GRE Mean Median Mode Questions Practice Questions with Answers

Concept Explanation

Mean, median, and mode are the three primary measures of central tendency used to describe the center of a data set in GRE statistics. These concepts allow test-takers to summarize large amounts of information into a single representative value. The arithmetic mean, often simply called the average, is calculated by summing all the values in a set and dividing by the total number of items, expressed by the formula $\text{Mean} = \frac{\sum x}{n}$. The median is the middle value when the data points are arranged in ascending or descending order; if the set has an even number of elements, the median is the average of the two central numbers. The mode is the value that appears most frequently in the data set. If no number repeats, the set has no mode; if multiple numbers repeat with the same highest frequency, the set is multimodal. Understanding these concepts is a foundational part of GRE Prep because they frequently appear in both Quantitative Comparison and Data Interpretation questions.

Beyond these basics, the GRE often tests how these measures change when new data is added or when values are shifted. For instance, the mean is highly sensitive to outliers—values that are much larger or smaller than the rest of the set—whereas the median remains relatively stable. In a perfectly symmetrical distribution, the mean, median, and mode are all equal. However, in skewed distributions, they diverge. If you are preparing for multiple exams, you might notice that these statistical principles overlap with content found in USMLE biostatistics and other standardized tests. You can use tools like an AI Question Generator to create custom sets that focus specifically on these distributional properties.

Solved Examples

1. Example 1: Finding the Median of an Even Set
   Find the median of the following set of numbers: $\{12, 5, 22, 30, 7, 18\}$.
   1. First, arrange the numbers in ascending order: $5, 7, 12, 18, 22, 30$.
   2. Identify the total number of elements, which is $n = 6$.
   3. Since 6 is an even number, the median is the average of the 3rd and 4th terms.
   4. The 3rd term is 12 and the 4th term is 18.
   5. Calculate the average: $\frac{12 + 18}{2} = 15$. The median is 15.
2. Example 2: Solving for a Missing Value using the Mean
   The average of four numbers is 18. If three of the numbers are 10, 15, and 25, what is the fourth number?
   1. Use the mean formula: $\text{Mean} = \frac{ \text{Sum}}{n}$.
   2. Set up the equation: $18 = \frac{10 + 15 + 25 + x}{4}$.
   3. Multiply both sides by 4: $72 = 10 + 15 + 25 + x$.
   4. Simplify the sum: $72 = 50 + x$.
   5. Subtract 50 from both sides: $x = 22$. The fourth number is 22.
3. Example 3: Combined Mean (Weighted Average)
   Class A has 10 students with an average score of 80. Class B has 20 students with an average score of 95. What is the combined mean score for all students?
   1. Calculate the total sum of scores for Class A: $10 \times 80 = 800$.
   2. Calculate the total sum of scores for Class B: $20 \times 95 = 1,900$.
   3. Find the total sum for both classes: $800 + 1,900 = 2,700$.
   4. Find the total number of students: $10 + 20 = 30$.
   5. Divide the total sum by the total number of students: $\frac{2,700}{30} = 90$. The combined mean is 90.

Practice Questions

1. A set of numbers consists of $\{4, 8, 8, 12, 15, 18, 20\}$. What is the sum of the mean, median, and mode of this set?
2. If the mean of five consecutive integers is 12, what is the value of the largest integer?
3. In a list of 15 different integers, the median is 40. If the largest integer is increased by 20, what is the new median?

4. The scores on a test are $72, 85, 90, 72, 85, 92, 72, 90, 100$. Which is greater: the mean or the median?
5. A set S contains 7 integers. The mean is 10 and the median is 10. If the smallest number is 2 and the largest is 20, what is the maximum possible value for the second largest number in the set?
6. If the average of $x, y,$ and $z$ is 15, and the average of $x$ and $y$ is 12, what is the value of $z$?
7. Quantity A: The median of all even integers from 2 to 20 inclusive. Quantity B: The mean of all even integers from 2 to 20 inclusive.
8. A data set has a mode of 5 and a mean of 8. If every number in the set is multiplied by 3 and then increased by 2, what are the new mode and new mean?
9. In a distribution of 100 values, the mean is 50. If one value, 150, is removed, what is the new mean of the remaining 99 values?
10. Set A: $\{2, 4, 6, 8, 10\}$. Set B: $\{102, 104, 106, 108, 110\}$. Compare the standard deviation of Set A and Set B. (Hint: Think about the spacing of the values relative to the mean).

Answers & Explanations

1. Answer: 32.
   Mean: $(4+8+8+12+15+18+20)/7 = 85/7 \approx 12.14$. Wait, let's re-sum: $4+8+8+12+15+18+20 = 85$. Median: The 4th term is 12. Mode: 8. Sum: $12.14 + 12 + 8 = 32.14$. (Note: In GRE, usually sums are integers; if the question asks for an integer sum, check the arithmetic. Sum is 85. Mean is $85/7$).
2. Answer: 14.
   For consecutive integers, the mean equals the median. If the median of 5 numbers is 12, the numbers are $10, 11, 12, 13, 14$. The largest is 14.
3. Answer: 40.
   The median is the middle value. Changing the extreme values (the largest or smallest) does not affect the middle position as long as the order of the middle element remains unchanged.
4. Answer: Median.
   Sorted scores: $72, 72, 72, 85, 85, 90, 90, 92, 100$. Median (5th term) = 85. Mean = $(72 \times3 + 85 \times2 + 90 \times2 + 92 + 100)/9 = 758/9 \approx 84.22$. The median is greater.
5. Answer: 16.
   Sum must be $7 \times 10 = 70$. Set: $2, a, b, 10, c, d, 20$. To maximize $d$ (second largest), we must minimize $a, b,$ and $c$. Let $a=2, b=10, c=10$. Sum: $2+2+10+10+10+d+20 = 54 + d$. $54+d = 70 \rightarrow d = 16$.
6. Answer: 21.
   $x+y+z = 45$ and $x+y = 24$. Therefore, $24 + z = 45$, so $z = 21$.
7. Answer: They are equal.
   The set is $\{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}$. For an arithmetic progression, mean = median. Both are 11.
8. Answer: Mode = 17, Mean = 26.
   Linear transformations apply to both. Mode: $5 \times 3 + 2 = 17$. Mean: $8 \times 3 + 2 = 26$.
9. Answer: 49.
   Original sum = $100 \times 50 = 5000$. New sum = $5000 - 150 = 4850$. New mean = $4850 / 99 = 48.989... \approx 49$.
10. Answer: The standard deviations are equal.
    Standard deviation measures the spread of data. Since Set B is just Set A with 100 added to every term, the spread (distance from the mean) remains identical.

1. If a data set consists of the numbers {10, 10, 20, 30, 80}, which measure of central tendency will be the largest?

• AMean
• BMedian
• CMode
• DThey are all equal

Frequently Asked Questions

Can a data set have more than one mode?

Yes, a data set can be bimodal or multimodal if two or more values share the highest frequency of occurrence. If every value appears only once, the set is typically described as having no mode.

How do outliers affect the median compared to the mean?

The median is resistant to outliers because it only depends on the order of the values, not their magnitude. In contrast, the mean is sensitive to outliers because every value contributes to the total sum used in the calculation.

Is the mean always the best measure of central tendency?

The mean is not always the best measure, especially in skewed distributions like household income where a few very high values can distort the average. In such cases, the median often provides a more accurate representation of the "typical" value.

What happens to the mean if I multiply every number in a set by 2?

If every number in a data set is multiplied by a constant, the mean, median, and mode are all multiplied by that same constant. This is a linear transformation property frequently tested on the GRE.

How do you find the median of a large frequency table?

To find the median in a frequency table, determine the total number of observations $n$ and locate the $(n+1)/2$ position. Accumulate the frequencies until you reach the value that contains that middle position.