# Displaying A Single Quantitative Variable ## Section A: Multiple Choice Questions
Q1: A dataset of test scores has the following values: 65, 70, 75, 80, 85, 90, 95. What is the range of this dataset? (a) 30 (b) 25 (c) 20 (d) 35
Solution:
Ans: (a) Explanation: The range is calculated as the difference between the maximum and minimum values. Maximum = 95, Minimum = 65. Range = 95 - 65 = 30.
Q2: Which measure of center is most affected by extreme values (outliers) in a dataset? (a) Median (b) Mode (c) Mean (d) Interquartile range
Solution:
Ans: (c) Explanation: The mean is calculated by summing all values and dividing by the count, so extreme values significantly affect it. The median and mode are resistant measures that are not heavily influenced by outliers. The interquartile range is a measure of spread, not center.
Q3: A histogram shows the distribution of heights in a classroom. If the histogram has a tail extending to the right, the distribution is: (a) Symmetric (b) Skewed left (c) Skewed right (d) Uniform
Solution:
Ans: (c) Explanation: When a distribution has a tail extending to the right, it is skewed right (or positively skewed). This means there are a few unusually high values pulling the tail in that direction. Skewed left would have a tail extending to the left, symmetric would have no tail, and uniform would show equal frequencies across all intervals.
Q4: The five-number summary of a dataset includes all of the following EXCEPT: (a) Minimum (b) First quartile (Q1) (c) Mean (d) Maximum
Solution:
Ans: (c) Explanation: The five-number summary consists of the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum. The mean is not part of the five-number summary; it is a separate measure of center.
Q5: A boxplot of student test scores shows the box extending from 60 to 80, with a line at 70. What does the line at 70 represent? (a) Mean (b) Mode (c) Median (d) Range
Solution:
Ans: (c) Explanation: In a boxplot, the line inside the box represents the median (Q2), which is the middle value of the dataset. The edges of the box represent Q1 and Q3. The mean is not typically shown on a boxplot.
Q6: Which of the following statements about standard deviation is TRUE? (a) It measures the center of the data (b) It is always larger than the range (c) It measures the spread of data around the mean (d) It is not affected by outliers
Solution:
Ans: (c) Explanation:Standard deviation measures how spread out the values are from the mean. It does not measure center (option a is incorrect), it is not always larger than the range (option b is incorrect), and it is affected by outliers (option d is incorrect).
Q7: A dotplot is most useful for displaying quantitative data when: (a) The dataset is very large with thousands of values (b) The dataset is small to moderate in size (c) The data has many repeated decimal values (d) You need to show the five-number summary
Solution:
Ans: (b) Explanation: A dotplot works best with small to moderate datasets because each value is represented by a dot. For very large datasets, dotplots become cluttered and difficult to read. Histograms are better for large datasets, and boxplots are better for showing the five-number summary.
Q8: The interquartile range (IQR) is calculated as: (a) Q3 - Q1 (b) Maximum - Minimum (c) Q2 - Q1 (d) Q3 - Q2
Solution:
Ans: (a) Explanation: The interquartile range (IQR) is calculated as \(Q3 - Q1\), which represents the spread of the middle 50% of the data. Option (b) describes the range. Options (c) and (d) represent half of the IQR in different portions of the distribution.
## Section B: Fill in the Blanks Q9: The __________ is the value that appears most frequently in a dataset.
Solution:
Ans: mode Explanation: The mode is defined as the value that occurs most often in a dataset. A dataset can have one mode, more than one mode, or no mode at all.
Q10: In a __________ distribution, the mean, median, and mode are all approximately equal.
Solution:
Ans: symmetric Explanation: In a symmetric distribution, the data is evenly distributed around the center, causing the mean, median, and mode to have similar values. This is characteristic of a normal or bell-shaped distribution.
Q11: A __________ is a graphical display that shows the distribution of quantitative data using bars to represent frequencies or relative frequencies of intervals.
Solution:
Ans: histogram Explanation: A histogram displays quantitative data by grouping values into intervals (bins) and using the height of bars to show how many observations fall in each interval.
Q12: The first quartile (Q1) is the value below which __________% of the data falls.
Solution:
Ans: 25 Explanation: The first quartile (Q1) marks the 25th percentile of the data, meaning 25% of the values in the dataset are less than or equal to Q1.
Q13: A value that is significantly different from other values in a dataset is called a(n) __________.
Solution:
Ans: outlier Explanation: An outlier is an observation that lies an abnormal distance from other values in the dataset. Outliers can significantly affect measures like the mean and range.
Q14: The __________ is calculated by summing all values in a dataset and dividing by the number of values.
Solution:
Ans: mean Explanation: The mean (or arithmetic average) is found by adding all data values together and dividing by the total number of observations: \(\bar{x} = \frac{\sum x}{n}\).
## Section C: Word Problems Q15: The ages of students in a statistics club are: 14, 15, 15, 16, 16, 16, 17, 18, 20. Find the median age.
Solution:
Ans: The dataset has 9 values (odd number). The median is the middle value when data is ordered. Position of median = \(\frac{9 + 1}{2} = 5\)th value The 5th value in the ordered list is 16. Final Answer: 16 years
Q16: A set of quiz scores has a mean of 78 and a standard deviation of 6. If a student scored 90 on the quiz, how many standard deviations above the mean is this score?
Solution:
Ans: To find how many standard deviations above the mean: Number of standard deviations = \(\frac{\text{Score} - \text{Mean}}{\text{Standard Deviation}}\) \(= \frac{90 - 78}{6} = \frac{12}{6} = 2\) Final Answer: 2 standard deviations above the mean
Q17: The following data represents the number of hours 10 students spent studying for an exam: 2, 3, 3, 4, 5, 5, 6, 7, 8, 12. Calculate the interquartile range (IQR) for this dataset.
Solution:
Ans: First, find Q1 and Q3. The data is already ordered: 2, 3, 3, 4, 5, 5, 6, 7, 8, 12 For 10 values: Q1 is at position 2.75, so Q1 = 3 + 0.75(4 - 3) = 3.75 Q3 is at position 8.25, so Q3 = 7 + 0.25(8 - 7) = 7.25 IQR = Q3 - Q1 = 7.25 - 3.75 = 3.5 Final Answer: 3.5 hours
Q18: A histogram of daily temperatures in January shows that most temperatures are clustered between 20°F and 35°F, with a few days having temperatures around 5°F. Describe the shape of this distribution and identify which measure of center (mean or median) would be more appropriate to use.
Solution:
Ans: The distribution is skewed left because there are a few unusually low temperatures (around 5°F) creating a tail on the left side. The median would be more appropriate because it is resistant to outliers and better represents the typical temperature. The mean would be pulled down by the extreme low values. Final Answer: Skewed left distribution; median is more appropriate
Q19: The five-number summary for the number of books read by students over summer is: Minimum = 2, Q1 = 5, Median = 8, Q3 = 12, Maximum = 20. Using the 1.5 × IQR rule, determine if the maximum value of 20 is an outlier.
Solution:
Ans: First, calculate IQR: IQR = Q3 - Q1 = 12 - 5 = 7 Upper fence = Q3 + 1.5 × IQR = 12 + 1.5(7) = 12 + 10.5 = 22.5 Since the maximum value (20) is less than 22.5, it is not an outlier. Final Answer: No, 20 is not an outlier
Q20: A class recorded the following test scores: 72, 75, 78, 80, 82, 85, 88, 90, 92, 95. Calculate the mean and the range of the test scores.
Solution:
Ans: Mean: Sum = 72 + 75 + 78 + 80 + 82 + 85 + 88 + 90 + 92 + 95 = 837 Number of values = 10 Mean = \(\frac{837}{10} = 83.7\)
Range: Range = Maximum - Minimum = 95 - 72 = 23
Final Answer: Mean = 83.7; Range = 23
The document Worksheet (with Solutions): Displaying A Single Quantitative Variable is a part of the Grade 10 Course High School Statistics.
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