Common Mistakes
Learn to recognize and avoid the most frequent errors students make in statistics, from hypothesis testing to probability calculations.
Learn
Understanding common mistakes is just as important as knowing the correct methods. This lesson highlights typical errors, explains why they happen, and shows you how to avoid them on tests like the SAT and ACT.
Why Students Make Statistical Errors
- Confusing similar concepts: Standard deviation vs. standard error, population vs. sample
- Misinterpreting questions: One-tailed vs. two-tailed tests
- Calculation errors: Forgetting to take square roots, using wrong formulas
- Logic errors: Misunderstanding what p-values and confidence intervals mean
- Context errors: Not relating answers back to the real-world problem
Examples of Common Mistakes
Mistake 1: Confusing Standard Deviation and Standard Error
The Error: Using standard deviation when standard error is needed (or vice versa).
Example: "A sample of 100 has mean 50 and sd = 10. Find the probability the sample mean exceeds 52."
Wrong Approach:
z = (52 - 50) / 10 = 0.2 (Using sd = 10 directly)
Correct Approach:
Standard Error = 10 / sqrt(100) = 1
z = (52 - 50) / 1 = 2.0
Key Rule: Use standard error (sd / sqrt(n)) when dealing with sample means; use standard deviation when dealing with individual values.
Mistake 2: Misinterpreting P-Values
The Error: Stating that p-value is the probability the null hypothesis is true.
Wrong Interpretation:
"A p-value of 0.03 means there's a 3% chance the null hypothesis is true."
Correct Interpretation:
"A p-value of 0.03 means that IF the null hypothesis is true, there's a 3% probability of obtaining results at least as extreme as what was observed."
Key Rule: P-value assumes H0 is true; it doesn't tell us the probability that H0 is actually true.
Mistake 3: Wrong Direction for Hypothesis Tests
The Error: Setting up a one-tailed test when two-tailed is needed, or vice versa.
Example: "Test if a new teaching method produces different results than the traditional method."
Wrong Setup:
H0: mean = 75, Ha: mean > 75 (one-tailed)
Correct Setup:
H0: mean = 75, Ha: mean is not equal to 75 (two-tailed)
Key Rule: "Different" or "changed" = two-tailed; "greater," "less," "increased," "decreased" = one-tailed.
Mistake 4: Misunderstanding Confidence Intervals
The Error: Saying there's a 95% probability the true mean is in the interval.
Wrong Interpretation:
"There is a 95% probability that the population mean is between 45 and 55."
Correct Interpretation:
"We are 95% confident that our interval (45, 55) contains the true population mean." OR "If we repeated this sampling process many times, 95% of the resulting intervals would contain the true mean."
Key Rule: The confidence level refers to the method, not the specific interval.
Practice: Spot the Mistake
For each problem, identify the error and provide the correct solution.
Problem 1
Student's work: "Sample of n = 64, sample mean = 100, sample sd = 16. For a 95% CI: 100 +/- 1.96 * 16 = (68.64, 131.36)"
Show the Mistake
Error: Used sd instead of standard error.
Correct: SE = 16 / sqrt(64) = 2
CI = 100 +/- 1.96 * 2 = (96.08, 103.92)
Problem 2
Student's work: "H0: mean = 50, Ha: mean > 50. Test statistic z = -1.8. Since |-1.8| > 1.645, reject H0."
Show the Mistake
Error: For a right-tailed test, a negative z-score never leads to rejection.
Correct: z = -1.8 is in the opposite direction of Ha. Fail to reject H0. The sample mean is actually below 50, not above it.
Problem 3
Student's work: "To find P(X < 85) where mean = 100 and sd = 15: z = (100 - 85) / 15 = 1.0"
Show the Mistake
Error: Subtracted in wrong order (should be x - mean, not mean - x).
Correct: z = (85 - 100) / 15 = -1.0
Problem 4
Student's conclusion: "p-value = 0.08. Since this is greater than alpha = 0.05, we accept the null hypothesis."
Show the Mistake
Error: Said "accept" instead of "fail to reject."
Correct: We fail to reject the null hypothesis. We never "accept" H0; we only fail to find sufficient evidence against it.
Problem 5
Student's work: "95% CI for proportion: p-hat = 0.4, n = 100. SE = sqrt(0.4 * 0.4 / 100) = 0.04"
Show the Mistake
Error: Used p * p instead of p * (1-p).
Correct: SE = sqrt(0.4 * 0.6 / 100) = sqrt(0.0024) = 0.049
Problem 6
Student's work: "For a two-tailed test at alpha = 0.05, critical z-values are +/- 1.645"
Show the Mistake
Error: Used one-tailed critical value for two-tailed test.
Correct: For two-tailed test at alpha = 0.05, critical values are +/- 1.96 (0.025 in each tail).
Problem 7
Student's work: "Sample size for 3% margin of error at 95% confidence: n = (1.96)^2 * 0.5 * 0.5 / 3 = 0.32"
Show the Mistake
Error: Divided by 3 instead of (0.03)^2.
Correct: n = (1.96)^2 * 0.25 / (0.03)^2 = 0.9604 / 0.0009 = 1068
Problem 8
Student's interpretation: "The correlation coefficient r = 0.85, so studying more causes higher grades."
Show the Mistake
Error: Interpreted correlation as causation.
Correct: There is a strong positive association between study time and grades. Correlation does not imply causation.
Problem 9
Student's work: "Type I error probability is beta; Type II error probability is alpha."
Show the Mistake
Error: Switched alpha and beta.
Correct: Alpha = P(Type I error) = P(reject H0 when H0 is true). Beta = P(Type II error) = P(fail to reject H0 when H0 is false).
Problem 10
Student's work: "Population mean is between 40 and 60 with 95% confidence. Therefore, there's a 5% chance the mean is outside this range."
Show the Mistake
Error: The population mean is a fixed value, not random. It either is or isn't in the interval.
Correct: The 95% confidence refers to the procedure. If we repeated sampling many times, 95% of the intervals constructed would contain the true mean. For this specific interval, the mean is either in it or not.
Check Your Understanding
Quick reference checklist to avoid common mistakes:
Before Submitting Any Statistics Problem:
- Did I use standard error (not sd) for problems involving sample means?
- Is my z-score calculated as (value - mean) / sd?
- Did I use the correct tail direction for my hypothesis test?
- Am I using the right critical value (z vs. t, one-tail vs. two-tail)?
- Did I square the margin of error in sample size formulas?
- Am I saying "fail to reject" instead of "accept" for H0?
- Have I interpreted the result in context?
- Did I use p*(1-p), not p*p, for proportion standard errors?
Next Steps
- Create a personal error log for statistics problems
- Review any topics where you frequently make mistakes
- Take the Unit Quiz to test your understanding
- Return to this lesson before major tests as a refresher