Distributions & the Normal Curve
A distribution is simply the shape you get when you lay every value in a data set out and count how often each one appears. The GRE cares about two shapes above all others: the messy real-world frequency distribution you read off a table or histogram, and the tidy, symmetric normal distribution — the bell curve. The normal curve is worth memorising because ETS tests it with one fixed tool, the 68-95-99.7 rule, which tells you what fraction of the data sits within one, two and three standard deviations of the mean. Almost every normal-distribution question on the GRE is that rule in disguise. This chapter builds the reading skills for frequency tables, the geometry of the bell curve, the percentage bands you must know cold, and the language of correlation that scatterplots speak.
Topics
⚡ GRE shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- For any normal question, sketch the bell and label the mean and ±1, ±2, ±3 SD lines. Write 34 / 13.5 / 2.35 into the slices and just add.
- Symmetry is free information: the mean splits a normal curve into two 50% halves, so "above the mean" is always 50%.
- Convert an awkward value to a z-score, (value − mean)/SD, to see instantly which band it lands in.
- Between +1 SD and +2 SD sits (95 − 68)/2 = 13.5%; beyond +2 SD sits (100 − 95)/2 = 2.5%. Memorise these two off-cuts.
- For correlation STRENGTH compare |r|; the sign only tells you the direction (up-slope vs down-slope).
- On a frequency table, the mean weights each value by its count — never average the distinct values alone.
⚠️ Common mistakes & traps
GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Averaging the distinct values in a frequency table instead of weighting each by its frequency.
- Forgetting to halve a symmetric band — "above +1 SD" is 16%, not 32%.
- Reading a histogram bar’s height as a data value rather than a count.
- Treating a strong correlation as proof of causation — the GRE almost always wants "no, a third factor".
- Judging correlation strength by the sign, so mistaking r = 0.3 as stronger than r = −0.8.
- Assuming the 68-95-99.7 rule applies to a skewed or non-normal distribution — it only holds for the bell curve.
📈 GRE exam insight & question patterns
Quantitative Comparison — a normal distribution has mean 40, SD 5. Quantity A: percent of data above 50. Quantity B: 2.5%. Which is greater?
Equal. 50 = mean + 2 SD, and the tail beyond +2 SD is (100 − 95)/2 = 2.5%.
How does the GRE usually test the normal distribution?
Almost always through the 68-95-99.7 rule plus symmetry — never a z-table. Mark the SD gridlines and add the labelled slices.
A scatterplot question asks what a strong positive correlation implies about causation. What is the safe answer?
It implies nothing about causation. Correlation measures how two variables move together, not whether one causes the other.
Numeric Entry — in a normal set with mean 500 and SD 100, enter the percent scoring between 400 and 600.
400 and 600 are ±1 SD, so about 68%.
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
- A frequency distribution counts how often each value occurs; relative frequency is its share, cumulative frequency the running total.
- Find a grouped mean by weighting each value by its frequency, not by averaging the distinct values.
- The normal distribution is symmetric and set entirely by its mean and standard deviation; mean = median = mode.
- The 68-95-99.7 rule: about 68% within ±1 SD, 95% within ±2 SD, 99.7% within ±3 SD — halve the bands using symmetry.
- A z-score, (value − mean)/SD, tells you how many standard deviations from the mean a value sits.
- Scatterplots show positive, negative or no correlation; r runs from −1 to +1 and its magnitude is the strength.
- Correlation is never proof of causation — expect the GRE to reward the "third factor" answer.
Chapter test
🏆 Vidaara GRE success checklist
You have truly mastered Distributions & the Normal Curve when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 8 cards |
Formula Reference Sheet
Reading a distribution
| Relative frequency | (frequency of a value) / (total count) |
|---|---|
| Cumulative frequency | running total of counts up to a value |
| Mean of a frequency table | Σ(value × frequency) / Σ(frequency) |
The normal curve (memorise)
| Within ±1 SD of the mean | ≈ 68% of the data |
|---|---|
| Within ±2 SD | ≈ 95% |
| Within ±3 SD | ≈ 99.7% (nearly all) |
| Each half is symmetric | mean = median = mode; 50% on each side |
| z-score (standardised value) | (value − mean) / standard deviation |
Correlation
| Correlation coefficient r | always between −1 and +1 |
|---|---|
| Positive / negative / none | r > 0 up-trend, r < 0 down-trend, r ≈ 0 no line |