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Descriptive Statistics Quiz 49 (10 MCQs)

This set of multiple-choice questions evaluates understanding of descriptive statistics, including measures of central tendency (mean, median, mode) and dispersion (absolute deviations, mean deviation). It covers concepts such as arithmetic mean, median calculation, data sorting, and relative frequency. The questions assess the ability to calculate and interpret summary measures and construct pie charts.

Quiz Instructions

Select an option to see the correct answer instantly.

1. It is the sum of the data divided by the number of data values.

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

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Easy

Concept notes: The mean is the sum of all data values divided by the number of data values.

Common Mistakes: A common mistake is confusing the mean with the median or mode. The median is the middle value when the data is ordered, and the mode is the most frequently occurring value.

Explanations: The mean is calculated by summing all the data values and then dividing by the number of data values. This matches the description given in the question.

Option Analysis:

Option A: Correct. The mean is the sum of the data divided by the number of data values.
Option B: Incorrect. The median is the middle value when the data is ordered.
Option C: Incorrect. The mode is the most frequently occurring value.
Option D: Incorrect. The correct answer is provided in the options.

2. Roger has the following data:12, 10, 3, 5, 6Susan has the following data:5, 9, 4, 5, 6Who has the data with the highest median and what is it?

Correct Answer: Correct answer is: (A) Roger-6.

Exam Relevance: AP Statistics, SAT Math, GRE Quantitative

Difficulty: Easy

Concept notes: The median is the middle value in a sorted list of numbers. To find the median, first sort the data in ascending order and then identify the middle value.

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 determine the median for Roger's data: 1. Sort Roger's data: 3, 5, 6, 10, 12 2. The middle value is 6. To determine the median for Susan's data: 1. Sort Susan's data: 4, 5, 5, 6, 9 2. The middle value is 5. Since 6 is greater than 5, Roger's data has the highest median, which is 6.

Option Analysis:

Option A: Correct. Roger's median is 6, which is higher than Susan's median of 5.
Option B: Incorrect. Susan's median is 5, not 7.
Option C: Incorrect. Roger's median is 6, not 8.
Option D: Incorrect. Susan's median is 5, not 5.5.

3. The relative frequencies (expressed as percentage) of the eight classes as given below: 12.6, 18.2, 17.5, 20.3, 2.8, 4.2, 9.8, 14.7. Find the central angle for IV class.

Correct Answer: Correct answer is: (B) $73^o$.

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

Difficulty: Moderate

Concept notes: In descriptive statistics, relative frequencies can be used to construct a pie chart where each class's central angle is proportional to its relative frequency. The central angle for a class is calculated by multiplying the relative frequency (as a percentage) by 360 degrees.

Common Mistakes: A common mistake is to forget to convert the relative frequency from a percentage to a decimal before multiplying by 360 degrees.

Explanations: To find the central angle for the IV class, we use the relative frequency of 20.3%. The formula to convert this percentage to a central angle is: \[ \text{Central Angle} = \left( \frac{\text{Relative Frequency}}{100} \right) \times 360^\circ \] Substituting the given relative frequency: \[ \text{Central Angle} = \left( \frac{20.3}{100} \right) \times 360^\circ = 0.203 \times 360^\circ = 73.08^\circ \] Rounding to the nearest degree, the central angle is $73^\circ$.

Option Analysis:

Option A: $58^\circ$ is incorrect because it does not match the calculated central angle.
Option B: $73^\circ$ is correct as it matches the calculated central angle.
Option C: $62^\circ$ is incorrect because it does not match the calculated central angle.
Option D: $105^\circ$ is incorrect because it does not match the calculated central angle.

4. Given the following data:1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Correct Answer: Correct answer is: (C) The average of the data is 5.5.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Moderate

Concept notes: The average (mean) of a set of numbers is calculated by summing all the numbers and then dividing by the count of numbers.

Common Mistakes: A common mistake is to assume the average is the middle number in a sequence, which is incorrect unless the sequence is symmetrically distributed around the mean.

Explanations: To find the average of the data set 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, we first sum all the numbers: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55. Next, we divide the sum by the number of data points, which is 10: 55 / 10 = 5.5. Thus, the average of the data is 5.5.

Option Analysis:

Option A: The average is not 5 because the sum of the data divided by 10 is 5.5, not 5.
Option B: The average is not 6 because the sum of the data divided by 10 is 5.5, not 6.
Option C: The average is 5.5 because the sum of the data divided by 10 is 5.5.
Option D: The average is not 55 because 55 is the sum of the data, not the average.

Mnemonic: Sum and divide, not just the middle.

5. A numerical value used as a summary measure for a sample, such as sample mean, is known as a

Correct Answer: Correct answer is: (C) Sample statistic.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Easy

Concept notes: A sample statistic is a numerical value that summarizes a characteristic of a sample. It is used to estimate or describe a feature of the sample data.

Common Mistakes: A common misunderstanding is confusing a sample statistic with a population parameter. A sample statistic is calculated from a subset of the population, while a population parameter is a numerical value that describes the entire population.

Explanations: A sample statistic is a summary measure calculated from a sample of data. For example, the sample mean is a sample statistic that provides a summary of the central tendency of the sample data. It is distinct from a population parameter, which describes the entire population.

Option Analysis:

Option A: A population parameter is a numerical value that describes a characteristic of the entire population, not a sample.
Option B: The term "sample parameter" is not commonly used; the correct term is "sample statistic."
Option C: A sample statistic is a numerical value that summarizes a characteristic of a sample, such as the sample mean.
Option D: The population mean is a parameter that describes the entire population, not a sample.

6. What term describes the middle of a sorted list of numbers?

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

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Easy

Concept notes: The median is the middle value in a sorted list of numbers. It is a measure of central tendency that divides the data into two equal halves.

Common Mistakes: A common mistake is confusing the median with the mean (average) or mode (most frequent value).

Explanations: The median is the value that separates the higher half from the lower half of a data sample. When the list of numbers is sorted, the median is the middle number. If there is an even number of observations, the median is the average of the two middle numbers.

Option Analysis:

Option A: The mean is the average of the numbers, not the middle value.
Option B: The mode is the most frequently occurring number, not the middle value.
Option C: The median is the middle value in a sorted list of numbers.
Option D: The range is the difference between the highest and lowest values, not the middle value.

7. Since the data is symmetrical, which measure of center is the best use?

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

Exam Relevance: AP Statistics, GRE, GMAT

Difficulty: Easy

Concept notes: In symmetrical data distributions, the mean is the best measure of center because it takes into account all values in the dataset and is not affected by the symmetry.

Common Mistakes: A common misunderstanding is that the median or mode might be better measures of center in symmetrical data, but the mean is actually the most appropriate because it uses all data points.

Explanations: In a symmetrical distribution, the mean, median, and mode are all equal. However, the mean is the most appropriate measure of center because it uses all the data points in its calculation, providing a more comprehensive representation of the dataset. The mean is the sum of all values divided by the number of values, which makes it the best choice for symmetrical data.

Option Analysis:

Option A: Mean is the correct choice because it uses all data points and is not affected by symmetry.
Option B: Mode is the value that appears most frequently, but it does not necessarily represent the center of symmetrical data.
Option C: Median is the middle value, but it does not use all data points and is not the best measure for symmetrical data.
Option D: Range is a measure of spread, not a measure of center.

8. It is a a numerical measure that describes a characteristic of a sample.

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

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Easy

Concept notes: A statistic is a numerical measure that describes a characteristic of a sample. A parameter, on the other hand, describes a characteristic of a population.

Common Mistakes: A common misunderstanding is confusing a statistic with a parameter. A statistic is based on a sample, while a parameter is based on the entire population.

Explanations: A statistic is a numerical measure that describes a characteristic of a sample. This is the correct definition and aligns with the given statement. A parameter, which describes a characteristic of a population, is not the correct answer because the question specifically refers to a sample.

Option Analysis:

Option A: Correct. A statistic is a numerical measure that describes a characteristic of a sample.
Option B: Incorrect. A parameter describes a characteristic of a population, not a sample.
Option C: Incorrect. A sample is a subset of a population, not a numerical measure.
Option D: Incorrect. A population is the entire group of interest, not a numerical measure.

9. The Arithmetic mean of numerical values of the deviations of items from some average value is called

Correct Answer: Correct answer is: (C) Mean deviation.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Moderate

Concept notes: The mean deviation is a measure of dispersion that calculates the average of the absolute deviations from a central value, such as the mean or median.

Common Mistakes: A common misunderstanding is confusing mean deviation with standard deviation, which is the square root of the variance and measures the average squared deviation from the mean.

Explanations: The mean deviation is defined as the arithmetic mean of the absolute deviations of the items from some average value, typically the mean or median. This makes it a direct measure of the average distance of each data point from the central value. The other options, such as standard deviation and range, do not fit this definition.

Option Analysis:

Option A: Standard deviation is the square root of the variance and measures the average squared deviation from the mean, not the arithmetic mean of the absolute deviations.
Option B: Range is the difference between the maximum and minimum values in a dataset, not the arithmetic mean of the deviations.
Option C: Mean deviation is the arithmetic mean of the absolute deviations from a central value, which matches the given definition.
Option D: This option is incorrect as the correct answer is mean deviation.

10. Find the mean:2, 12, 1, 5, 4

Correct Answer: Correct answer is: (B) 4.8.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Easy

Concept notes: The mean is calculated by summing all the values and 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 of the numbers 2, 12, 1, 5, 4, follow these steps: 1. Sum the values: 2 + 12 + 1 + 5 + 4 = 24. 2. Count the number of values: There are 5 values. 3. Divide the sum by the number of values: 24 / 5 = 4.8.

Option Analysis:

Option A: 24 is the sum of the values, not the mean.
Option B: 4.8 is the correct mean.
Option C: 11 is not the mean of the given values.
Option D: 13 is not the mean of the given values.

Frequently Asked Questions

What is descriptive statistics?

Descriptive statistics involves summarizing and organizing data to make it more understandable. It includes measures like the mean, median, and mode.

How is the mean calculated in descriptive statistics?

The mean is calculated by summing all the data values and then dividing by the number of data values. This provides an average value for the dataset.

What is the median in descriptive statistics?

The median is the middle value in a sorted list of data. If there is an even number of data points, the median is the average of the two middle values.

What is the purpose of a pie chart in descriptive statistics?

A pie chart is used to show the relative frequency or proportion of different categories within a dataset. It helps visualize how the total data is divided among categories.

What is the difference between a sample statistic and a population parameter?

A sample statistic is a measure calculated from a subset of the population, while a population parameter is a measure calculated from the entire population. Sample statistics are used to estimate population parameters.

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