Study the box plot below showing test scores for two classes, A and B.
Which statement is correct?
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Memory Point: BOX WIDTH = IQR (spread). Median line position = center. Whisker length = range outside middle 50%. Skewed right → right whisker longer.
Section 3 — Standard Deviation (Sample vs Population)
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Standard Deviation Concept Easy
Two classes both have a mean test score of 75. Class X has a standard deviation of 2, and Class Y has a standard deviation of 15. What can you conclude?
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Memory Point: SD = "Average Distance from Mean." Small SD → clustered together. Large SD → widely spread. SD is NEVER negative.
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Sample vs Population Standard Deviation Medium
A researcher measures the heights of all 300 students at a school. She calculates the standard deviation. Which formula and notation should she use?
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Memory Point: POPULATION → divide by N, symbol σ (sigma) | SAMPLE → divide by n−1, symbol s | "All students = population = σ"
σ = √ Σ(xi − μ)²N
|
s = √ Σ(xi − x̄)²n − 1
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Standard Deviation Calculation — Word Problem Medium
A coffee shop records the number of customers per hour over 5 hours: 20, 25, 30, 25, 20. The mean is 24. What is the sample standard deviation? (Round to one decimal place.)
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Memory Point: SAMPLE SD STEPS → ① Find deviations (x − x̄) ② Square them ③ Sum them ④ Divide by (n−1) ⑤ Square root → "Dev-Square-Sum-Divide-Root"
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Standard Deviation — Common Mistake Hard
If every value in a dataset is increased by 10, what happens to the mean and standard deviation?
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Memory Point: SHIFT vs SCALE → Add constant: Mean shifts, SD unchanged (relative distances don't change). Multiply constant: Both mean and SD scale. "Shift moves center, not spread."
Section 4 — Normal Distribution
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Empirical Rule (68-95-99.7) Easy
The heights of adult males are normally distributed with a mean of 70 inches and a standard deviation of 3 inches. Approximately what percentage of men are between 64 and 76 inches tall?
In a normal distribution with μ = 100 and σ = 15, what is the probability that a randomly selected value is above 115?
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Memory Point: SYMMETRY TRICK → Normal curve is symmetric at mean. P(above μ) = 0.5. Use 68-95-99.7 to find tails. One tail at +1σ = (100% − 68%) ÷ 2 = 16%
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Normal Distribution Word Problem Hard
A factory produces bolts whose lengths follow a normal distribution with μ = 10 cm and σ = 0.2 cm. Quality control rejects bolts that are shorter than 9.6 cm or longer than 10.4 cm.
Approximately what percentage of bolts are accepted?
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Memory Point: Count how many σ away the cutoffs are: (10 − 9.6) ÷ 0.2 = 2σ below; (10.4 − 10) ÷ 0.2 = 2σ above → Use ±2σ = 95% rule.
Section 5 — Standard Normal Distribution & Z-Scores
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Z-Score Calculation Medium
Emma scored 88 on a history test. The class mean was 75, and the standard deviation was 8. What is Emma's z-score?
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Memory Point: Z = (x − μ) ÷ σ → "Distance from mean, measured in standard deviations." Positive z = above average. Negative z = below average.
z = (x − μ) / σ
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Z-Score Comparison — Word Problem Medium
Jake scored 82 on Math (μ = 78, σ = 4) and 76 on English (μ = 70, σ = 8).
On which test did Jake perform relatively better compared to his classmates?
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Memory Point: Z-SCORE COMPARISON → Same scale, different subjects? Compute both z-scores. Higher z = better relative performance. "Z-score = fair comparison tool."
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Standard Normal Distribution — Z-Table Hard
Scores on a standardized exam are normally distributed with μ = 500, σ = 100. Using the fact that P(Z < 1.25) ≈ 0.8944, what is the approximate probability that a randomly chosen student scores between 500 and 625?
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Memory Point: BETWEEN TWO VALUES → P(a < X < b) = P(Z < z₂) − P(Z < z₁). Use symmetry: P(Z < 0) = 0.5 always for standard normal.
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Inverse Normal — Percentile Hard
A cereal company fills boxes with a mean of 20 oz (σ = 0.5 oz), normally distributed. The company wants only the top 2.5% of boxes by weight to be classified as "overfilled." What is the minimum weight for an "overfilled" box?
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Memory Point: INVERSE NORMAL → Know z, find x: x = μ + z·σ. Top 2.5% means z ≈ +1.96 (from 95% rule: 2.5% in each tail at ±1.96σ).
Section 6 — Advanced Word Problems (Exam Level)
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Combined Concepts — Tricky Word Problem Expert
A data set has a mean of 50 and a standard deviation of 10. Each data value is multiplied by 2, then 5 is added.
What are the new mean and standard deviation?
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Memory Point: LINEAR TRANSFORM y = ax + b → New mean = a·μ + b → New SD = |a|·σ (b does NOT affect SD). "Multiply scales both. Add shifts only mean."
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Skewness & Measures of Center Expert
In a right-skewed distribution (like household incomes in the US), which of the following relationships is always true?
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Memory Point: SKEW DIRECTION → Right skew: long tail points right → Mean > Median > Mode. Left skew: long tail points left → Mode > Median > Mean. "Tail pulls the mean."
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Full Integration — Multi-Step Word Problem Expert
At a hospital, patient waiting times are normally distributed with μ = 45 min and σ = 12 min. The hospital's goal is to ensure that no more than 16% of patients wait longer than a certain threshold time T.
What is the threshold time T, and what z-score corresponds to it?