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Descriptive Statistics Quiz 28 (10 MCQs)

This set of multiple-choice questions evaluates understanding of data classification, measurement scales, and mean calculation. Concepts covered include arithmetic mean, descriptive statistics, frequency distribution, and correlation coefficient. The material tests the ability to identify ordinal data, compute means, and interpret statistical measures and relationships between variables.

Quiz Instructions

Select an option to see the correct answer instantly.

1. A researcher is analyzing the average weight of a sample of mice.The weights in grams are: 20, 22, 24, 26, and 28.What is the average weight of the mice?

Correct Answer: Correct answer is: (B) 24 g.

Exam Relevance: AP Statistics, GRE, GMAT

Difficulty: Easy

Concept notes: The average weight of a sample is calculated by summing all the weights and dividing by the number of observations.

Common Mistakes: A common mistake is to assume the average is the middle value without performing the calculation.

Explanations: To find the average weight, sum the weights: 20 + 22 + 24 + 26 + 28 = 120. Then, divide by the number of mice, which is 5. So, the average weight is 120 / 5 = 24 g.

Option Analysis:

Option A: 22 g is incorrect because it is not the result of the sum of the weights divided by the number of mice.
Option B: 24 g is correct because it is the result of the sum of the weights divided by the number of mice.
Option C: 25 g is incorrect because it is not the result of the sum of the weights divided by the number of mice.
Option D: 26 g is incorrect because it is not the result of the sum of the weights divided by the number of mice.

2. Calculating by adding all of the scores together and dividing by the number of scores.

Correct Answer: Correct answer is: (A) Mean.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Easy

Concept notes: The mean is the average of a set of numbers, calculated by summing all the values and then dividing by the number of values.

Common Mistakes: A common mistake is confusing the mean with other measures of central tendency, such as the median or mode.

Explanations: The mean is calculated by adding all the scores together and dividing by the number of scores. This matches the description given in the question.

Option Analysis:

Option A: Correct. The mean is calculated by summing all the scores and dividing by the number of scores.
Option B: Incorrect. The median is the middle value when the scores are arranged in order.
Option C: Incorrect. The mode is the most frequently occurring score.
Option D: Incorrect. The range is the difference between the highest and lowest scores.
Option E: Incorrect. The standard deviation measures the spread of the scores around the mean.

3. The mean of 10 test scores was calculated as 70, but it turns out that one of the scores was misread as 45 instead of 55. What is the correct average of the test scores?

Correct Answer: Correct answer is: (C) 71.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Moderate

Concept notes: The mean of a set of numbers is the sum of the numbers divided by the count of the numbers. When an error is found in one of the numbers, the correct mean can be recalculated by adjusting the sum and then dividing by the count.

Common Mistakes: A common mistake is to simply add the difference between the incorrect and correct score to the original mean, which does not account for the total sum of the scores.

Explanations: 1. The original mean of the 10 test scores is 70. 2. The sum of the original scores is \( 70 \times 10 = 700 \). 3. One score was misread as 45 instead of 55, so the correct sum should be \( 700 - 45 + 55 = 710 \). 4. The correct mean is \( \frac{710}{10} = 71 \).

Option Analysis:

Option A: 65 is incorrect because it does not account for the adjustment in the sum of the scores.
Option B: 80 is incorrect because it overestimates the adjustment needed.
Option C: 71 is correct because it correctly recalculates the mean after adjusting the sum of the scores.
Option D: 75 is incorrect because it overestimates the adjustment needed.

4. On a scatterplot there is no relationship between the data sets

Correct Answer: Correct answer is: (C) No Correlation.

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

Difficulty: Moderate

Concept notes: In descriptive statistics, correlation measures the strength and direction of a relationship between two variables. No correlation indicates that there is no relationship between the variables.

Common Mistakes: A common misunderstanding is to confuse no correlation with a negative or positive correlation, or to think that no correlation means the data points are randomly scattered without any discernible pattern.

Explanations: No correlation means that the data points on a scatterplot do not show any clear pattern or trend. The variables are not related, and changes in one variable do not predict changes in the other. This is indicated by a scatterplot where the points are randomly distributed without any discernible pattern.

Option Analysis:

Option A: Positive correlation indicates a relationship where an increase in one variable is associated with an increase in the other. This does not apply to the scenario described.
Option B: Negative correlation indicates a relationship where an increase in one variable is associated with a decrease in the other. This does not apply to the scenario described.
Option C: No correlation indicates that there is no relationship between the variables, which matches the scenario described.
Option D: Increasing relation suggests a positive correlation, which does not apply to the scenario described.

5. The relative frequency of a class is computed by

Correct Answer: Correct answer is: (D) Dividing the frequency of the class by the sample size.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Moderate

Concept notes: The relative frequency of a class is a measure of how often a particular class occurs in relation to the total number of observations. It is calculated by dividing the frequency of the class by the total sample size.

Common Mistakes: A common mistake is to confuse the relative frequency with other measures such as the midpoint or the frequency alone, without considering the total sample size.

Explanations: The relative frequency of a class is calculated by dividing the frequency of the class by the sample size. This provides a proportion that indicates the frequency of the class relative to the total number of observations. This measure is useful for comparing the frequency of different classes within the same dataset.

Option Analysis:

Option A: This option is incorrect because the midpoint of the class is not used in the calculation of relative frequency. The midpoint is a measure of central tendency within the class, not a measure of frequency.
Option B: This option is incorrect because dividing the frequency of the class by the midpoint does not provide a meaningful measure of relative frequency. The midpoint is not related to the total sample size.
Option C: This option is incorrect because dividing the sample size by the frequency of the class would not provide a meaningful measure of relative frequency. The correct approach is to divide the frequency of the class by the sample size.
Option D: This option is correct because it correctly describes the calculation of relative frequency, which is the frequency of the class divided by the sample size.

6. What is the mean of the set of data?4, 8, 3, 10, 4, 6, 0

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

Exam Relevance: AP Statistics, SAT, GRE, GMAT

Difficulty: Easy

Concept notes: The mean of a set of data is calculated by summing all the values and then dividing by the number of values.

Common Mistakes: A common mistake is to confuse the mean with other measures of central tendency, such as the median or mode.

Explanations: To find the mean, we first sum all the values in the set: 4 + 8 + 3 + 10 + 4 + 6 + 0 = 35. Next, we divide the sum by the number of values, which is 7. Therefore, the mean is 35 / 7 = 5.

Option Analysis:

Option A: 4 is incorrect because the sum of the values divided by the number of values does not equal 4.
Option B: 5 is correct because the sum of the values divided by the number of values equals 5.
Option C: 6 is incorrect because the sum of the values divided by the number of values does not equal 6.
Option D: 7 is incorrect because the sum of the values divided by the number of values does not equal 7.

7. The mean marks obtained by a class of 40 students is 65. The mean marks of half of the students is found to be 45. The mean marks of the remaining students is .....

Correct Answer: Correct answer is: (A) 85.

Exam Relevance: SAT, GRE, GMAT, ACT

Difficulty: Moderate

Concept notes: The mean marks of a group can be calculated by dividing the total marks by the number of students. The problem involves finding the mean marks of a subset of students given the overall mean and the mean of another subset.

Common Mistakes: A common mistake is to assume that the mean of the remaining students can be found by simply averaging the given means, which is incorrect. The correct approach involves calculating the total marks for each subset and then finding the mean of the remaining students.

Explanations: 1. Calculate the total marks for the entire class: \[ \text{Total marks for the class} = \text{Mean marks} \times \text{Number of students} = 65 \times 40 = 2600 \] 2. Calculate the total marks for the first half of the students: \[ \text{Total marks for the first half} = \text{Mean marks of the first half} \times \text{Number of students in the first half} = 45 \times 20 = 900 \] 3. Calculate the total marks for the remaining half of the students: \[ \text{Total marks for the remaining half} = \text{Total marks for the class} - \text{Total marks for the first half} = 2600 - 900 = 1700 \] 4. Calculate the mean marks for the remaining half of the students: \[ \text{Mean marks for the remaining half} = \frac{\text{Total marks for the remaining half}}{\text{Number of students in the remaining half}} = \frac{1700}{20} = 85 \]

Option Analysis:

Option A: Correct. The mean marks of the remaining students is 85.
Option B: Incorrect. The mean marks of the remaining students is not 60.
Option C: Incorrect. The mean marks of the remaining students is not 70.
Option D: Incorrect. The mean marks of the remaining students is not 65.

8. An item with choice of answer 'Excellent', 'Good', 'Average', 'As Expected', 'BelowAverage', is an example of ..... level of measurement.

Correct Answer: Correct answer is: (A) Ordinal.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Moderate

Concept notes: The level of measurement for data is categorized into nominal, ordinal, interval, and ratio. Ordinal data involves categories that have a natural, meaningful order but do not have a consistent interval between categories.

Common Mistakes: A common mistake is confusing ordinal data with nominal data, which does not have a natural order.

Explanations: The choices 'Excellent', 'Good', 'Average', 'As Expected', 'Below Average' have a natural order from best to worst. This order is meaningful, but the differences between each category are not necessarily equal. Therefore, this data is classified as ordinal.

Option Analysis:

Option A: Correct. The choices have a natural order, making it ordinal data.
Option B: Incorrect. Nominal data does not have a natural order.
Option C: Incorrect. Ranking is not a standard level of measurement in statistics.
Option D: Incorrect. Qualitative data refers to non-numeric data, but does not specify the level of measurement.

Mnemonic: Orderly Ranks, Not Equal (ORNE)

9. A frequency distribution is a tabular summary of data showing the

Correct Answer: Correct answer is: (D) Number of items in several classes.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Easy

Concept notes: A frequency distribution is a statistical tool that organizes and summarizes data by showing the number of occurrences (frequency) of each value or range of values (classes) within a dataset.

Common Mistakes: A common misunderstanding is that a frequency distribution shows percentages or fractions instead of the actual number of items. However, the primary purpose is to count the number of items in each class.

Explanations: A frequency distribution is a tabular summary that shows the number of items in several classes. It is a way to organize data by counting how many times each value or range of values appears in the dataset. This method provides a clear and concise overview of the data distribution.

Option Analysis:

Option A: This option is incorrect because a frequency distribution does not show the fraction of items in several classes. It shows the actual count.
Option B: This option is incorrect because a frequency distribution does not show the percentage of items in several classes. It shows the actual count.
Option C: This option is incorrect because a frequency distribution does not show the relative percentage of items in several classes. It shows the actual count.
Option D: This option is correct because a frequency distribution shows the number of items in several classes.

10. A measure of the extent to which two factors vary together, and thus of how well either factor predicts the other is called a

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

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

Difficulty: Moderate

Concept notes: Correlation measures the strength and direction of the relationship between two variables.

Common Mistakes: A common misunderstanding is confusing correlation with causation. Just because two variables are correlated does not mean one causes the other.

Explanations: Correlation is a statistical measure that quantifies the degree to which two variables move in relation to each other. It ranges from -1 to 1, where -1 indicates a perfect negative relationship, 0 indicates no relationship, and 1 indicates a perfect positive relationship. This measure is used to understand how well one variable can predict the other, making it the correct answer.

Option Analysis:

Option A: Measure of central tendency refers to measures like mean, median, and mode, which describe the center of a data set, not the relationship between two variables.
Option B: Coefficient is a general term that can refer to various statistical measures, but it is not specific to the relationship between two variables.
Option C: Inferential statistics involves making inferences about a population based on sample data, not the relationship between two variables.
Option D: Correlation measures the extent to which two factors vary together and how well one factor predicts the other, making it the correct answer.

Frequently Asked Questions

What is the arithmetic mean in descriptive statistics?

The arithmetic mean, often referred to as the average, is a statistical measure that represents the sum of all values in a dataset divided by the number of items in the dataset.

How is the correlation coefficient used in descriptive statistics?

The correlation coefficient measures the strength and direction of the relationship between two variables. A value of 0 indicates no correlation, while values closer to 1 or -1 indicate a strong positive or negative relationship, respectively.

What is a frequency distribution?

A frequency distribution is a tabular summary that shows the number of occurrences of each value or range of values in a dataset. It helps in understanding the pattern and spread of data.

What is the significance of the level of measurement in descriptive statistics?

The level of measurement determines the type of statistical analysis that can be performed on a dataset. It includes nominal, ordinal, interval, and ratio levels, each allowing different types of calculations and interpretations.

How do scatterplots help in descriptive statistics?

Scatterplots are graphical representations that show the relationship between two variables. They help in identifying patterns, trends, and outliers in the data, which can be useful for further analysis.

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