GRE · Data Analysis · TOPIC LESSON
Read relative position and distribution shape without confusing rank, probability and a raw score.
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BEFORE YOU START
Original educational lesson. Work through the examples before revealing the practice answers. This original lesson supports current GRE General Test study; it does not reproduce official questions, certify a writing score or predict test content.
LEARN THROUGH ORIGINAL PRACTICE
All teaching passages, writing prompts, examples and exercises in this lesson are original. No official ETS questions, prompts or explanations are reproduced. Current structure and observed public sample metadata are separate from this original teaching; no official score or question probability is implied.
01 / UNDERSTAND & APPLY
A percentile describes position relative to a reference group, not the percentage of questions answered correctly. A score near the 80th percentile lies above roughly 80% of the reference observations, subject to the stated treatment of ties and ranking convention. It does not mean a raw score of 80 or 80% accuracy.
Percentile definitions can differ in small data sets and when there are ties. If a question gives a convention or percentile table, use it. Do not invent an exact number strictly below a tied score from a percentile label alone.
02 / UNDERSTAND & APPLY
The median is the second quartile Q₂. Q₁ and Q₃ mark the lower and upper quartile positions. For a computation on a short list, follow the stated method, since methods for splitting an odd-sized list or interpolating ranks can differ. A common classroom method takes medians of the lower and upper halves, excluding the overall median when n is odd.
Quartiles describe ordered positions rather than equal numerical lengths. One quarter of observations may occupy a narrow range while another quarter spans a wide range.
ORIGINAL WORKED EXAMPLE
For 1,3,4,7,8,10,13,15, define Q₁ and Q₃ as the medians of the lower and upper halves. Find Q₁,Q₂,Q₃.
Answer Q₁=3.5, Q₂=7.5, Q₃=11.5
03 / UNDERSTAND & APPLY
The interquartile range IQR=Q₃−Q₁ measures spread of the middle half by position and is less sensitive to extreme endpoints than the full range. A basic boxplot displays a five-number summary: minimum, Q₁, median, Q₃, maximum. Its box spans Q₁ to Q₃, with a line at the median.
Some boxplots use whiskers ending at the most extreme non-outlier values and plot outliers separately. Read the stated convention. The length of a box or whisker is a numerical interval, not the number of observations in that interval.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
ORIGINAL WORKED EXAMPLE
A basic boxplot has minimum 2, Q₁=5, median 9, Q₃=14 and maximum 20. Find range and IQR.
Answer Range 18; IQR 9
04 / UNDERSTAND & APPLY
A symmetric distribution has matching shape around its center. A right-skewed distribution has a longer tail toward larger values; a left-skewed one has a longer tail toward smaller values. In many unimodal skewed distributions the mean is drawn toward the long tail, but a rough visual rule does not prove every comparison for arbitrary data.
A distribution may be unimodal, bimodal, nearly uniform or irregular. Do not assume a bell shape just because a mean and standard deviation are given. These two summaries alone do not fix its shape or tail probabilities.
05 / UNDERSTAND & APPLY
A discrete probability distribution lists possible values x with probabilities p(x). Every probability lies between 0 and 1, and all probabilities sum to 1. Probabilities of disjoint values add. A numerical outcome and its probability are different quantities: the outcome can exceed 1 while its probability cannot.
Expected value is the probability-weighted mean E(X)=Σx·p(x). It need not be a possible individual outcome. It describes a long-run average under repeated comparable trials, not a guarantee for the next trial.
ORIGINAL WORKED EXAMPLE
X takes values 0,2,5 with probabilities 0.2,0.5,p. Find p and E(X).
Answer p=0.3; E(X)=2.5
06 / UNDERSTAND & APPLY
A normal distribution has a smooth symmetric bell shape, with mean and median at its center. Its total area is 1, and probability corresponds to area over an interval, not curve height at a point. For an ideal continuous normal variable, a single exact value has probability zero even though intervals around it have positive probability.
Approximately 68% of a normal distribution lies within 1 standard deviation of its mean, 95% within 2, and 99.7% within 3. These are approximate model facts, not universal facts about all distributions. Symmetry divides the probability outside a centered interval equally between its two tails.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
ORIGINAL WORKED EXAMPLE
A variable is normally distributed with mean 50 and standard deviation 4. Approximately what percentage exceeds 58?
Answer Approximately 2.5%
07 / UNDERSTAND & APPLY
The standardized value z=(x−μ)/σ expresses how many standard deviations x lies from the mean, when σ>0. Positive z is above the mean; negative z below. Equal z-values mean equal relative distances, but equal percentile ranks require the distributions to have the same shape, such as both being normal.
Do not turn a z-value directly into a percentage without distribution information. A value two standard deviations above a mean is not “200% above the mean,” and a z-value of 0.8 is not an 80th percentile by definition.
ORIGINAL WORKED EXAMPLE
In distribution A, mean 60 and standard deviation 5, a value is 70. In B, mean 100 and standard deviation 12, a value is 118. Which is farther above its own mean in standard-deviation units?
Answer Distribution A
08 / UNDERSTAND & APPLY
A sample distribution describes the observations available. It does not automatically represent a whole population if selection is biased. A relationship in data does not by itself establish that one variable caused another. Inferential statistics and formal hypothesis tests are outside this GRE lesson scope.
When information is incomplete, state what the supplied summaries do determine and what they leave open. Boxplots reveal quartile positions but not every observation, exact mean, detailed modes or individual probabilities.
09 / UNDERSTAND & APPLY
All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.
A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.
A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.
10 / UNDERSTAND & APPLY
For normal-model calculations, use the stated approximate 68–95–99.7 rule.
ORIGINAL PRACTICE · 1
15
IQR=27−12=15. The median is not required, and this is not necessarily the full range.
ORIGINAL PRACTICE · 2
C
A percentile is a relative rank measure. Accuracy, raw score and pass probability are different quantities.
ORIGINAL PRACTICE · 3
4
The weighted mean is 1·0.4+6·0.6=4. An unweighted average of 3.5 would ignore the unequal probabilities.
ORIGINAL PRACTICE · 4
A, D and E
Each valid list has entries between 0 and 1 summing to 1. B sums to 1.2; C contains a negative entry despite summing to 1.
ORIGINAL PRACTICE · 5
74
Subtract one standard deviation: 80−6=74. A distance below the mean has a negative standardized value, not a negative raw value necessarily.
ORIGINAL PRACTICE · 6
B: 34%
About 68% is within one standard deviation on both sides. Symmetry assigns half, 34%, to the interval from the mean to one standard deviation above it.
ORIGINAL PRACTICE · 7
D
The normal upper-tail estimate near 16% cannot be assumed. Equal probabilities at 35 and 45 give 0% strictly above 45. Probabilities 0.8 at 37.5 and 0.2 at 50 have mean 40 and variance 25, and give 20% above 45. Both satisfy the stated summaries.
ORIGINAL PRACTICE · 8
12
The upper half is 9,12,14, whose median is 12. The overall median (7+9)/2=8 is a different statistic.
KEEP THE BIG IDEAS
READ WITH CONTEXT
The teaching notes, passages, worked explanations and practice are original. Official references identify the assessment framework and skill scope; they do not endorse this lesson or license official question reproduction.
GRE analytics distinguish current test structure from a small independently classified public sample selection. Writing instruction has no observed prompt-frequency denominator. These are not historical administered-exam frequencies.