normalcdf & invNorm on the TI-84 — Complete AP Statistics Guide
The two most-tested calculator functions in AP Statistics are normalcdf and invNorm. This guide explains exactly what they do, what each parameter means, and how to write the answer in AP exam format.
What is normalcdf?
normalcdf calculates the probability (area under the normal curve) between two values.
Syntax:
normalcdf(lower bound, upper bound, μ, σ)
| Parameter | Meaning | Example |
|---|---|---|
| lower bound | Left edge of the region | -1 |
| upper bound | Right edge of the region | 1 |
| μ (mu) | Population mean | 0 |
| σ (sigma) | Population standard deviation | 1 |
How to access normalcdf on the TI-84
- Press 2ND → VARS (this opens the DISTR menu)
- Select 2: normalcdf(
- Enter:
lower, upper, μ, σ - Press ENTER
normalcdf Examples
Example 1: Standard Normal (Z-distribution)
Question: Find P(−1 < Z < 1) for a standard normal distribution.
Keystrokes:
2ND → VARS → 2
normalcdf(-1, 1, 0, 1)
Result: 0.6826894921
AP-style answer: "There is approximately a 68.3% probability that a randomly selected value falls within one standard deviation of the mean."
Example 2: Real-world Normal Distribution
Question: SAT scores are normally distributed with μ = 1060 and σ = 195. Find the probability that a randomly selected student scores between 900 and 1200.
Keystrokes:
normalcdf(900, 1200, 1060, 195)
Result: 0.5765
AP-style answer: "The probability that a randomly selected student scores between 900 and 1200 is approximately 0.5765 (57.65%)."
Example 3: One-tailed probability (above a value)
Question: Find P(X > 1300) for SAT scores (μ = 1060, σ = 195).
Keystrokes:
normalcdf(1300, 1E99, 1060, 195)
Use
1E99as the upper bound when there is no upper limit (represents +∞).
Type: 1 → 2ND → EE → 99
Result: 0.1094
AP-style answer: "The probability that a randomly selected student scores above 1300 is approximately 0.1094."
Example 4: Below a value (left-tail)
Question: Find P(X < 900) for SAT scores.
Keystrokes:
normalcdf(-1E99, 900, 1060, 195)
Result: 0.2061
What is invNorm?
invNorm is the inverse of normalcdf — it finds the value (x) that corresponds to a given probability (area to the left).
Syntax:
invNorm(area, μ, σ)
| Parameter | Meaning |
|---|---|
| area | Probability to the left of x (always between 0 and 1) |
| μ | Population mean |
| σ | Population standard deviation |
How to access invNorm
- Press 2ND → VARS
- Select 3: invNorm(
- Enter:
area, μ, σ - Press ENTER
invNorm Examples
Example 1: Find a percentile
Question: Find the 90th percentile of SAT scores (μ = 1060, σ = 195).
Keystrokes:
invNorm(0.90, 1060, 195)
Result: 1310.0
AP-style answer: "The 90th percentile of SAT scores is approximately 1310. A student needs to score at least 1310 to be in the top 10% of test-takers."
Example 2: Find a critical value (Z*)
Question: Find the critical value z* for a 95% confidence interval.
For a 95% CI, you need 2.5% in each tail, so the area to the left of z* is 0.975.
Keystrokes:
invNorm(0.975, 0, 1)
Result: 1.96
This is why the 95% CI formula uses z* = 1.96.
Example 3: Qualifying scores
Question: A company accepts only applicants scoring in the top 15% on a skills test with μ = 72 and σ = 8. What is the minimum qualifying score?
Keystrokes:
invNorm(0.85, 72, 8)
Top 15% = bottom 85%, so area = 0.85
Result: 80.3
AP-style answer: "An applicant must score at least 80.3 to qualify (be in the top 15%)."
Common Mistakes on the AP Exam
| Mistake | Correct approach |
|---|---|
| Using the wrong tail | normalcdf always calculates the area between lower and upper. For one-tail use 1E99 or -1E99 |
| Forgetting μ and σ | Always include all 4 parameters. If using standard normal, enter 0, 1 |
| Using area > 0.5 for invNorm when you want a right-tail value | Use 1 − probability as the area, or use invNorm(1 - p, μ, σ) |
| Rounding too early | Report at least 4 decimal places unless the problem specifies otherwise |
| Confusing z-scores with raw values | invNorm(p, 0, 1) gives a z-score; invNorm(p, μ, σ) gives a raw value |
AP Exam Writing Tips
For normalcdf questions, always state:
"normalcdf(lower, upper, μ, σ) = [value]"
For invNorm questions, always show your setup:
"invNorm(0.90, 1060, 195) = 1310"
The AP graders expect to see the calculator syntax written out — don't just write the answer.
Practice Using the Online TI-84
Try normalcdf and invNorm on our free TI-84 Plus CE calculator — the functions behave identically to the physical device.
Quick practice:
- Find P(Z > 2.33) →
normalcdf(2.33, 1E99, 0, 1) - Find the 25th percentile of a N(50, 10) distribution →
invNorm(0.25, 50, 10)