore → Practice → Review is informed by the IES learning guide. This implementation has not yet been evaluated with learners.
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