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Histogram Quiz 8 (10 MCQs)

This set of multiple-choice questions evaluates your understanding of histogram analysis, frequency distribution, and data interpretation. Skills tested include data range calculation, class width determination, histogram construction, and interpretation, including skewness identification and symmetry in data. Concepts such as statistical dispersion, interquartile range, and data representation are also covered.

Quiz Instructions

Select an option to see the correct answer instantly.

1. The histogram has a tail to the:

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

Exam Relevance: AP Statistics, GRE Quantitative Reasoning, GMAT Data Sufficiency

Difficulty: Moderate

Concept notes: A histogram with a tail to the left is known as left-skewed or negatively skewed. This means that the majority of the data is concentrated on the right side of the histogram, with a few smaller values extending to the left.

Common Mistakes: A common misunderstanding is to confuse the direction of the tail with the direction of the skew. The tail points in the direction of the skew, so a left tail indicates left skewness.

Explanations: In a histogram, the tail to the left indicates that the distribution has a longer tail on the left side. This means that the majority of the data points are clustered towards the right, with fewer data points extending to the left. This is characteristic of a left-skewed distribution.

Option Analysis:

Option A: Correct. The histogram has a tail to the left, indicating left skewness.
Option B: Incorrect. A tail to the right would indicate right skewness, which is not the case here.

2. Which of the following statements about this distribution is not correct?

Correct Answer: Correct answer is: (C) The IQR is $ 35.

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

Difficulty: Moderate

Concept notes: The interquartile range (IQR) is a measure of statistical dispersion, being equal to the difference between the third quartile (Q3) and the first quartile (Q1). In a histogram, the IQR can be visually estimated by identifying the range between the first and third quartiles.

Common Mistakes: A common mistake is to assume that the IQR can be directly read from the histogram without identifying the quartiles. The IQR is not a fixed value but depends on the distribution of the data.

Explanations: The IQR is calculated as the difference between the third quartile (Q3) and the first quartile (Q1). Without specific values for Q1 and Q3, it is not possible to confirm that the IQR is $35. The statement that the IQR is $35 is not correct based on the information provided.

Option Analysis:

Option A: The histogram is skew right. This statement could be correct if the histogram shows a longer tail on the right side.
Option B: The median is less than the $20. This statement could be correct if the median value is below $20.
Option C: The IQR is $35. This statement is not correct as the IQR cannot be confirmed to be $35 without specific quartile values.
Option D: The mean is greater than the median. This statement could be correct if the distribution is skewed right, which often results in the mean being greater than the median.

3. The table shows the number of photocopies made during one day at each of the 25 schools in a school district. Which histogram displays all the data in the table correctly?

Correct Answer: Correct answer is: (A) .

Exam Relevance: AP Statistics, SAT Math, GRE Quantitative Reasoning

Difficulty: Moderate

Concept notes: A histogram is a graphical representation of data using bars of different heights. Each bar represents a range of values, and the height of the bar indicates the frequency of data points within that range.

Common Mistakes: A common mistake is misinterpreting the ranges or frequencies, leading to incorrect placement or height of the bars.

Explanations: The correct histogram (Option A) accurately represents the number of photocopies made at each school. Each bar corresponds to a range of photocopies, and the height of each bar correctly reflects the number of schools that fall within that range. This ensures that the histogram accurately displays the distribution of photocopies across the 25 schools.

Option Analysis:

Option A: Correctly displays the data with appropriate ranges and frequencies.
Option B: Incorrectly displays the ranges or frequencies, leading to a misrepresentation of the data.
Option C: Incorrectly displays the ranges or frequencies, leading to a misrepresentation of the data.
Option D: Incorrectly displays the ranges or frequencies, leading to a misrepresentation of the data.

4. Emily is making a histogram for the data: 12, 94, 61, 24, 77, 36, 56, 83, 7, 34, 45, 75, 86, 42, 91, 78, and 96. She wants to make 10 classes. What should the class width be?

Correct Answer: Correct answer is: (B) 9.

Exam Relevance: AP Statistics, IB Mathematics, SAT Math, GRE Quantitative

Difficulty: Moderate

Concept notes: The class width in a histogram is determined by dividing the range of the data by the number of classes. The range is the difference between the maximum and minimum values in the dataset.

Common Mistakes: A common mistake is to round the class width to a convenient number without considering the actual range and number of classes, which can lead to incorrect class intervals.

Explanations: To determine the class width, first find the range of the data. The maximum value is 96 and the minimum value is 7. The range is calculated as: \[ \text{Range} = 96 - 7 = 89 \] Next, divide the range by the number of classes (10): \[ \text{Class Width} = \frac{89}{10} = 8.9 \] Since class width should be a whole number, we round 8.9 to the nearest whole number, which is 9. Therefore, the class width should be 9.

Option Analysis:

Option A: 8 is incorrect because it does not account for the rounding of 8.9 to the nearest whole number.
Option B: 9 is correct because it is the rounded value of 8.9.
Option C: 10 is incorrect because it is not the nearest whole number to 8.9.
Option D: 17 is incorrect because it is not the result of dividing the range by the number of classes.

5. Describe the shape of this new histogram as approximately symmetric, skewed left, or skewed right? (a)

Correct Answer: Correct answer is: (Choice A) Approximately symmetrical.

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

Difficulty: Moderate

Concept notes: A histogram is considered approximately symmetrical if the distribution of data is roughly the same on both sides of the center. This means that the left and right sides of the histogram mirror each other, with the peak of the distribution in the middle.

Common Mistakes: A common mistake is to confuse a slightly skewed histogram with a symmetrical one. It is important to carefully observe the shape and distribution of the bars to determine symmetry.

Explanations: The histogram is described as approximately symmetrical, meaning that the distribution of data is roughly the same on both sides of the center. This implies that the left and right sides of the histogram mirror each other, with the peak of the distribution in the middle. Therefore, the correct answer is that the histogram is approximately symmetrical.

Option Analysis:

Option A: Correct. The histogram is approximately symmetrical.
Option B: Incorrect. The histogram is not skewed left.
Option C: Incorrect. The histogram is not skewed right.
Option D: Incorrect. The histogram is approximately symmetrical, not none of the above.

6. The line plot shows the number of points David scored in 30 games. Which histogram represents the same set of data?

Correct Answer: Correct answer is: (A) .

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Common Mistakes: A common mistake is to confuse the histogram with a bar graph. In a histogram, the bars are adjacent and represent continuous data, whereas in a bar graph, the bars are separate and represent categorical data.

Explanations: The line plot shows the number of points David scored in 30 games. To convert this data into a histogram, we need to group the scores into intervals and count the number of scores in each interval. The correct histogram (Option A) should accurately represent these intervals and their corresponding frequencies as shown in the line plot.

Option Analysis:

Option A: This histogram correctly represents the data from the line plot by grouping the scores into intervals and showing the frequency of scores in each interval.
Option B: This histogram does not accurately represent the data from the line plot. The intervals or frequencies may be incorrect.
Option C: This histogram does not accurately represent the data from the line plot. The intervals or frequencies may be incorrect.
Option D: This histogram does not accurately represent the data from the line plot. The intervals or frequencies may be incorrect.

7. What does it mean if a histogram starts at a value greater than 0?

Correct Answer: Correct answer is: (B) It may misrepresent the data, as it does not show the full range of values.

Exam Relevance: AP Statistics, GRE Quantitative Reasoning, GMAT Data Sufficiency

Difficulty: Moderate

Concept notes: A histogram is a graphical representation of data distribution. It shows the frequency of data points within specified intervals. If a histogram starts at a value greater than 0, it may not accurately represent the full range of data, potentially leading to misinterpretation.

Common Mistakes: A common misunderstanding is that a histogram starting at a value greater than 0 is always correct, as it may seem to focus on the relevant data range. However, this can hide important information about the data's full range.

Explanations: A histogram that starts at a value greater than 0 may misrepresent the data because it does not show the full range of values. This can lead to incorrect conclusions about the data distribution, as important information about the lower values is omitted. It is crucial for a histogram to accurately represent the entire range of data to avoid misinterpretation.

Option Analysis:

Option A: This is incorrect because a histogram starting at a value greater than 0 does not necessarily accurately represent the data, as it may omit important lower values.
Option B: This is correct because a histogram starting at a value greater than 0 may misrepresent the data by not showing the full range of values, which can lead to incorrect conclusions.
Option C: This is incorrect because a histogram starting at a value greater than 0 does not indicate a normal distribution; it simply omits lower values.
Option D: This is incorrect because while it suggests that there are no values below the starting point, this is not necessarily true and can be misleading.

8. This histogram shows how many windows students have in their homes. How many students have exactly 7 windows?

Correct Answer: Correct answer is: (C) Can not be determined.

Exam Relevance: AP Statistics, SAT Math, GRE Quantitative Reasoning

Difficulty: Moderate

Concept notes: A histogram displays data in intervals or bins, not individual values. Therefore, it is not possible to determine the exact number of students with exactly 7 windows from a histogram.

Common Mistakes: A common mistake is assuming that the histogram provides exact counts for each individual value, when in fact it groups data into intervals.

Explanations: A histogram groups data into intervals or bins, and the height of each bar represents the frequency of data points within that interval. Since the histogram does not provide information on individual values, it is impossible to determine the exact number of students with exactly 7 windows.

Option Analysis:

Option A: Incorrect because the histogram does not provide exact counts for individual values.
Option B: Incorrect because the histogram does not provide exact counts for individual values.
Option C: Correct because the histogram groups data into intervals, and it is not possible to determine the exact number of students with exactly 7 windows.
Option D: Incorrect because the histogram does not provide exact counts for individual values.

9. According to the histogram, everyone saw at least one movie.

Correct Answer: Correct answer is: (C) Not enough information.

Exam Relevance: AP Statistics, GRE, GMAT, SAT

Difficulty: Moderate

Concept notes: A histogram is a graphical representation of data distribution, showing the frequency of data points within intervals. The question asks whether everyone saw at least one movie based on the histogram.

Common Mistakes: A common mistake would be to assume that the histogram provides complete information about each individual's movie-watching habits, when in fact, it only shows the distribution of the number of movies seen.

Explanations: The histogram only shows the distribution of the number of movies seen by a group of people. It does not provide individual data points, so we cannot determine if every single person saw at least one movie. Therefore, the correct answer is that there is not enough information to make this determination.

Option Analysis:

Option A: This option suggests that everyone saw at least one movie, which cannot be concluded from the histogram alone.
Option B: This option suggests that not everyone saw at least one movie, which also cannot be concluded from the histogram alone.
Option C: This option correctly states that there is not enough information to determine if everyone saw at least one movie.
Option D: This option suggests that none of the above is correct, which is not accurate since option C is correct.

10. What does it mean if a histogram is symmetric?

Correct Answer: Correct answer is: (A) A symmetric histogram indicates that the data is evenly distributed around a central value, with similar frequencies on both sides.

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

Difficulty: Easy

Concept notes: A symmetric histogram is characterized by its mirror-like symmetry around a central point, indicating that the data is evenly distributed around this central value.

Common Mistakes: A common misunderstanding is that symmetry in a histogram implies a normal distribution, which is not necessarily true. Symmetry only indicates that the frequencies are similar on both sides of the central value.

Explanations: A symmetric histogram shows that the data points are evenly distributed around a central value, meaning the frequencies of data points on the left side of the central value are similar to those on the right side. This symmetry does not imply a specific type of distribution, such as normal, but rather a balanced distribution of data.

Option Analysis:

Option A: Correct. Symmetric histograms have similar frequencies on both sides of the central value.
Option B: Incorrect. A symmetric histogram does not show a single peak on one side; it shows a balanced distribution around a central value.
Option C: Incorrect. A symmetric histogram does not indicate skewness; skewness is characterized by an imbalance in the distribution.
Option D: Incorrect. Symmetry in a histogram does not imply the absence of outliers; outliers can still exist in a symmetric distribution.

Frequently Asked Questions

What is a histogram?

A histogram is a graphical representation of data distribution, showing the frequency of data points within specified intervals or bins.

How do you interpret the skewness of a histogram?

Skewness in a histogram indicates the asymmetry of the data distribution. A left-skewed histogram has a longer tail on the left side, while a right-skewed histogram has a longer tail on the right side.

What is the significance of the class width in a histogram?

The class width in a histogram determines the size of the intervals used to group data. It affects the shape of the histogram and can influence the interpretation of the data distribution.

How does a histogram help in understanding data dispersion?

A histogram helps visualize the spread of data points, showing how data is distributed across different intervals. This can highlight the central value and the range of data, indicating the degree of dispersion.

What is the difference between a symmetric and a skewed histogram?

A symmetric histogram has a mirror image on either side of the central value, indicating an even distribution. A skewed histogram, on the other hand, has a longer tail on one side, indicating an uneven distribution.

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