Descriptive Statistics (Mean, Median, SD) - Quantitative Reasoning

SECTION 1

Historical Context & Motivation

Long before standardized tests or data-science pipelines existed, scholars wrestled with a deceptively simple question: how do you reduce a collection of measurements to a single, representative number? The answer gave rise to descriptive statistics — a family of techniques designed to summarize, organize, and communicate the essential features of a data set. On the GRE Quantitative section, questions about the mean, median, and standard deviation appear frequently in Quantitative Comparison, Data Interpretation, and Problem Solving formats. Understanding the historical roots of these measures illuminates why each was invented and when each is most appropriate.

~1670 • The Arithmetic Mean in Astronomy

Christiaan Huygens and later Edmond Halley routinely averaged astronomical observations to reduce measurement error, establishing the arithmetic mean as the standard tool for combining repeated measurements.

1774 • Laplace and the Method of Least Squares

Pierre-Simon Laplace formalized the idea that the mean minimizes the sum of squared deviations — a theoretical justification that would later inspire the concept of variance and standard deviation.

1883 • Galton Introduces the Median

Francis Galton popularized the median as a robust measure of central tendency in his studies of heredity, noting its resistance to extreme values that plagued averaged biological data.

1893 • Karl Pearson Coins 'Standard Deviation'

Karl Pearson introduced the term standard deviation to replace the cumbersome 'mean error,' giving the statistical community a clean, universal measure of spread.

1947–Present • Modern Standardized Testing

The Educational Testing Service (ETS) begins producing the GRE, embedding descriptive statistics questions that test candidates' ability to interpret means, medians, and standard deviations in real-world data contexts.

The central question that unifies these developments is both practical and profound: given a finite set of numerical observations, what single value best represents the "typical" case, and how much do the individual values deviate from that center? Answering the first part requires a measure of central tendency (mean or median), and answering the second requires a measure of spread (standard deviation). Together, these statistics form the bedrock of quantitative reasoning on the GRE.

SECTION 2

Core Principles & Definitions

Descriptive statistics distill raw data into interpretable summaries. Before diving into formulas, it is essential to grasp the foundational ideas that govern how and why we compute the mean, median, and standard deviation. Each statistic captures a different aspect of a data set's "shape," and the GRE routinely tests whether you understand those distinctions — not just whether you can calculate a number.

Arithmetic Mean (Average)

The sum of all data values divided by the count of values. It uses every data point, which makes it sensitive to outliers. On the GRE, the mean is the default measure of center.

Median (Middle Value)

The value that divides an ordered data set into two equal halves. For an even number of data points, the median is the average of the two central values. It is resistant to outliers, making it preferred for skewed distributions.

Standard Deviation (SD)

A measure of how spread out the data values are around the mean. A small SD means values cluster tightly; a large SD means they are widely dispersed. The GRE tests your ability to compare SDs visually from distributions.

Variance

The square of the standard deviation. Variance (σ²) is used primarily in intermediate calculations; the standard deviation (σ) restores the result to the original units of the data, which is why it is reported more often.

Skewness & Outlier Effects

When a distribution is symmetric, the mean and median coincide. When it is skewed, the mean is pulled toward the tail while the median remains stable — a relationship the GRE frequently probes.

✦

KEY TAKEAWAY

Think of the mean as the balance point of a data set — the spot where a number line would teeter perfectly level if you placed equal weights at every data value. The median, by contrast, is the physical midpoint — the value that splits the count of data points exactly in half. If a single billionaire walks into a room of college students, the mean salary skyrockets while the median barely moves. The standard deviation, meanwhile, tells you how tightly the weights cluster around that balance point; a small SD is like all weights stacked in the center, while a large SD spreads them to the ends.

SECTION 3

Visual Explanation — Mean, Median, and Skew

The relationship between the mean and median is best understood visually. In a perfectly symmetric distribution, both statistics coincide at the peak. However, when data is skewed — as it commonly is in GRE data-interpretation scenarios involving income, prices, or reaction times — the mean shifts toward the longer tail. The diagram below shows three common distribution shapes and marks the positions of the mean and median in each.

In a symmetric distribution, the mean and median are equal. In a right-skewed distribution (long tail on the right), the mean is pulled above the median. In a left-skewed distribution, the mean is pulled below the median.

This visual relationship is one of the most commonly tested concepts on the GRE. A Quantitative Comparison question might present two distributions and ask which has the greater mean relative to the median. If you can recognize the direction of skew from a histogram or frequency chart, you can immediately determine whether the mean lies above or below the median — no calculation required. Remember the mnemonic: the mean chases the tail. In a right-skewed distribution, the tail extends to the right, so the mean shifts right (i.e., increases) relative to the median, and vice versa.

SECTION 4

Mathematical Framework

Let us formalize the three statistics using standard notation. Suppose we have a data set of n observations: x₁, x₂, …, xₙ. Throughout GRE preparation, you should be comfortable both applying these formulas mechanically and interpreting the results conceptually.

ARITHMETIC MEAN

x̄ = (x₁ + x₂ + ⋯ + xₙ) / n = (Σxᵢ) / n

Where x̄ is the sample mean, Σxᵢ denotes the sum of all observations, and n is the total number of data points. The mean treats every value with equal weight.

MEDIAN

Median = x₍₍ₙ₊₁₎/₂₎ if n is odd; Median = (x₍ₙ/₂₎ + x₍ₙ/₂₊₁₎) / 2 if n is even

After sorting the data in ascending order, the median is the value at position (n + 1)/2 for an odd count, or the average of the two middle values for an even count. The subscript notation x₍ₖ₎ denotes the k-th value in the sorted list.

POPULATION STANDARD DEVIATION

σ = √[ (1/N) × Σ(xᵢ − μ)² ]

Where μ is the population mean, N is the population size, and each term (xᵢ − μ)² is a squared deviation from the mean. On the GRE, unless explicitly stated otherwise, assume you are computing the population SD (dividing by N, not n − 1).

VARIANCE

σ² = (1/N) × Σ(xᵢ − μ)²

Variance is simply the standard deviation squared. It measures dispersion in squared units. Although the GRE rarely asks for variance directly, understanding this relationship helps when you need to reason about how adding a constant to every value (no change in SD) or multiplying by a constant (SD scales by |c|) affects spread.

💡

GRE TIP

The GRE will never ask you to compute a standard deviation from scratch for a large data set — the arithmetic would be too time-consuming. Instead, expect questions that ask you to compare standard deviations of two distributions visually, or to reason about how a transformation (adding a constant, multiplying by a constant, removing an outlier) affects the SD.

SECTION 5

Standard Deviation — A Closer Look

Standard deviation is the measure that GRE test-takers find most conceptually challenging, largely because they conflate range with spread. Two data sets can have identical ranges yet vastly different standard deviations. The key insight is that SD depends on how every value relates to the mean, not merely on the gap between the maximum and minimum. The following diagram walks through the computation of SD for a small data set, illustrating each squared deviation as a colored area.

Each bar represents a data value. The dashed red lines show negative deviations from the mean (μ = 5), and dashed cyan lines show positive deviations. Squaring each deviation and averaging yields the variance (4); taking the square root gives the standard deviation (2).

Notice that the two data points farthest from the mean (2 and 9) contribute disproportionately to the total sum of squared deviations: 9 + 16 = 25 out of 32 total. This is precisely why the standard deviation is sensitive to outliers — squaring amplifies large deviations far more than small ones. On the GRE, if a question states that one data value is changed or removed, consider whether that value is close to the mean (minimal impact on SD) or far from it (significant impact).

ℹ

TRANSFORMATION RULES

If every data value is transformed by xᵢ → a × xᵢ + b, then the new mean is a × (old mean) + b, but the new SD is |a| × (old SD). Adding a constant shifts the center without affecting spread; multiplying by a constant scales both the center and the spread.

SECTION 6

Worked Example

The following problem mirrors the style you will encounter in GRE Problem Solving. We will compute the mean, median, and standard deviation of a small data set and then determine what happens when a new value is added.

GRE-Style Problem: Adding a Data Point

1
Step 1 — State the Data

A researcher records the number of hours 6 students studied for an exam: {3, 5, 7, 7, 8, 12}. We need the mean, median, and standard deviation of this data set.
2
Step 2 — Compute the Mean

Sum = 3 + 5 + 7 + 7 + 8 + 12 = 42. Number of values = 6. Therefore, x̄ = 42 / 6 = 7.

Mean = 7
3
Step 3 — Find the Median

The data is already sorted: {3, 5, 7, 7, 8, 12}. With n = 6 (even), the median is the average of the 3rd and 4th values: (7 + 7) / 2 = 7.

Median = 7
4
Step 4 — Compute the Standard Deviation

First, find each squared deviation from the mean (μ = 7): (3 − 7)² = 16, (5 − 7)² = 4, (7 − 7)² = 0, (7 − 7)² = 0, (8 − 7)² = 1, (12 − 7)² = 25. Sum of squared deviations = 16 + 4 + 0 + 0 + 1 + 25 = 46. Variance = 46 / 6 ≈ 7.667. Standard deviation = √7.667 ≈ 2.77.

σ ≈ 2.77
5
Step 5 — Add a New Value and Analyze

A 7th student studied for exactly 7 hours. The new data set is {3, 5, 7, 7, 7, 8, 12}. New sum = 42 + 7 = 49, new mean = 49 / 7 = 7 — the mean is unchanged because the new value equals the old mean. New median = the 4th value in sorted order = 7 — the median is also unchanged. For the new SD, the added squared deviation is (7 − 7)² = 0, so the new sum of squared deviations remains 46. New variance = 46 / 7 ≈ 6.571. New SD = √6.571 ≈ 2.56. The standard deviation decreased because the added value is right at the mean, pulling the average deviation down.

New σ ≈ 2.56 (decreased)

🔑

INSIGHT

Adding a value equal to the mean never changes the mean itself, but it always decreases the standard deviation (unless all values are already identical). This is because the denominator in the variance formula increases while the numerator stays the same. Conversely, adding a value far from the mean will increase the SD. This principle is a frequent trap in GRE Quantitative Comparison questions.

SECTION 7

Strengths, Limitations, and When to Use Each

The GRE expects you to know not just how to compute these statistics but also when each is the most appropriate descriptor. The table below synthesizes the strengths and limitations of each measure. Understanding these trade-offs is the key to answering qualitative reasoning questions about data analysis.

Statistic	Strengths	Limitations
Mean	Uses every data point; algebraically tractable; minimizes the sum of squared deviations; foundation for most advanced statistical methods.	Sensitive to outliers and skewed data; may not represent a 'typical' value when the distribution is heavily skewed.
Median	Robust to outliers; always represents the 'middle' of the data regardless of shape; preferred for income, home price, and other skewed distributions.	Ignores the actual magnitudes of extreme values; harder to manipulate algebraically; less useful in further statistical inference.
Standard Deviation	Measures spread in the same units as the data; foundational for confidence intervals, z-scores, and the normal distribution; sensitive to the overall shape.	Sensitive to outliers (due to squaring); can be misleading for multimodal or heavily skewed distributions; requires the mean as a reference point.

Comparison of Mean, Median, and Standard Deviation

🎯

GRE STRATEGY NOTE

Think of the mean and median as complementary sensors. A weather station using only a thermometer (mean) would be misled by a single volcanic heat spike; one using only a barometer (median) might miss gradual warming. The standard deviation is the instrument's sensitivity dial — a high SD means the readings fluctuate wildly, while a low SD means they are remarkably consistent. On the GRE, always consider which instrument the question is asking you to read, and be ready to explain why one is more informative than the other in a given context.

SECTION 8

Connection to Inferential Statistics and Beyond

Descriptive statistics are the building blocks of more advanced analytical tools. While the GRE Quantitative section focuses on description rather than inference, understanding where these concepts lead will strengthen your intuition and help you navigate the handful of harder questions that border on inferential reasoning.

Descriptive Concept	Advanced Extension	What It Enables
Mean (x̄)	Sampling distribution of the mean; Central Limit Theorem	Confidence intervals and hypothesis testing about population means, even for non-normal populations.
Standard Deviation (σ)	Z-scores; the 68-95-99.7 rule (Empirical Rule)	Determining how unusual a particular value is; comparing values from different distributions.
Median	Percentiles and quartiles; Interquartile Range (IQR)	Box-and-whisker plots; outlier detection using the 1.5 × IQR rule.
Variance (σ²)	Analysis of Variance (ANOVA); Regression sum of squares	Partitioning total variability into explained and unexplained components; model building.

From Descriptive to Inferential Statistics

The most directly GRE-relevant extension is the z-score, defined as z = (x − μ) / σ. A z-score tells you how many standard deviations a particular value lies above or below the mean. If a GRE question states that a distribution is approximately normal and asks about the percentage of values within certain bounds, recall the Empirical Rule: approximately 68% of values fall within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. This rule converts abstract standard-deviation values into concrete probabilities and is one of the most efficient tools for Data Interpretation questions.

📝

LOOKING AHEAD

In graduate coursework, the descriptive tools you master here form the first step of any empirical analysis — from simple t-tests to multivariate regression. A solid grasp of mean, median, and SD now will pay dividends throughout your quantitative career.

SECTION 9

Practice Problems

Test your understanding with these five problems, calibrated from conceptual to critical-thinking difficulty. Work each problem completely before reading the answer.

Question 1: CONCEPTUAL

A data set has a mean of 50 and a median of 42. What can you conclude about the shape of the distribution — is it symmetric, right-skewed, or left-skewed? Explain your reasoning.

Show answer

The distribution is right-skewed. When the mean exceeds the median, it indicates that the right tail of the distribution contains values large enough to pull the mean upward while the median, being the positional middle value, remains relatively unaffected. This is the classic signature of right (positive) skew. In everyday terms, a few unusually high values inflate the average beyond the typical (median) value — the same pattern seen in income distributions where a few high earners raise the mean above the median.

Question 2: BASIC CALCULATION

Find the mean, median, and population standard deviation of the data set: {10, 14, 14, 18, 20, 26}.

Show answer

Mean: (10 + 14 + 14 + 18 + 20 + 26) / 6 = 102 / 6 = 17. Median: the average of the 3rd and 4th values in the sorted list = (14 + 18) / 2 = 16. For SD, compute squared deviations from μ = 17: (10 − 17)² = 49, (14 − 17)² = 9, (14 − 17)² = 9, (18 − 17)² = 1, (20 − 17)² = 9, (26 − 17)² = 81. Sum = 158. Variance = 158 / 6 ≈ 26.33. SD = √26.33 ≈ 5.13. The mean (17) exceeds the median (16), consistent with a slight right skew caused by the relatively high value of 26.

Question 3: INTERMEDIATE

List A consists of the values {1, 2, 3, 4, 5}. List B consists of the values {1, 2, 3, 4, 15}. Without computing the exact standard deviations, determine which list has the greater SD and explain why. Then verify by computing both SDs.

Show answer

List B has the greater standard deviation. Replacing the value 5 with 15 greatly increases the deviation of the largest value from the mean, and because deviations are squared, this single change has an outsized effect. Verification: List A mean = 3, squared deviations: 4 + 1 + 0 + 1 + 4 = 10, variance = 10/5 = 2, SD = √2 ≈ 1.41. List B mean = 5, squared deviations: 16 + 9 + 4 + 1 + 100 = 130, variance = 130/5 = 26, SD = √26 ≈ 5.10. As expected, List B's SD is roughly 3.6 times that of List A, driven primarily by the outlier value 15.

Question 4: APPLIED

A company's eight employees earn the following annual salaries (in thousands): {35, 38, 40, 42, 44, 48, 50, 200}. The company claims its 'average salary' is $62,125. A job candidate argues the 'typical salary' is $43,000. Who is correct, and which statistic does each party use? If the $200k earner leaves and is replaced by someone earning $46k, how do the mean and median change?

Show answer

The company is citing the mean: (35 + 38 + 40 + 42 + 44 + 48 + 50 + 200) / 8 = 497 / 8 = $62,125. The candidate is citing the median: average of the 4th and 5th sorted values = (42 + 44) / 2 = $43,000. Both are technically correct, but the median is a more representative summary because the $200k outlier severely inflates the mean. After the replacement, the new sum = 497 − 200 + 46 = 343, new mean = 343 / 8 = $42,875 — a dramatic drop. The new sorted set is {35, 38, 40, 42, 44, 46, 48, 50}, and the new median = (42 + 44) / 2 = $43,000, unchanged. This illustrates the median's robustness to changes in extreme values.

Question 5: CRITICAL THINKING

Prove that for any data set, adding a new value equal to the current mean x̄ does not change the mean. Then determine under what condition adding a value equal to the mean would also leave the standard deviation unchanged.

Show answer

Let the original data set have n values with sum S, so x̄ = S/n. After adding the value x̄, the new sum is S + x̄ = S + S/n = S(n + 1)/n, and the new count is n + 1. The new mean = [S(n + 1)/n] / (n + 1) = S/n = x̄. Thus the mean is preserved. For the standard deviation, the new variance equals Σ(xᵢ − x̄)² / (n + 1), where the numerator is unchanged (the added term is (x̄ − x̄)² = 0) but the denominator has grown from n to n + 1. Therefore the new variance = (n/(n + 1)) × σ², which is strictly less than σ² unless σ² = 0. The SD is unchanged only if all original values are identical (σ = 0), because 0 × n/(n + 1) = 0. In every other case, adding a value at the mean reduces the standard deviation. This result formalizes the intuition that injecting a perfectly 'average' data point makes the overall distribution slightly less variable.

SUMMARY

Lesson Summary

Summary

The three pillars of descriptive statistics — the arithmetic mean, the median, and the standard deviation — provide complementary snapshots of any data set. The mean (x̄ = Σxᵢ / n) leverages every observation and serves as the algebraic center, while the median offers a robust, outlier-resistant measure of the typical value. The standard deviation (σ = √[Σ(xᵢ − μ)² / N]) quantifies spread around the mean by averaging squared deviations and returning to the original units via the square root.

For GRE success, remember these core principles: in a symmetric distribution the mean equals the median; in a skewed distribution the mean is pulled toward the tail. Adding a constant to every value shifts the mean and median but leaves the SD unchanged; multiplying every value by a constant c multiplies the SD by |c|. A data point added at the mean preserves the mean but reduces the SD, while an extreme outlier inflates both the mean and the SD. Master these relationships, and you will handle any descriptive-statistics question the GRE presents with confidence.