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Descriptive Statistics Quiz 43 (10 MCQs)

This set of multiple-choice questions evaluates understanding of descriptive statistics, including mean, median, mode, and data summarization techniques. Concepts covered include measures of central tendency, frequency distribution, and correlation calculation. Students will demonstrate their ability to calculate and interpret statistical measures, summarize data, and understand linear relationships.

Quiz Instructions

Select an option to see the correct answer instantly.

1. The table shown is comparing a person's foot length to their height in cm. Calculate the correlation (r) and what it means.

Correct Answer: Correct answer is: (C) R = 0.986; Strong Positive.

Exam Relevance: AP Statistics, IB Mathematics, GRE Quantitative Reasoning

Difficulty: Moderate

Concept notes: The correlation coefficient (r) measures the strength and direction of the linear relationship between two variables. A value of r close to 1 indicates a strong positive correlation, meaning that as one variable increases, the other variable also tends to increase.

Common Mistakes: A common mistake is to confuse the magnitude of the correlation coefficient with its interpretation. A value of r = 0.986 is very close to 1, indicating a strong positive relationship, not a weak one.

Explanations: The correlation coefficient (r) is calculated using the data from the table. The value of r = 0.986 indicates a very strong positive correlation between foot length and height. This means that as foot length increases, height also tends to increase significantly. The value of r being close to 1 confirms a strong positive relationship.

Option Analysis:

Option A: Incorrect because R = 5.5 is not a valid correlation coefficient value. Correlation coefficients range from -1 to 1.
Option B: Incorrect because R = 0.986 is a strong positive correlation, not a weak one.
Option C: Correct because R = 0.986 indicates a strong positive correlation.
Option D: Incorrect because R = 25.05 is not a valid correlation coefficient value. Correlation coefficients range from -1 to 1.

2. A scientist records the growth of a plant over five days as 5 cm, 7 cm, 6 cm, 8 cm, and 4 cm. What is the average daily growth?

Correct Answer: Correct answer is: (A) 6 cm.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

Concept notes: The average daily growth is calculated by summing the daily growth values and dividing by the number of days.

Common Mistakes: A common mistake is to miscount the number of days or to incorrectly sum the values.

Explanations: To find the average daily growth, sum the growth values: 5 cm + 7 cm + 6 cm + 8 cm + 4 cm = 30 cm. Then, divide the total by the number of days, which is 5. Therefore, the average daily growth is 30 cm / 5 = 6 cm.

Option Analysis:

Option A: Correct. The average daily growth is 6 cm.
Option B: Incorrect. 5 cm is not the correct average.
Option C: Incorrect. 7 cm is not the correct average.
Option D: Incorrect. 6.5 cm is not the correct average.

3. The temperature in Michigan City on this date in the last ten years were: 68, 68, 69, 70, 72, 70, 68, 75, 55, 67. What is the mean temperature on this date over the last ten years? (Round to nearest whole number.)

Correct Answer: Correct answer is: (C) 68.

Exam Relevance: AP Statistics, SAT Math, GRE Quantitative

Difficulty: Moderate

Concept notes: The mean temperature is calculated by summing all the temperatures and dividing by the number of observations.

Common Mistakes: A common mistake is to confuse the mean with the median or mode, or to miscalculate the sum or division.

Explanations: To find the mean temperature, we first sum all the temperatures: 68 + 68 + 69 + 70 + 72 + 70 + 68 + 75 + 55 + 67 = 682. Next, we divide the sum by the number of observations, which is 10: 682 / 10 = 68.2. Rounding 68.2 to the nearest whole number gives us 68.

Option Analysis:

Option A: 60 is incorrect because it is significantly lower than the calculated mean.
Option B: 61 is incorrect because it is lower than the calculated mean.
Option C: 68 is correct as it is the rounded mean of the given temperatures.
Option D: 69 is incorrect because it is higher than the calculated mean.

4. Consists of the collection, organization, summarization, and presentation of data

Correct Answer: Correct answer is: (B) Descriptive.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Easy

Concept notes: Descriptive statistics involves the collection, organization, summarization, and presentation of data.

Common Mistakes: A common misunderstanding is confusing descriptive statistics with inferential statistics, which involves making inferences or predictions about a larger population based on a sample.

Explanations: Descriptive statistics focuses on summarizing and presenting data in a meaningful way. This includes organizing data into tables, charts, and graphs, as well as calculating measures such as mean, median, and mode. The goal is to provide a clear and concise summary of the data without making any inferences or predictions about a larger population.

Option Analysis:

Option A: Inferential statistics involves making inferences or predictions about a population based on sample data, which is not the focus of the question.
Option B: Descriptive statistics involves the collection, organization, summarization, and presentation of data, which matches the description in the question.
Option C: Data refers to the raw information collected, but does not encompass the processes of organizing, summarizing, and presenting the data.
Option D: A variable is a characteristic or attribute that can take on different values, but does not describe the processes involved in descriptive statistics.

5. Write down your answer to this question, you will need it for the following problem!The mean of a set of numbers is 123. The sum of the numbers is 2,214. Which of the equations below could help you determine how many numbers are in the set?

Correct Answer: Correct answer is: (B) $\frac{2,214}{x}=123$.

Exam Relevance: AP Statistics, SAT, GRE, GMAT

Difficulty: Moderate

Concept notes: The mean of a set of numbers is calculated by dividing the sum of the numbers by the count of the numbers. The equation that represents this relationship is $\text{mean} = \frac{\text{sum of the numbers}}{\text{count of the numbers}}$.

Common Mistakes: A common mistake is to confuse the roles of the mean, sum, and count in the equation. For example, some might incorrectly set up the equation with the mean in the numerator or the sum in the denominator.

Explanations: The mean of a set of numbers is given by the formula $\text{mean} = \frac{\text{sum of the numbers}}{\text{count of the numbers}}$. Here, the mean is 123 and the sum of the numbers is 2,214. We need to find the count of the numbers, denoted as $x$. Therefore, the equation that correctly represents this relationship is $\frac{2,214}{x} = 123$.

Option Analysis:

Option A: This equation $\frac{x}{123}=2,214$ incorrectly places the count of the numbers in the numerator and the mean in the denominator, which does not represent the relationship between the mean, sum, and count.
Option B: This equation $\frac{2,214}{x}=123$ correctly represents the relationship between the sum of the numbers, the count of the numbers, and the mean.
Option C: This equation $\frac{123}{x}=2,214$ incorrectly places the mean in the numerator and the count of the numbers in the denominator, which does not represent the relationship between the mean, sum, and count.
Option D: This equation $\frac{x}{2,214}=123$ incorrectly places the count of the numbers in the numerator and the sum of the numbers in the denominator, which does not represent the relationship between the mean, sum, and count.

6. In a frequency distribution, the class may be:

Correct Answer: Correct answer is: (D) Inclusive or exclusive.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Moderate

Concept notes: In descriptive statistics, class intervals in a frequency distribution can be either inclusive or exclusive.

Common Mistakes: A common misunderstanding is that class intervals can only be singular or plural, which is incorrect. Class intervals are defined by their range and can be inclusive or exclusive.

Explanations: In descriptive statistics, class intervals in a frequency distribution can be defined as inclusive or exclusive. An inclusive class interval includes both the lower and upper limits, while an exclusive class interval includes the lower limit but excludes the upper limit. This distinction is important for accurately representing and analyzing data.

Option Analysis:

Option A: Class intervals are not described as singular or plural. This option is incorrect.
Option B: Class intervals are not described as subjective or objective. This option is incorrect.
Option C: Class intervals are not described as individual or discrete. This option is incorrect.
Option D: Class intervals can be inclusive or exclusive, which is the correct description. This option is correct.

Mnemonic: Inclusive: "Include both ends, like a hug." Exclusive: "Exclude the upper end, like a handshake."

7. Descriptive statistics is used to

Correct Answer: Correct answer is: (A) Describe the Characteristics of sample.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Easy

Concept notes: Descriptive statistics involves summarizing and organizing data to describe the basic features of the data in a study. It provides simple summaries about the sample and the measures.

Common Mistakes: A common misunderstanding is that descriptive statistics can be used to make predictions or generalize results, which is not accurate. Descriptive statistics only describe the data at hand.

Explanations: Descriptive statistics is used to describe the characteristics of a sample. It involves measures such as mean, median, mode, range, and standard deviation, which provide a summary of the data's central tendency and variability. This approach does not involve making predictions or generalizations beyond the sample data.

Option Analysis:

Option A: Correct. Descriptive statistics is used to describe the characteristics of a sample.
Option B: Incorrect. Predicting the association between variables is more related to inferential statistics.
Option C: Incorrect. Generalizing the results is a function of inferential statistics, not descriptive statistics.
Option D: Incorrect. Comparing groups is typically done using inferential statistics, not descriptive statistics.

8. If the average of 8 numbers is 14, how much would the average of 5 more numbers need to be to raise the overall average to 19?

Correct Answer: Correct answer is: (D) 27.

Exam Relevance: GRE, GMAT, SAT, ACT

Difficulty: Moderate

Concept notes: The question involves calculating the average of a combined set of numbers based on the averages of two separate sets.

Common Mistakes: A common mistake is to assume that the average of the new set of numbers should be the same as the desired overall average, which is not correct.

Explanations: 1. Calculate the total sum of the initial 8 numbers: \[ \text{Sum of 8 numbers} = 8 \times 14 = 112 \] 2. Let the average of the 5 new numbers be $ x $. The sum of these 5 numbers is: \[ \text{Sum of 5 new numbers} = 5x \] 3. The overall average of the 13 numbers (8 original + 5 new) should be 19. Therefore, the total sum of the 13 numbers is: \[ \text{Sum of 13 numbers} = 13 \times 19 = 247 \] 4. Set up the equation for the total sum: \[ 112 + 5x = 247 \] 5. Solve for $ x $: \[ 5x = 247 - 112 \] \[ 5x = 135 \] \[ x = \frac{135}{5} = 27 \] Thus, the average of the 5 new numbers must be 27 to raise the overall average to 19.

Option Analysis:

Option A: 13 is incorrect because it does not satisfy the equation derived from the total sum requirement.
Option B: 19 is incorrect because it is the desired overall average, not the average of the new numbers.
Option C: 24 is incorrect because it does not satisfy the equation derived from the total sum requirement.
Option D: 27 is correct as it satisfies the equation derived from the total sum requirement.

9. Find the median from the following data ..... 5, 3, 6, 5, 3, 9, 4

Correct Answer: Correct answer is: (B) 5.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

Concept notes: The median is the middle value in a sorted list of numbers. If the list has an odd number of observations, the median is the middle number. If the list has an even number of observations, the median is the average of the two middle numbers.

Common Mistakes: A common mistake is to confuse the median with the mean (average) or mode (most frequent value). The median specifically refers to the middle value in a sorted list.

Explanations: To find the median, first sort the data: 3, 3, 4, 5, 5, 6, 9. Since there are 7 numbers (an odd number), the median is the middle number, which is the 4th number in the sorted list. The 4th number is 5.

Option Analysis:

Option A: 8 is incorrect because it is not the middle value in the sorted list.
Option B: 5 is correct because it is the middle value in the sorted list.
Option C: 6 is incorrect because it is not the middle value in the sorted list.
Option D: 7 is incorrect because it is not the middle value in the sorted list.

10. Also referred as statistical averages tell us the point about which items have a tendency to cluster. The 3 most common statistical averages are (arithmetic) mean, median and mode.

Correct Answer: Correct answer is: (C) Measures of central tendency.

Exam Relevance: AP Statistics, GRE, GMAT, SAT, ACT

Difficulty: Moderate

Concept notes: Measures of central tendency are statistical measures that identify a single value as representative of an entire distribution. They aim to provide a description of the center of a data set.

Common Mistakes: A common mistake is confusing measures of central tendency with measures of dispersion, which describe the spread of the data rather than the central point.

Explanations: Measures of central tendency, such as the mean, median, and mode, are used to describe the central point or typical value of a dataset. These measures tell us the point around which the data tends to cluster. The arithmetic mean is the sum of all values divided by the number of values, the median is the middle value when the data is ordered, and the mode is the most frequently occurring value. These measures are essential for understanding the central location of a dataset.

Option Analysis:

Option A: Measures of dispersion describe the spread or variability of the data, not the central point.
Option B: Measures of asymmetry (skewness) describe the degree of asymmetry in the distribution of data, not the central point.
Option C: Measures of central tendency, such as mean, median, and mode, describe the central point around which data tends to cluster.
Option D: Measures of relationship describe the association between two or more variables, not the central point of a single dataset.

Frequently Asked Questions

What is the purpose of descriptive statistics?

Descriptive statistics summarize and describe the main features of a dataset, providing a clear overview of the data's characteristics.

How do you calculate the mean of a dataset?

The mean is calculated by summing all the values in the dataset and then dividing by the number of values.

What is the difference between inclusive and exclusive class intervals?

Inclusive class intervals include the upper limit in the class, while exclusive class intervals do not include the upper limit, making the next class start right after the previous one ends.

What does a strong positive correlation coefficient indicate?

A strong positive correlation coefficient indicates a high degree of linear relationship between two variables, where an increase in one variable is associated with an increase in the other.

How is frequency distribution used in descriptive statistics?

Frequency distribution organizes data into intervals or categories, showing how often each value or range of values occurs, which helps in understanding the data's distribution pattern.

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