Mastering GRE Quant Statistics: Mean, Median, and Standard Deviation

GRE Quant: Statistics - Mean, Median, Standard Deviation

What the Examiner is Testing

The GRE Quantitative section assesses your ability to interpret and analyze numerical data sets, particularly your understanding of central tendency and dispersion. You'll be expected to compute and compare these statistical measures, often in the context of real-world scenarios or data presented graphically.

The Method

When faced with a statistics problem involving mean, median, or standard deviation, follow these steps:

1. Understand the Data Set: Carefully read the problem to identify the raw data points and any associated units. Determine if the data is discrete or continuous, and if there are any outliers.
2. Order the Data (for Median): If the median is required, arrange the data set in ascending or descending order. This is a crucial preliminary step for accurate median calculation.
3. Calculate the Mean (\(\mu\) or \(\bar{x}\)): Sum all the data points and divide by the total number of data points, \(N\).
   $$ \mu = \frac{\sum_{i=1}^{N} x_i}{N} $$
4. Calculate the Median:
   • If \(N\) is odd, the median is the middle value in the ordered data set. Its position is \((N+1)/2\).
   • If \(N\) is even, the median is the average of the two middle values in the ordered data set. Their positions are \(N/2\) and \((N/2)+1\).
5. Calculate the Standard Deviation (\(\sigma\)):
   • Step 5a: Calculate Deviations from the Mean: For each data point \(x_i\), find its deviation from the mean: \((x_i - \mu)\).
   • Step 5b: Square the Deviations: Square each deviation: \((x_i - \mu)^2\).
   • Step 5c: Sum the Squared Deviations: Sum all the squared deviations: \(\sum_{i=1}^{N} (x_i - \mu)^2\).
   • Step 5d: Calculate the Variance (\(\sigma^2\)): Divide the sum of squared deviations by \(N\) (for population standard deviation, which is typically assumed on the GRE unless specified otherwise for sample standard deviation, where you'd divide by \(N-1\)).
     $$ \sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N} $$
   • Step 5e: Take the Square Root: The standard deviation is the square root of the variance.
     $$ \sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}} $$
6. Interpret and Compare: Once calculated, interpret what these measures tell you about the data. For comparative questions, clearly state your findings.

Fully Worked Example

A research team recorded the daily high temperatures (in degrees Celsius) for a week in a specific location: \(18^\circ C, 22^\circ C, 19^\circ C, 25^\circ C, 20^\circ C, 22^\circ C, 21^\circ C\). Calculate the mean, median, and standard deviation of these temperatures.

1. Understand the Data Set: We have 7 daily high temperatures in degrees Celsius. \(N=7\).
   Data: \(18, 22, 19, 25, 20, 22, 21\) (\(^\circ C\)).

2. Order the Data:
   Ordered Data: \(18, 19, 20, 21, 22, 22, 25\) (\(^\circ C\)).

3. Calculate the Mean (\(\mu\)):
   $$ \mu = \frac{18+19+20+21+22+22+25}{7} $$
   $$ \mu = \frac{147}{7} $$
   $$ \mu = 21^\circ C $$

4. Calculate the Median:
   Since \(N=7\) (odd), the median is the \((7+1)/2 = 4^{th}\) value in the ordered list.
   Ordered Data: \(18, 19, 20, \underline{21}, 22, 22, 25\)
   $$ \text{Median} = 21^\circ C $$

5. Calculate the Standard Deviation (\(\sigma\)):

   • Step 5a & 5b: Deviations from the Mean and Squared Deviations:

     • \((18 - 21)^2 = (-3)^2 = 9\)
     • \((19 - 21)^2 = (-2)^2 = 4\)
     • \((20 - 21)^2 = (-1)^2 = 1\)
     • \((21 - 21)^2 = (0)^2 = 0\)
     • \((22 - 21)^2 = (1)^2 = 1\)
     • \((22 - 21)^2 = (1)^2 = 1\)
     • \((25 - 21)^2 = (4)^2 = 16\)

   • Step 5c: Sum the Squared Deviations:
     $$ \sum (x_i - \mu)^2 = 9 + 4 + 1 + 0 + 1 + 1 + 16 = 32 $$

   • Step 5d: Calculate the Variance (\(\sigma^2\)):
     $$ \sigma^2 = \frac{32}{7} \approx 4.5714 $$

   • Step 5e: Take the Square Root:
     $$ \sigma = \sqrt{4.5714} \approx 2.138^\circ C $$

   Results: The mean temperature is \(21^\circ C\), the median temperature is \(21^\circ C\), and the standard deviation is approximately \(2.14^\circ C\).

Three Mistakes That Lose Marks

1. Incorrectly Ordering Data for Median: Failing to sort the data set before finding the middle value(s) is a common error, leading to an incorrect median. Always order the data first.
2. Confusing Population vs. Sample Standard Deviation: While GRE typically assumes population standard deviation (division by \(N\)), some problems might subtly imply a sample (division by \(N-1\)). Misinterpreting this can lead to a slight but incorrect numerical answer, especially if the options are close. Pay attention to context.
3. Calculation Errors with Negative Deviations: When calculating standard deviation, forgetting to square negative deviations correctly (e.g., \((-3)^2 = -9\) instead of \(9\)) is a frequent slip. Remember that squared terms are always non-negative.

30-Second Recap

Mean is the average, median is the middle value (after ordering), and standard deviation measures data spread around the mean. Always order data for the median. For standard deviation, calculate deviations from the mean, square them, sum, divide by \(N\), then take the square root. Watch out for ordering errors, \(N\) vs. \(N-1\), and squaring negatives.