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Descriptive Statistics
Mean, Median & SD

Master the three pillars of data summary: central tendency and spread. Learn the formulas, build intuition, and avoid the traps the GRE sets.

GRE Quantitative

What Are Descriptive Statistics?

Descriptive statistics summarize a dataset using just a few numbers. Instead of listing every value, we capture two key ideas:

Central Tendency
Where is the "center" of the data? Answered by the mean and median.
Spread / Variability
How far do values stray from the center? Answered by the standard deviation.

The Arithmetic Mean

The mean is the sum of all values divided by the count. It is the "balance point" of a dataset.

Every single data point contributes to the mean. This makes it sensitive to outliers: one extreme value can pull the mean far from the bulk of the data.

The Median

The median is the middle value when the data is sorted in order. It splits the dataset into two equal halves.

1
Sort all values from smallest to largest.
2
Odd \( n \): The median is the value at position \( \frac{n+1}{2} \).
3
Even \( n \): The median is the average of the values at positions \( \frac{n}{2} \) and \( \frac{n}{2}+1 \).

Visualizing Mean vs Median

2 3 4 5 6 20 (outlier) Median = 5 Mean = 6 outlier pulls mean right

Standard Deviation: Measuring Spread

Standard deviation (\( \sigma \) for a population, \( s \) for a sample) measures how far each data point typically falls from the mean.

A small SD means data points cluster tightly around the mean. A large SD means they are widely scattered.

Computing Standard Deviation Step by Step

1
Find the mean \( \bar{x} \) of the dataset.
2
Subtract the mean from each value: \( x_i - \bar{x} \) (these are the deviations).
3
Square each deviation: \( (x_i - \bar{x})^2 \).
4
Average the squared deviations (divide by \( N \) for population, \( n-1 \) for sample). This gives the variance.
5
Take the square root of the variance to get the standard deviation.

Visualizing Low vs High SD

Low SD (tight cluster) mean High SD (wide spread) mean small SD large SD Same mean, different spreads SD captures how "wide" the distribution is

Mean vs Median: When to Use Each

Mean Median
Affected by outliers? Yes, strongly No (resistant)
Best for Symmetric data Skewed data
Uses all data values? Yes Only position
Skewed right \( \text{Mean} > \text{Median} \) Closer to center
Skewed left \( \text{Mean} < \text{Median} \) Closer to center
Symmetric \( \text{Mean} = \text{Median} \)

Key Properties of Standard Deviation

Always ≥ 0
SD equals zero only when all values are identical. Otherwise it is positive.
Add a constant
Adding \( c \) to every value does not change the SD. Spread stays the same.
Multiply by \( k \)
Multiplying every value by \( k \) multiplies the SD by \( |k| \).

How to Apply These Concepts on the GRE

1
Read the question carefully: Is it asking for the mean, median, or SD? Or a comparison?
2
Sort the data if computing the median. For the mean, just sum and divide.
3
Check for outliers: If present, the median is more reliable than the mean.
4
For SD comparisons: Visually assess which dataset is more tightly clustered. You rarely need exact computation.
5
Apply transformation rules: Adding shifts the mean but not the SD. Multiplying scales both.

Worked Example: Finding the Mean

A student scores the following on five quizzes: 72, 85, 90, 68, 95. Find the mean score.

Worked Example: Finding the Median

Find the median of this dataset: 14, 7, 23, 9, 18, 12.

1
Sort: 7, 9, 12, 14, 18, 23
2
Count: \( n = 6 \) (even), so average positions 3 and 4.
3
Values at positions 3 and 4: 12 and 14.
4
Median: \( \frac{12 + 14}{2} = \) 13.

Worked Example: Computing SD

Find the population standard deviation for: 2, 4, 4, 4, 5, 5, 7, 9.

Common Mistakes to Avoid

✕ Forgetting to Sort
To find the median, you must sort the data first. Picking the middle of an unsorted list gives the wrong answer.
✕ Confusing \( N \) and \( n-1 \)
Population SD divides by \( N \). Sample SD divides by \( n-1 \). On the GRE, read carefully which is asked.
✕ Adding Shifts SD
Adding a constant to every value changes the mean but not the SD. Many students incorrectly assume both change.

Summary Comparison

Mean \( \bar{x} \) Median SD \( \sigma \)
Measures Center (balance) Center (middle) Spread
Formula \( \frac{\sum x_i}{n} \) Middle value \( \sqrt{\frac{\sum(x_i-\bar{x})^2}{N}} \)
Outlier sensitive? Yes No Yes
Add \( c \) to all \( \bar{x} + c \) Median \( + c \) No change
Multiply by \( k \) \( k\bar{x} \) \( k \times \) Median \( |k| \cdot \sigma \)

Question 1 of 5

The mean of the dataset 3, 7, 8, 12, 15 is:

Question 2 of 5

Find the median of: 5, 2, 9, 1, 8, 3.

Question 3 of 5

A dataset has a mean of 50 and a standard deviation of 8. If every value is multiplied by 3, what is the new standard deviation?

Question 4 of 5

In a right-skewed distribution, which of the following is true?

Question 5 of 5

A dataset has a standard deviation of 6. If 10 is added to every value, the new standard deviation is: