Digital SAT Math Practice Questions - Medium : One-variable data: distributions and measures of centre and spread

SAT MAth Practice questions – all topics

Problem-solving and Data Analysis Weightage: 15% Questions: 5-7
- Ratios, rates, proportional relationships, and units
- Percentages
- One-variable data: distributions and measures of centre and spread
- Two-variable data: models and scatterplots
- Probability and conditional probability
- Inference from sample statistics and margin of error
- Evaluating statistical claims: observational studies and Experiments

SAT MAth and English – full syllabus practice tests

[Calc] Question Medium

The dot plot represents a data set.

What is the median of the 19 values in the data set?

▶️Answer/Explanation

Ans:42

First, let’s count the dots above each value on the x-axis:

36: 2 dots
38: 3 dots
40: 2 dots
42: 3 dots
44: 4 dots
46: 3 dots
48: 2 dots

With 19 values, the median will be the 10th value when arranged in order. Counting to the 10th value:

So, the median value is 42.

[Calc] Question Medium

A list of 10 data values is shown below.
\[
4,6,7,2,8,9,6,3,3,3
\]

What is the mean of the data?

▶️Answer/Explanation

Ans:51/10

To find the mean of the given data, you sum up all the values and then divide by the total number of values.

Given data: $4, 6, 7, 2, 8, 9, 6, 3, 3, 3$

Sum of the data: $4 + 6 + 7 + 2 + 8 + 9 + 6 + 3 + 3 + 3 = 51$

Number of data values: $10$

Mean $ = \frac{\text{Sum of data}}{\text{Number of data values}} = \frac{51}{10} = 5.1$

So, the mean of the data is $5.1$.

[Calc] Question Medium

Data set P: 12, 18, 19, 19, 19, 19, 19, 21, 21, 22, 22
Data set $P$ contains the lengths, in inches, of 11 objects. The length 12 inches is removed from data set $\mathrm{P}$ to create data set $\mathrm{N}$, which contains the lengths, in inches, of 10 objects. Which statement best compares the mean $q$ and the median $r$ of data set $\mathrm{P}$ with the mean $s$ and the median $t$ of data set $\mathrm{N}$ ?
A) $q<s ; r>t$
B) $q=s ; r>t$
C) $q<s ; r=t$
D) $q=s ; r=t$

▶️Answer/Explanation

Ans:C

Data set $P$: 12, 18, 19, 19, 19, 19, 19, 21, 21, 22, 22

When 12 inches is removed, data set $N$ becomes: 18, 19, 19, 19, 19, 19, 21, 21, 22, 22

Mean Comparison:
Mean of $P$ ($q$):
\[
q = \frac{\sum P}{11} = \frac{12 + 18 + 19 + 19 + 19 + 19 + 19 + 21 + 21 + 22 + 22}{11} = \frac{211}{11} = 19.18
\]
Mean of $N$ ($s$):
\[
s = \frac{\sum N}{10} = \frac{18 + 19 + 19 + 19 + 19 + 19 + 21 + 21 + 22 + 22}{10} = \frac{199}{10} = 19.9
\]

Since the smallest value (12) was removed, the mean increased:
\[
q < s
\]

Median Comparison:
Median of $P$ ($r$):
Middle value of sorted list of 11 items, $r = 19$
Median of $N$ ($t$):
Middle value of sorted list of 10 items, average of 5th and 6th values:
\[
t = \frac{19 + 19}{2} = 19
\]

Since the median value (19) was not affected by removing the smallest value:
\[
r = t
\]

So, the correct answer is:
\[
\boxed{q < s; r = t}
\]

[Calc] Question Medium

$
0,2,3,4,5,5,5,6,6,7
$

The given list shows a baseball team’s score for each of its first 10 games. In the eleventh game, the team had score of 18. Which of the following best describes the mean and the median of the team’s scores for the first 11 games compared to the first 10 games?

A) The mean increased and the median remained unchanged.
B) The median increased and the mean remained unchanged.
C) Both the mean and the median remained unchanged.
D) Both the mean and the median increased.

▶️Answer/Explanation

Ans:A

Initial Scores (10 games):
\[0, 2, 3, 4, 5, 5, 5, 6, 6, 7\]

Mean of the first 10 games:
\[
\text{Mean} = \frac{\sum \text{scores}}{\text{number of games}} = \frac{0 + 2 + 3 + 4 + 5 + 5 + 5 + 6 + 6 + 7}{10} = \frac{43}{10} = 4.3
\]

Median of the first 10 games:
Since there are an even number of games, the median is the average of the 5th and 6th scores:
\[
\text{Median} = \frac{5 + 5}{2} = 5
\]

Scores with the 11th game:
\[0, 2, 3, 4, 5, 5, 5, 6, 6, 7, 18\]

Mean of the first 11 games:
\[
\text{Mean} = \frac{\sum \text{scores}}{\text{number of games}} = \frac{43 + 18}{11} = \frac{61}{11} \approx 5.545
\]

Median of the first 11 games:
Since there are now an odd number of games, the median is the middle score (6th score):
\[
\text{Median} = 5
\]

Comparing the changes:
The mean increased from 4.3 to approximately 5.545.
The median remained unchanged at 5.

[Calc] Question Medium

The tables show the frequencies of data values for two data sets.

Which statement best compares the mean $a$ and standard deviation $b$ of data set $\mathrm{P}$ with the mean $c$ and standard deviation $d$ of data set $\mathrm{Q}$ ?
A) $a<c ; b<d$
B) $a<c ; b=d$
C) $a>c ; b=d$
D) $a>c ; b>d$

▶️Answer/Explanation

To compare the mean and standard deviation of data set P with those of data set Q, we need to calculate the mean and standard deviation for both sets.

For data set P:
Mean ($a$) can be calculated as the sum of (Value * Frequency) divided by the total frequency.
Standard deviation ($b$) can be calculated based on the formula for standard deviation.

For data set Q:
Mean ($c$) and standard deviation ($d$) can be calculated similarly.

However, without calculating these values, we can infer some information from the given data:
Data set Q has fewer unique values than data set P.
The frequencies of the values in data set Q seem to be similar to those in data set P, but with a shift (e.g., the frequency distribution in Q seems to be a subset of that in P).

From this, we can conclude that:
The mean of data set Q ($c$) might be slightly lower than that of data set P ($a$) due to the smaller values.
The standard deviation of both data sets ($b$ and $d$) might be similar, considering the similarity in the frequency distributions.

Therefore, the best comparison would be:
\[ \boxed{B) \, a<c ; b=d} \]

[Calc] Question medium

A data set consists of 100 values. The least value is 30, and the greatest two values are 80 and 125. If the value 125 is removed from the original data set to create a new data set with 99
values, which statement must be true?
A. The means of the two data sets will be the same.
B. The mean of the new data set will be less than the mean of the original data set.
C. The medians of the two data sets will be the same.
D. The median of the new data set will be less than the median of the original data set.

▶️Answer/Explanation

Ans: B

If the least value is 30 and the greatest two values are 80 and 125, then the median value will remain the same in both datasets since the values are being added or removed at the extremes. However, when the value 125 is removed from the dataset, the mean of the new dataset will be less than the mean of the original dataset because 125 is a higher value. Therefore, the correct answer is option B.

[Calc] Question Medium

A biologist selected a sample of adult female Karner blue butterflies at random from a local population. The mean forewing length of the butterflies in the sample is 1.5 centimeters. The margin of error associated with this estimate for the population mean is 1 centimeter. If the biologist wants an estimate that has a smaller margin of error associated with it and can be generalized to the entire local population, which of the following changes should be made when the study is repeated?
A) Using a different tool to measure the butterflies
B) Measuring the butterflies at two different times of the day and comparing the results
C) Selecting and measuring only the butterflies that look the smallest
D) Selecting and measuring a larger random sample of the butterflies

▶️Answer/Explanation

To reduce the margin of error associated with the estimate for the population mean, the biologist should select and measure a larger random sample of the butterflies. This will increase the sample size, leading to a smaller margin of error for the estimate of the population mean.

Therefore, the answer is:
\[ \boxed{D) \, \text{Selecting and measuring a larger random sample of the butterflies}} \]

[Calc] Question medium

A forestry department measures tree trunk diameter, in inches, at a constant height from the ground for each tree growing in a certain area. The data for 95 of these trees are summarized in the histogram. The first bar represents trees with a trunk diameter less than 6 inches. The second bar represents trees with a trunk diameter of at least 6 inches but less than 12 inches, and so on.

Which of the following is a possible value for the median trunk diameter, in inches, of these trees?
A. 10
B. 15
C. 20
D. 25

▶️Answer/Explanation

Ans: C

To determine the median trunk diameter, we need to find the value at the 50 th percentile, or the middle value, of the ordered dataset. Since there are 95 trees, the median will be the trunk diameter of the 48th tree when the diameters are sorted in ascending order (because $\frac{95+1}{2}=48$ ).

From the histogram:
The first bar (0 to 6 inches) contains 5 trees.
he second bar (6 to 12 inches) contains approximately 10 trees.
The third bar (12 to 18 inches) contains approximately 30 trees.

Summing up the trees in these categories:
Trees with diameters less than 18 inches: $5+10+30=45$ trees.

The 48th tree lies in the fourth bar (18 to 24 inches), because after accounting for the first 45 trees, we need to count 3 more trees to reach the 48th tree. The fourth bar (18 to 24 inches) contains approximately 25 trees, which starts at the 46 th tree and ends at the 70 th tree.

Thus, the 48th tree must fall within the 18 to 24 inches range.
So, the possible value for the median trunk diameter is: C. 20

[Calc] Question medium

Which of the following dot plots represents the data set with the greatest standard deviation?

▶️Answer/Explanation

Ans: D

The standard deviation is a measure of how spread out the values in a data set are. To find which dot plot represents the data set with the greatest standard deviation, we should look for the plot where the values are most spread out from the mean.

Let’s analyze each option:

Option A: The data points are clustered around the values 4, 5, 6, 7, and 8. This indicates a smaller spread around the mean.
Option B: The data points are spread out across the values 3 through 5. This indicates a moderate spread.
Option C: The data points are spread out almost evenly across the entire range from 2 to 5. This indicates a medium spread around the mean.
Option D: The data points are spread out from 1 and 3 to 5 , with some points at 8. This indicates a Wide spread.

Since option D has data points that are spread out almost evenly across the entire range from 0 to 9, it represents the data set with the greatest standard deviation.

[Calc] Question Medium

The table shows the yearly snowfall, in centimeters (cm), in Toronto for 9 years. Year Snowfall (cm)

What was the median yearly snowfall, in cm, in Toronto for these years?

A. 32.4
B. 114.9
C. 115.5
D. 184.1

▶️Answer/Explanation

Ans: B

To find the median yearly snowfall, we first need to arrange the snowfall values in ascending order:

\[ 32.4, 45.6, 89.0, 114.1, 114.9, 129.6, 134.9, 162.6, 216.5 \]

Since there are 9 data points, the median will be the middle value. If the number of data points is odd, the median is the middle value. If the number of data points is even, the median is the average of the two middle values.

In this case, since the number of data points is odd (9), the median will be the fifth value in the sorted list, which is $114.9$.

So, the median yearly snowfall in Toronto for these years is:

\[ \boxed{\text{B) } 114.9 \, \mathrm{cm}} \]

[Calc] Question medium

The box plots shown summarize the number of tanker and dry-bulk merchant fleets for 18 countries. Which of the following measures must be greater for the number of tanker merchant fleets than for the number of dry-bulk merchant fleets?

The median
The range

A) I only

B) II only

C) I and II

D) Neither I nor II

▶️Answer/Explanation

C) I and II

For the median (I): The horizontal line within the tanker boxplot is clearly higher than the horizontal line within the dry bulk boxplot, indicating the median number of tanker fleets is greater.

For the range (II): The range is the length of the boxplot whiskers, representing the difference between the minimum and maximum values. While the maximum whisker values are not shown, the minimum whisker for tankers appears lower than the minimum for dry bulk. And the overall tanker boxplot looks more spread out horizontally compared to dry bulk.

So based on the graphical evidence, both the median (I) and the range (II) seem greater for the tanker merchant fleets compared to dry bulk.

[Calc] Question medium

Aditi and Bella each attempted the long jump five times during a track meet, and their distances are shown in the table. The mean distance for Bella’s attempts was 0.3 meter greater than the mean distance for Aditi’s attempts. What is the value of x?

▶️Answer/Explanation

4.5

To find the mean distance for Aditi’s attempts, we sum up all her distances and divide by the number of attempts:
\[ \text{Mean distance for Aditi} = \frac{4.2 + 3.8 + 3.2 + 4.0 + 4.3}{5} = \frac{19.5}{5} = 3.9 \text{ meters} \]

For Bella, we already know the mean distance is $0.3$ meters greater than Aditi’s mean distance:
\[ \text{Mean distance for Bella} = 3.9 + 0.3 = 4.2 \text{ meters} \]
Therefore, $\frac{x + 4.4 + 3.7 + 3.8 + 4.6}{5} = 4.2$.

Therefore, $x= 21-16.5\Rightarrow 4.5$.

[Calc] Question Medium

The bar graph shows the number of commercials Albert saw each day that he watched television last week. For these five days, how much greater is the mean number of commercials per day than the median number of commercials per day?
A) 2
B) 3
C) 5
D) 6

▶️Answer/Explanation

[Calc] Question Medium

The graph above shows the distribution of the number of years of experience for 25 teachers enrolled in an advanced-degree program at a particular university. If a 26th teacher with 2 years of experience is added to the program and to the data set, what will be the effect on the mean and median of the data set?
A) The mean and median will both decrease.
B) The mean and median will both remain the same.
C) The mean will decrease and the median will remain the same.
D) The mean will remain the same and the median will decrease.

▶️Answer/Explanation

Question

In a tournament for 64 teams, each game is played between two teams. Each team plays one game in the first round. For all rounds, the winning team of each game advances to play a game in the next round, and the losing team is eliminated from the tournament. How many teams remain to play in the fourth round of the tournament? 2.14

▶️Answer/Explanation

Question

Data set R and data set S are represented in the dot plots shown. The mean for each data set is 50.0. Which of the following best describes the relationship between the standard deviation of data set R and the standard deviation of data set S?

Both data sets have the same standard deviation.
The standard deviation of data set R is greater than the standard deviation of data set S.
The standard deviation of data set R is less than the standard deviation of data set S.
The sum of the standard deviations of the two data sets is 100.

▶️Answer/Explanation

Question

A fish hatchery has three tanks for holding fish before they are introduced into the wild. Ten fish weighing less than 5 ounces are placed in tank A. Eleven fish weighing at least 5 ounces but no more than 13 ounces are placed in tank B. Twelve fish weighing more than 13 ounces are placed in tank $\mathrm{C}$. Which of the following could be the median of the weights, in ounces, of these 33 fish?
A. 4.5
B. 8
C. 13.5
D. 15

▶️Answer/Explanation

Ans: B

Question

A data set of 27 different numbers has a mean of 33 and a median of 33 . A new data set is created by adding 7 to each number in the original data set that is greater than the median and subtracting 7 from each number in the original data set that is less than the median. Which of the following measures does NOT have the same value in both the original and the new data sets?
A. Median
B. Mean
C. Sum of the numbers
D. Standard deviation

▶️Answer/Explanation

Ans: D

Questions

The results of an international survey of contact lens fittings during a given time period are summarized in the table and bar graph above. The table shows the number of total fittings and the mean age, in years, of the patients who were fitted for contact lenses during the time period. The total fittings consisted of new contact lens fittings and refittings. The bar graph shows the percent of the patients who received new fittings and the percent who received refittings.

What is the range, in years, of the mean ages of the patients surveyed who had contact lens fittings in the countries shown?

8.0
8.4
9.7
10.3

▶️Answer/Explanation

Ans: C

Questions

Of the following, which best approximates the number of patients surveyed who received refittings in New Zealand?

274
358
447
585
▶️Answer/Explanation
Ans: C

Question

Of 100 people who played a certain video game, 85 scored more than 0 but less than 10,000 points, 14 scored between 10,000 and 100,000 points, and the remaining player scored 5,350,000 points. Which of the following statements about the mean and median of the 100 scores is true?
A. The mean is greater than the median.
B. The median is greater than the mean.
C. The mean and the median are equal.
D. There is not enough information to determine whether the mean or the median is greater.

▶️Answer/Explanation

Ans: A

Question

In spring 2015, three separate studies on the fitness level of tenth graders were conducted in the city of Mistwick. In each study, every student in a group of tenth graders took the same fitness test and received a score on it. The possible scores on the fitness test are the whole numbers from 50 to 100, inclusive. The distribution of the scores for each of the studies is shown in the table below.

The participants for the studies were selected as follows.
• For Study I, 100 tenth graders were selected at random from all tenth graders in Mistwick.
• For Study II, 200 tenth graders were selected at random from all tenth graders in Mistwick.
• For Study III, 300 tenth graders from Mistwick volunteered to participate.
No tenth grader participated in more than one of the three studies.

What percent of all the scores reported in the three studies were in the 50-59 range?

24%
25%
26%
27%
▶️Answer/Explanation
Ans: D

Questions

A group of 10 students played a certain game. Every player received a score equal to an integer from 1 to 10 , inclusive. For the 10 players, the mean score was 4. If more than half of the players received a score greater than 5 , which of the following is true about the mean score of the remaining players?
A. It must be less than 4 .
B. It must be equal to 4 .
C. It must be between 4 and 5 .
D. It must be greater than 5 .

▶️Answer/Explanation

Ans: A

Questions

For a particular office building with 1,420 employees, Tia and Amir each conducted a survey about the average one-way commute times, in minutes, between the employees’ home and office. Both Tia and Amir selected employees at random, mailed out surveys, and collected
data from the returned surveys. For both surveys, respondents were asked to report their average commute times to the nearest 5 minutes. Tia collected data from 150 employees, and Amir collected data from 85 employees. The results from Tia’s and Amir’s returned surveys are summarized below.

If $T$ is the median commute time of the employees who responded to Tia’s survey and $A$ is the median commute time of the employees who responded to Amir’s survey, what is the value of $T – A$ ?

10
8
5
0
▶️Answer/Explanation
Ans: C

Questions

Which of the following box plots could represent Amir’s survey data?

▶️Answer/Explanation

Ans: B

Questions

Which of the following is true about the standard deviations of the two data sets in the table above?

The standard deviation of data set B is larger than the standard deviation of data set A.
The standard deviation of data set A is larger than the standard deviation of data set B.
The standard deviation of data set A is equal to the standard deviation of data set B.
There is not enough information available to compare the standard deviations of the two data sets.
▶️Answer/Explanation
Ans: A