5.1 Descriptive Statistics & Data Measures

5.1 Descriptive Statistics & Data Measures

Descriptive statistics forms a cornerstone of the GRE Quantitative Reasoning section. The exam tests your capacity to summarize data sets, evaluate central tendencies, measure variability, understand how mathematical transformations alter data summaries, and apply properties of symmetrical distributions such as the normal curve.

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Measures of Central Tendency

Central tendency measures describe the center or typical value of a numerical data set. The three primary metrics tested on the GRE are the arithmetic mean, median, and mode.

1. Arithmetic Mean & Weighted Average

The arithmetic mean (commonly referred to simply as the average) of a set of $n$ numbers is the sum of all values divided by $n$:

$$\text{Mean} = \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{x_1 + x_2 + \dots + x_n}{n}$$

Key Properties of the Mean:

• Sum Formula: $\text{Sum} = \text{Mean} \times n$. In many GRE word problems, finding the sum of elements is the essential first step.
• Balance Point: The sum of the deviations of each data point from the mean is always zero: $\sum (x_i - \bar{x}) = 0$.
• Sensitivity to Outliers: Extreme values heavily pull the mean toward them.

Weighted Average

When combining groups of different sizes or weights, you cannot simply average their individual means. Instead, use the weighted average formula:

$$\text{Weighted Mean} = \frac{w_1 x_1 + w_2 x_2 + \dots + w_k x_k}{w_1 + w_2 + \dots + w_k}$$

Worked Example (Weighted Mean): Class A has 20 students with a test mean of 80. Class B has 30 students with a test mean of 90. What is the combined mean for all 50 students?

$$\text{Combined Mean} = \frac{(20 \times 80) + (30 \times 90)}{20 + 30} = \frac{1600 + 2700}{50} = \frac{4300}{50} = 86$$

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2. Median & Mode

Median ($Q_2$)

The median is the middle number when a data set is arranged in ascending or descending order.

• Odd $n$: The median is the single middle value located at position $\frac{n+1}{2}$. For example, in $\{3, 7, 9, 12, 15\}$, $n=5$, and the median is $9$.
• Even $n$: The median is the average of the two central numbers at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$. For example, in $\{4, 8, 11, 17, 20, 25\}$, $n=6$, and the median is $\frac{11 + 17}{2} = 14$.
• Robustness: Unlike the mean, the median is resistant to extreme outliers. Adding an extreme value like $1,000$ to the set $\{3, 7, 9, 12, 15\}$ changes the mean drastically but changes the median only slightly.

Mode

The mode is the value that appears most frequently in a data set.

• A data set can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all (if every number appears with equal frequency).
• Example: In $\{2, 4, 4, 7, 9, 9, 9, 12\}$, the mode is $9$.

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Measures of Dispersion: Range & Standard Deviation

While central tendency pinpoints the middle of a data set, measures of dispersion describe how spread out the data values are.

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Step-by-Step Variance and Standard Deviation Calculation

Consider the data set $\{2, 4, 4, 4, 6\}$ ($n = 5$):

1. Find the Mean: $\bar{x} = \frac{2 + 4 + 4 + 4 + 6}{5} = \frac{20}{5} = 4$.
2. Compute Deviations from Mean ($x_i - \bar{x}$): $-2, 0, 0, 0, +2$.
3. Square Each Deviation ($(x_i - \bar{x})^2$): $4, 0, 0, 0, 4$.
4. Calculate Variance (Average of Squared Deviations): $\sigma^2 = \frac{4 + 0 + 0 + 0 + 4}{5} = \frac{8}{5} = 1.6$.
5. Calculate Standard Deviation: $\sigma = \sqrt{1.6} \approx 1.265$.

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Operational Properties of Standard Deviation on the GRE

GRE Quantitative Reasoning frequently tests standard deviation conceptually without requiring laborious arithmetic. Memorize these transformation rules:

1. Adding/Subtracting a Constant ($+k$ or $-k$): If a constant $k$ is added to (or subtracted from) every data point in a set:

   • Mean increases by $k$: $\bar{x}_{new} = \bar{x} + k$.
   • Median increases by $k$: $\text{Median}_{new} = \text{Median} + k$.
   • Standard Deviation remains UNCHANGED: $\sigma_{new} = \sigma$. (Shifting the entire data set along the number line does not alter the distance between data points).
   • Range remains UNCHANGED: $\text{Range}_{new} = \text{Range}$.

2. Multiplying/Dividing by a Constant ($\times k$ for $k > 0$): If every data point in a set is multiplied by a positive constant $k$:

   • Mean is multiplied by $k$: $\bar{x}_{new} = k \cdot \bar{x}$.
   • Median is multiplied by $k$: $\text{Median}_{new} = k \cdot \text{Median}$.
   • Standard Deviation is multiplied by $k$: $\sigma_{new} = k \cdot \sigma$.
   • Variance is multiplied by $k^2$: $\sigma^2_{new} = k^2 \cdot \sigma^2$.
   • Range is multiplied by $k$: $\text{Range}_{new} = k \cdot \text{Range}$.

3. Spread Comparison:

   • A data set whose values are clustered tightly around the mean has a smaller SD.
   • A data set whose values are spread far from the mean has a larger SD.

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The Normal Distribution & The 68-95-99.7 Empirical Rule

A normal distribution is a symmetric, bell-shaped continuous probability distribution. Its central axis of symmetry occurs at the mean $\mu$, which equals both the median and the mode (Mean = Median = Mode).

                      Normal Curve (68-95-99.7 Rule)
                                  | 
                               .  |  .
                             .    |    .
                           .      |      .
                        .   34%   |   34%  .
                     . 13.5%      |      13.5% .
                 . 2.35%          |          2.35% .
             ---------------------------------------------------
              μ-3σ   μ-2σ   μ-1σ  μ   μ+1σ   μ+2σ   μ+3σ

The 68-95-99.7 Empirical Rule Breakdown:

• $\mu \pm 1\sigma$: Approximately 68% of all data falls within 1 standard deviation of the mean (34% between $\mu - 1\sigma$ and $\mu$, 34% between $\mu$ and $\mu + 1\sigma$).
• $\mu \pm 2\sigma$: Approximately 95% of all data falls within 2 standard deviations of the mean (13.5% between $1\sigma$ and $2\sigma$ on each side).
• $\mu \pm 3\sigma$: Approximately 99.7% of all data falls within 3 standard deviations of the mean (2.35% between $2\sigma$ and $3\sigma$ on each side).
• Tails: Only 0.15% of data lies beyond $+3\sigma$, and 0.15% lies below $-3\sigma$.

Worked Example (Empirical Rule): Test scores are normally distributed with a mean of 72 and a standard deviation of 8. In a group of 1,000 test takers, how many students scored between 64 and 88?

• $64 = 72 - 8 = \mu - 1\sigma$.
• $88 = 72 + 2(8) = \mu + 2\sigma$.
• Between $\mu - 1\sigma$ and $\mu$: $34\%$.
• Between $\mu$ and $\mu + 2\sigma$: $95\% / 2 = 47.5\%$.
• Total percentage = $34\% + 47.5\% = 81.5\%$.
• Number of students = $81.5\% \times 1000 = 815$ students.

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GRE Exam Tips & Quantitative Comparison Strategies

1. QC Strategy - Estimating SD without Calculation: If Quantitative Comparison asks you to compare the SD of Set A $\{10, 20, 30, 40, 50\}$ vs Set B $\{10, 10, 30, 50, 50\}$, notice that Set B has more numbers further from its mean of 30 ($10, 10, 50, 50$) than Set A ($20, 40$). Thus, Set B has a larger standard deviation.
2. Mean vs Median Skewness Trap:
   • Right-Skewed (Positively Skewed): Long tail to the right $\implies \text{Mean} > \text{Median}$.
   • Left-Skewed (Negatively Skewed): Long tail to the left $\implies \text{Mean} < \text{Median}$.
   • Symmetric: $\text{Mean} = \text{Median}$.