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Descriptive Statistics Quiz 13 (10 MCQs)

This set of multiple-choice questions evaluates the ability to interpret bar charts, understand frequency, and analyze measures of central tendency such as mode, mean, and median. It also covers concepts like correlation, frequency distribution, and data dispersion, including range and mean absolute deviation. The material tests the understanding of statistical measures and their application in analyzing categorical data.

Quiz Instructions

Select an option to see the correct answer instantly.

1. Based on the graph provided, what is the median of the data?

Correct Answer: Correct answer is: (B) 14.5.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Moderate

Concept notes: The median is the middle value in a sorted list of numbers. 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 an ordered list.

Explanations: To find the median, we need to identify the middle value of the data set. If the data set has an even number of observations, the median is the average of the two middle values. Based on the graph, the data set has an even number of observations, and the two middle values are 14 and 15. The median is therefore the average of these two values, which is (14 + 15) / 2 = 14.5.

Option Analysis:

Option A: 14 is incorrect because it is one of the middle values, but not the median for an even number of observations.
Option B: 14.5 is correct because it is the average of the two middle values, 14 and 15.
Option C: 15 is incorrect because it is one of the middle values, but not the median for an even number of observations.
Option D: None of the above is incorrect because 14.5 is the correct median.

2. Correlations tell us

Correct Answer: Correct answer is: (C) How things are linked.

Exam Relevance: AP Statistics, GRE Quantitative Reasoning, GMAT Quantitative Section

Difficulty: Moderate

Concept notes: Correlations measure the strength and direction of the relationship between two variables. They do not explain causation or provide a detailed understanding of how things work.

Common Mistakes: A common misunderstanding is that correlations indicate causation, which is not true. Correlations only show how variables are linked, not how they work or why they are linked.

Explanations: Correlations are statistical measures that indicate the degree to which two variables move in relation to each other. They do not provide information on causation, the number of things being measured, or participants' feelings. Instead, they show the extent to which two variables are linked or associated.

Option Analysis:

Option A: Correlations do not explain how things work; they only show the relationship between variables.
Option B: Correlations do not indicate the number of things being measured; they focus on the relationship between two variables.
Option C: Correlations measure the relationship or linkage between two variables, making this the correct answer.
Option D: Correlations do not measure participants' feelings; they measure the relationship between variables.

3. If one bar is taller than another in a bar chart, what does that indicate about the data?

Correct Answer: Correct answer is: (A) The taller bar represents a larger value.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Easy

Concept notes: In descriptive statistics, bar charts are used to represent categorical data. The height of each bar corresponds to the frequency or value of the category it represents.

Common Mistakes: A common misunderstanding is that the color or position of the bars might indicate the value, rather than the height.

Explanations: In a bar chart, the height of each bar is directly proportional to the value it represents. Therefore, a taller bar indicates a larger value for the corresponding category.

Option Analysis:

Option A: Correct. The taller bar represents a larger value.
Option B: Incorrect. The color of the bars does not determine the values.
Option C: Incorrect. The taller bar represents a larger value, not a smaller one.
Option D: Incorrect. Bars in a bar chart do show values for categories.

4. Julius is analyzing survey responses from 100 users. He wants to find the most frequently selected interface design. What statistical measure should he use?

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

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Easy

Concept notes: The mode is the statistical measure that identifies the most frequently occurring value in a dataset. In this context, Julius is looking for the most frequently selected interface design from the survey responses.

Common Mistakes: A common mistake would be to use the median or mean, which are measures of central tendency but do not identify the most frequently occurring value.

Explanations: Julius needs to find the most frequently selected interface design from the survey responses. The mode is the appropriate measure because it identifies the value that appears most often in the dataset. The median and mean are not suitable as they represent the middle value and the average value, respectively, and do not indicate the most frequent selection.

Option Analysis:

Option A: Mode is the correct choice as it identifies the most frequently occurring value in the dataset.
Option B: Median is the middle value in a dataset when arranged in order, which does not indicate the most frequent selection.
Option C: Range is the difference between the highest and lowest values, which does not indicate the most frequent selection.
Option D: Mean is the average value of the dataset, which does not indicate the most frequent selection.

5. Observe the frequency distribution:Estimate the relative frequency of the fourth class.

Correct Answer: Correct answer is: (C) 0.189 or 18.9%.

Exam Relevance: AP Statistics, GRE, GMAT

Difficulty: Moderate

Concept notes: Relative frequency is the proportion of the total number of observations that fall into a specific class. It is calculated by dividing the frequency of the class by the total number of observations.

Common Mistakes: A common mistake is to confuse relative frequency with absolute frequency, which is the count of observations in a class, or to misinterpret the percentage as a raw count.

Explanations: To find the relative frequency of the fourth class, we need to divide the frequency of the fourth class by the total number of observations. If the frequency of the fourth class is 189 and the total number of observations is 1000, the relative frequency is calculated as follows: \[ \text{Relative frequency} = \frac{189}{1000} = 0.189 \] This can also be expressed as a percentage by multiplying by 100: \[ 0.189 \times 100 = 18.9\% \]

Option Analysis:

Option A: 69 is incorrect because it does not represent a relative frequency or a percentage.
Option B: 5.3 is incorrect because it does not represent a relative frequency or a percentage.
Option C: 0.189 or 18.9% is correct as it represents the relative frequency of the fourth class.
Option D: 0.54 or 54% is incorrect because it does not match the calculated relative frequency.

6. James collects the following data on the number of dolphins seen from a cruise ship over the last 5 days. 12, 15, 13, 22, 18What is the mean absolute deviation of the data?

Correct Answer: Correct answer is: (C) 3.2.

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

Difficulty: Moderate

Concept notes: The mean absolute deviation (MAD) is a measure of variability that represents the average distance between each data point and the mean of the dataset.

Common Mistakes: A common mistake is to confuse the mean absolute deviation with the standard deviation or to incorrectly calculate the mean of the dataset.

Explanations: To find the mean absolute deviation, follow these steps: 1. Calculate the mean of the dataset. 2. Find the absolute difference between each data point and the mean. 3. Calculate the average of these absolute differences. Step 1: Calculate the mean. Mean = (12 + 15 + 13 + 22 + 18) / 5 = 80 / 5 = 16. Step 2: Find the absolute differences. |12 - 16| = 4 |15 - 16| = 1 |13 - 16| = 3 |22 - 16| = 6 |18 - 16| = 2 Step 3: Calculate the average of these absolute differences. MAD = (4 + 1 + 3 + 6 + 2) / 5 = 16 / 5 = 3.2. Thus, the mean absolute deviation is 3.2.

Option Analysis:

Option A: Incorrect. This value is the mean of the dataset, not the mean absolute deviation.
Option B: Incorrect. This value is not the correct mean absolute deviation.
Option C: Correct. This value is the correct mean absolute deviation.
Option D: Incorrect. This value is not the correct mean absolute deviation.

7. The maximum length in feet of several whales is listed below. 46, 53, 33, 53, 79 If the maximum length of the Blue Whale, 98 feet, is added to this list, which of the following statements would be true?

Correct Answer: Correct answer is: (B) The mean would increase.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Moderate

Concept notes: The mean is the average of a set of numbers, calculated by summing all the values and dividing by the number of values. Adding a new value that is larger than the current maximum will increase the sum and thus the mean.

Common Mistakes: A common mistake is to assume that adding a new value will not significantly change the mean, especially if the new value is an outlier.

Explanations: The original list of whale lengths is 46, 53, 33, 53, 79. The sum of these lengths is 264. The mean of these lengths is 264 / 5 = 52.8 feet. When the Blue Whale's length of 98 feet is added, the new sum is 264 + 98 = 362. The new mean is 362 / 6 = 60.33 feet. Since 60.33 feet is greater than 52.8 feet, the mean increases.

Option Analysis:

Option A: The mode is the most frequently occurring value. Adding 98 feet does not change the mode, which remains 53. Thus, the mode would not decrease.
Option B: The mean is the sum of all values divided by the number of values. Adding a value larger than the current maximum increases the sum and the number of values, resulting in a higher mean.
Option C: The median is the middle value when the data is ordered. Adding 98 feet to the list does not change the middle value, which remains 53. Thus, the median would not decrease.
Option D: The mean would not decrease because adding a value larger than the current maximum increases the sum and the number of values, resulting in a higher mean.

8. What is the 'range' in the context of data analysis?

Correct Answer: Correct answer is: (A) The difference between the highest and lowest value in a data set.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Easy

Concept notes: The range is a measure of dispersion in descriptive statistics, indicating the spread of the data by calculating the difference between the maximum and minimum values.

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

Explanations: The range is calculated by subtracting the smallest value from the largest value in a data set. This provides a simple measure of the spread of the data. It is important to note that the range is sensitive to outliers, as it is based solely on the two extreme values.

Option Analysis:

Option A: Correct. The range is defined as the difference between the highest and lowest value in a data set.
Option B: Incorrect. The most frequently occurring value in a data set is the mode, not the range.
Option C: Incorrect. The middle value in a sorted data set is the median, not the range.
Option D: Incorrect. The average value of a data set is the mean, not the range.

Mnemonic: Range: R = Highest - Lowest

9. The heights of some students are given.158cm 172cm 164cm164cm 167cm 159cm What is the range of the heights?

Correct Answer: Correct answer is: (B) 14cm.

Exam Relevance: AP Statistics, SAT, GRE, GMAT

Difficulty: Easy

Concept notes: The range in descriptive statistics is the difference between the maximum and minimum values in a dataset.

Common Mistakes: A common mistake is to confuse the range with other measures of spread, such as the interquartile range or standard deviation.

Explanations: To find the range, identify the maximum and minimum values in the dataset. The maximum height is 172cm and the minimum height is 158cm. The range is calculated as follows: 172cm - 158cm = 14cm.

Option Analysis:

Option A: 13cm is incorrect because the difference between the maximum and minimum values is 14cm, not 13cm.
Option B: 14cm is correct because it is the difference between the maximum and minimum values in the dataset.
Option C: 164cm is incorrect because it is one of the values in the dataset, not the range.
Option D: 330cm is incorrect because it is not the difference between the maximum and minimum values in the dataset.

10. Identify the best average to use for the following data ..... FAVOURITE FOOTBALL TEAM: Arsenal, Chelsea, Chelsea, Arsenal, Southampton, Arsenal, Man Utd

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

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

Difficulty: Easy

Concept notes: The mode is the value that appears most frequently in a dataset. In this context, it is the most appropriate measure of central tendency for categorical data, such as favorite football teams.

Common Mistakes: A common mistake is to assume that the mean or median is appropriate for categorical data, which is incorrect as these measures are used for numerical data.

Explanations: The mode is the most suitable measure for categorical data like favorite football teams because it identifies the most frequently occurring category. In the given data, Arsenal appears most frequently, making it the mode.

Option Analysis:

Option A: The mean is not applicable for categorical data.
Option B: The median is not applicable for categorical data.
Option C: The mode is the most appropriate measure for categorical data.
Option D: The mode is a valid measure for this data, so this option is incorrect.

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 median calculated when there is an even number of observations?

When there is an even number of observations, the median is calculated by finding the average of the two middle values.

What is the range in descriptive statistics?

The range is a measure of data dispersion, calculated as the difference between the maximum and minimum values in a dataset.

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

A bar chart is used to display categorical data, showing the frequency or proportion of each category in a clear and visual manner.

What does the mean absolute deviation measure?

The mean absolute deviation measures the average distance between each data point and the mean, providing insight into the variability of the dataset.

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