GRE Quantitative Quiz: Descriptive Statistics

What this quiz covers

This quiz focuses on Descriptive Statistics, giving you a quick way to practice the rules, question types, and explanations that matter most for GRE Quantitative.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. The mean decreases and the median decreases.
2. The mean decreases and the median increases.
3. The mean increases and the median decreases.
4. The mean decreases and the median stays the same. (correct answer)
5. The mean stays the same and the median decreases.

Explanation: This question tests descriptive statistics by evaluating changes in mean and median after removing an outlier. Mean is the average, sensitive to extremes, while median is the middle value, resistant to outliers. Original set {3,3,4,4,4,5,20} sorted, median is 4 (fourth value), sum=3+3+4+4+4+5+20=43, mean=43/7≈6.143. Removing 20 gives {3,3,4,4,4,5}, sum=23, mean=23/6≈3.833 (decreases), median now average of 3rd and 4th: 4 and 4, so 4 (same). Thus, mean decreases, median stays the same, as removal of high outlier lowers average but middle unchanged. Tempting to think median changes, like in A, but with even count now, it's still 4. Mean vs median confusion might lead to thinking both decrease, but median is robust.

Question 2

1. Dataset $P$ has the greater median, because it has the largest value.
2. Dataset $Q$ has the greater mean, because it has no outlier.
3. The medians are equal, and dataset $P$ has the greater mean. (correct answer)
4. Dataset $P$ has the greater median and the greater mean.
5. The means are equal because 100 is an outlier and should be ignored.

Explanation: This question tests descriptive statistics by comparing mean and median in skewed datasets. Mean is pulled by outliers, median is middle value resistant to them. P {1,2,3,4,100} median 3 (third), mean (1+2+3+4+100)/5=110/5=22. Q {1,2,3,4,5} median 3, mean 15/5=3. So medians equal at 3, P has greater mean due to 100. The outlier inflates P's mean but not median, keeping medians same. Tempting to ignore outlier for mean, like E, but mean includes all values. Another error is thinking P has greater median due to largest value, like A.

Question 3

1. 8
2. $\frac{38}{3}$ (correct answer)
3. 10
4. 12
5. $\frac{40}{3}$

Explanation: This question tests descriptive statistics by finding mean of a subset after knowing overall mean. Mean is sum divided by count; if overall mean 10 for 4 numbers, sum=40. If one is 2, sum of remaining 3 is 40-2=38, mean=38/3≈12.667. Thus, the mean of the remaining 3 is 38/3, as subtracting the known value adjusts the sum accordingly. A common mistake is thinking it's still 10, like C, ignoring the removal. Confusion might lead to 40/3 like E, but that's including the 2 incorrectly.

Question 4

1. Dataset $R$ has the greater standard deviation.
2. Dataset $S$ has the greater standard deviation. (correct answer)
3. The standard deviations are equal because the means are equal.
4. Dataset $R$ has the greater median.
5. Dataset $S$ has the greater median because it has a larger range.

Explanation: This question tests descriptive statistics by comparing standard deviation based on spread, despite equal means. Standard deviation quantifies dispersion from mean; greater spread means higher SD. Both have mean 50, but R clustered near 50 (low SD), S from 10 to 90 (high spread, high SD). Thus, S has greater standard deviation due to wider range of values. A common error is thinking equal means imply equal SD, like C, but SD measures variability, not location. Another mistake is focusing on median instead of SD, like D or E.

Question 5

1. Dataset $A$, because its mean is 10.
2. Dataset $A$, because all values are equal.
3. Dataset $B$, because its values are more spread out. (correct answer)
4. Dataset $B$, because its median is 10.
5. They have equal standard deviation because they have the same mean.

Explanation: This question tests descriptive statistics by comparing variability in datasets with same mean. Standard deviation measures spread from the mean, higher when values are more dispersed. Dataset A {10,10,10,10,10} has mean 10, all values at mean, so SD=0. Dataset B {8,9,10,11,12} has mean (8+9+10+11+12)/5=50/5=10, but values spread out, so higher SD. B has greater standard deviation because its values deviate more from the mean, while A has none. A common mistake is thinking equal means imply equal SD, like in E, but SD is independent of mean value. Another error is focusing on median or equality instead of spread.

Question 6

1. 11
2. 12
3. 13
4. $\frac{82}{7}$ (correct answer)
5. $\frac{112}{7}$

Explanation: This question tests descriptive statistics by calculating mean after removing a value. Mean is sum over count; original 8 values mean 14, sum=112. Remove 30, sum=112-30=82, remaining 7 mean=82/7≈11.714. Thus, the mean is 82/7, as adjusting sum and count gives the new average. Tempting to think it's 112/7=16, like E, but that's without removing. Mean vs total confusion might lead to 11 or 12, like A or B.

Question 7

1. It decreases by 3.
2. It increases by 30.
3. It increases by 6.
4. It increases by 3. (correct answer)
5. It stays the same because only one score changed.

Explanation: This question tests descriptive statistics by assessing mean change after score correction. Mean is total sum over count; original mean 80 for 10 students, sum=800. Correcting from 60 to 90 increases sum by 30, new sum=830, new mean=830/10=83, increase by 3. The mean increases by 3, as the correction adds 30 to total, divided by 10. A common error is thinking increase by 30, like B, forgetting to divide by n. Another mistake is assuming no change since only one score, like E.

Question 8

1. The mean decreases, and the median increases.
2. The mean increases, and the median stays the same.
3. The mean decreases, and the median stays the same. (correct answer)
4. The mean stays the same because the mode is unchanged.
5. The mean decreases, and the median decreases.

Explanation: This question tests descriptive statistics by examining mean and median after replacing a value. Mean sensitive to changes, median to order; original {1,2,2,2,9} sum=16, mean=3.2, median 2. Replace 9 with 3: {1,2,2,2,3} sum=10, mean=2 (decreases), median 2 (same). Mean decreases, median stays same, as replacement lowers sum but middle unchanged. A tempting option is thinking median increases to 2.5 or something, but sorted it's 1,2,2,2,3, middle 2. Confusion with mode might suggest mean same, like D.

Question 9

1. The mean decreases and the median decreases.
2. The mean decreases and the median stays the same. (correct answer)
3. The mean stays the same and the median decreases.
4. The mean increases and the median stays the same.
5. The mean decreases and the median increases.

Explanation: This question tests descriptive statistics by examining changes in central tendency measures after data modification. The key concepts are mean, which is the average of the values, and median, which is the middle value in an ordered list. Originally, the set {6, 7, 7, 8, 8, 9, 20} has a sum of 65 and mean of 65/7 ≈ 9.29, with median 8 as the fourth value. Replacing 20 with 10 gives a new sum of 55 and mean of 55/7 ≈ 7.86, while the ordered set {6, 7, 7, 8, 8, 9, 10} still has median 8. Thus, the mean decreases, but the median stays the same, as the replacement affects the average but not the middle position. A tempting mistake is thinking the median changes, like in option A, but median depends on order, not extreme values. Confusion between mean and median often leads to incorrect choices, such as assuming both decrease when only the mean is pulled down by the lower value.

Question 10

1. The added value is 15.
2. At least 5 of the original values are equal to 15.
3. In the new dataset, the 5th and 6th values (when ordered) average to 15. (correct answer)
4. The mean of the original dataset is 15.
5. The standard deviation of the new dataset is 0.

Explanation: This question tests descriptive statistics by exploring properties of median in even and odd datasets. Median for odd n is middle value; for even, average of two middle. Original 9 values, median 15 (5th value). Adding one makes 10 values, median average of 5th and 6th =15. Thus, the 5th and 6th in ordered new set average to 15, which must be true regardless of added value's position. This holds because the median definition requires it for the new even count. A tempting incorrect option is A, assuming added value is 15, but it could be anything as long as middle two average 15. Another error is thinking original mean is 15, like D, confusing mean and median.

Question 11

1. At least three values are 20.
2. The largest value is greater than 20.
3. The sum of the 5 values is 90. (correct answer)
4. The range is at least 2.
5. The standard deviation is less than 2.

Explanation: This question tests descriptive statistics by deducing necessary conditions from mean and median. For 5 values, mean 18 implies sum=90, median 20 is third value. The third value is 20, but statement C says sum is 90, which must be true from mean. Yes, regardless of other values, sum must be 90. Other options like A (at least three 20) not necessarily, e.g. {10,15,20,25,20} sum=90, median 20, only two 20s. Or B largest >20 not must, could be {18,18,20,18,16} wait sum=90, but sorted 16,18,18,18,20 median 18 not 20. To have median 20, third ≥20, but sum=90 must. Yes, C must be true. Tempting is A, thinking mode or something, but not required. Another is D range at least 2, but could all be 18 except adjust, but for median 20 with mean 18, possible with range 0? If all 18, median 18 not 20. To have third 20 but mean 18, need some below, so range >0, but not necessarily >=2. E.g. {17.5,19.5,20,20,13} wait sum not 90. But possible small range, but not must <2. Anyway, C is must true.

Question 12

1. The first dataset has the greater median.
2. The first dataset has the greater mean.
3. The datasets have equal means, and the second dataset has the greater standard deviation.
4. The datasets have equal means, and the first dataset has the greater standard deviation. (correct answer)
5. The datasets have equal medians, and the first dataset has the smaller range.

Explanation: This question tests descriptive statistics by comparing mean, median, and standard deviation across datasets. Mean is average, median middle, SD spread; first {5,5,5,5,25} mean=(5*4+25)/5=45/5=9, median 5, high SD due to 25. Second {9,9,9,9,9} mean=9, median 9, SD=0. Means equal at 9, first has greater SD due to outlier causing more dispersion. First has greater SD, while medians differ (5 vs 9). Tempting to think second has greater SD, but it's constant. Mean equality might confuse with SD equality, but outlier increases SD.

Question 13

1. Median
2. Mode
3. Mean (correct answer)
4. Minimum
5. Interquartile range

Explanation: This question tests descriptive statistics by identifying which measure is most sensitive to an outlier. Mean is average, heavily influenced by extremes; median, mode, min, IQR are more robust. Data {12,12,13,13,40}, mean=(12+12+13+13+40)/5=90/5=18, without 40 mean=50/4=12.5, big change. Median 13 with or without, mode 12 and 13 bimodal, min 12, IQR (Q3-Q1) changes little. Mean is most affected by 40, as it pulls the average up significantly. A tempting choice is median, but it's unchanged at 13. Confusion arises from thinking mode changes, but outlier doesn't affect frequency much here.

Question 14

1. The mean increases by 5, and the standard deviation increases by 5.
2. The mean increases by 5, and the standard deviation stays the same. (correct answer)
3. The mean stays the same, and the standard deviation increases by 5.
4. The mean increases by 30, and the standard deviation stays the same.
5. The median decreases by 5, and the standard deviation stays the same.

Explanation: This question tests descriptive statistics by observing effects of shifting data on mean and standard deviation. Adding a constant to each value increases mean by that constant, but SD remains unchanged as spread is preserved. Original {2,4,6,8,10,12} mean= (2+4+6+8+10+12)/6=42/6=7, SD calculable but adding 5 gives {7,9,11,13,15,17} mean=12 (increase by 5), SD same since differences unchanged. Mean increases by 5, SD stays same, as shift doesn't affect variance. Tempting to think SD increases by 5, like A, but SD is not additive like mean. Confusion with total sum might suggest increase by 30, like D.

Question 15

1. 10
2. 12
3. 13
4. 15
5. 16 (correct answer)

Explanation: This question tests descriptive statistics by finding a value that achieves a new mean. Mean is sum divided by count; original 5 prices {10,12,12,13,15} sum=62, mean=12.4. Add sixth, new sum=62+x, new mean=(62+x)/6=13, so 62+x=78, x=16. Thus, 16 could be the price, solving for x gives exactly 13 mean. A common mistake is thinking x=13, like C, but that would make mean (62+13)/6=75/6=12.5, not 13. Confusion with median might suggest 15, but question is about mean.

Question 16

1. The mean decreases by 3.
2. The mean increases by 3.
3. The mean increases by $\frac{18}{7}$. (correct answer)
4. The mean increases by $\frac{18}{6}$.
5. The mean stays the same because 30 is an outlier.

Explanation: This question tests descriptive statistics by assessing how adding a value affects the mean. The mean is the sum divided by the number of values, a measure of central tendency. Original set of 6 numbers has mean 12, so sum = 6*12 = 72. Adding 30 makes new sum 72+30=102, new mean 102/7 ≈14.571, increase of ≈2.571, which is 18/7. The mean increases by (30 - 12)/7 = 18/7, as the added value is above the original mean, pulling it up. A common error is thinking the increase is 18/6=3, like in option D, but that's incorrect because the denominator is the new count of 7. Confusing the change per original items ignores the new total count.

Question 17

1. The mean increases by 3, and the median increases by 3. (correct answer)
2. The mean increases by 3, but the median stays the same.
3. The mean stays the same, but the median increases by 3.
4. The range increases by 3.
5. The standard deviation increases by 3.

Explanation: This question tests descriptive statistics by examining how adding a constant to all values affects various measures. When a constant is added to every value in a dataset, the mean and median both increase by that constant. Original data: 4, 6, 7, 9, 10, 14 has mean = 50/6 ≈ 8.33 and median = (7+9)/2 = 8. After adding 3 to each value: 7, 9, 10, 12, 13, 17 has mean = 68/6 ≈ 11.33 and median = (10+12)/2 = 11. Both increased by exactly 3. The standard deviation, which measures spread, remains unchanged because the relative distances between values stay the same. Students might incorrectly think only the mean changes or that measures of spread like standard deviation would also increase.