Multiple Choice

1. The topic of this unit introduces the idea of 'variability' in a data set. What does variability measure?

   The exact middle value of the data.

   The sum of all the data points.

   The largest value minus the smallest value.

   How spread out the data points are.

2. Before calculating the Mean Absolute Deviation (MAD), the first necessary step is to calculate the central point from which all measurements deviate. What is this central measure called?

   Median

   Mode

   Range

   Mean

3. A baker recorded the number of cakes sold daily: $\{5, 9, 11, 7\}$. Calculate the mean ($\bar{x}$) number of cakes sold.

   $8$

   $9$

   $7$

   $8.5$

4. If a student's test scores are $\{80, 85, 90, 95\}$, what is the mean score?

   $88$

   $85$

   $87.5$

   $90.5$

5. After calculating the mean ($\bar{x}$), the next step in finding the MAD is to determine the distance of each individual data point ($x$) from the mean. This distance calculation is called the:

   Absolute Deviation

   Variability Measurement

   Final Summation

   Central Tendency

6. A group of five friends has an average height of $60$ inches. If one friend is $65$ inches tall, what is the absolute deviation for that data point?

   $125$ inches

   $-5$ inches

   $0$ inches

   $5$ inches

7. When calculating absolute deviation, we always use the absolute value sign ($| |$). Why is it necessary to ensure the distance is positive?

   Because distance or spread cannot be a negative quantity.

   Because negative numbers cannot be used in averages.

   Because distance is always multiplied by the mean.

   To simplify the final calculation.

8. A data set of temperatures has a mean of $15^\circ$ F. What is the absolute deviation for a temperature reading of $12^\circ$ F?

   $3^\circ$ F

   $27^\circ$ F

   $1.25^\circ$ F

   $-3^\circ$ F

9. The first step in finding the MAD for the data set $\{2, 4, 6, 8\}$ is finding the mean. The mean is $5$. What is the complete list of absolute deviations for this data set?

   $\{2, 4, 6, 8\}$

   $\{3, 1, 1, 3\}$

   $\{-3, -1, 1, 3\}$

10. What does the acronym MAD stand for in the context of data analysis?

    Mean Absolute Deviation

    Most Accurate Data

    Mean Average Distance

    Maximum Absolute Deviation

11. A student calculated the absolute deviations for a data set of five numbers as $\{4, 1, 0, 3, 2\}$. What is the Mean Absolute Deviation (MAD)?

    $2.5$

    $1.5$

    $2$

    $10$

12. The data set for the number of pets owned by four neighbors is $\{10, 8, 12, 6\}$. The mean is $9$. The absolute deviations are $\{1, 1, 3, 3\}$. Calculate the MAD.

    $8$

    $2$

    $1.5$

    $4$

13. To find the MAD of a set of numbers, what is the final calculation required after listing all the absolute deviations?

    Sum the deviations and divide by the mean.

    Find the average (mean) of the list of absolute deviations.

    Multiply the deviations by the mean.

    Subtract the smallest deviation from the largest deviation.

14. Three weights are measured: $\{14$ lbs, $15$ lbs, $16$ lbs$\}$. The mean is $15$ lbs. What is the MAD for this data set?

    $2/3$ lb

    $1$ lb

    $0$ lbs

    $2/5$ lb

15. In a very spread-out data set $\{1, 10, 19\}$, the mean is $10$. Calculate the MAD.

    $6$

    $30$

    $9$

    $4$

16. If the Mean Absolute Deviation (MAD) for the price of coffee at different shops is $0.50$, what does this value specifically mean in a real-world context?

    All coffee shops sell coffee for the same price.

    The typical distance of any coffee price from the average price is $\$0.50$.

    The average price of coffee is $\$0.50$.

    The price of coffee never goes above $\$0.50$.

17. What does a relatively high MAD value indicate about a data set compared to a set with a low MAD value?

    The mean of the data set is larger.

    The data set has fewer numbers.

    The data points are tightly clustered around the mean.

    The data points are more spread out (greater variability).

18. Basketball team A has a MAD of $1.5$ points per game, and Basketball team B has a MAD of $4.2$ points per game. Which team is more consistent in its scoring?

    They are equally consistent since MAD is just an average.

    Neither, because MAD doesn't measure consistency.

    Team A, because a lower MAD indicates less variability.

    Team B, because $4.2$ is a higher number.

19. Two sets of delivery times, in minutes, are compared: Set X has a MAD of $8$ minutes. Set Y has a MAD of $3$ minutes. If a customer prefers reliable and fast delivery, which set of times shows greater reliability?

    Both are equally reliable.

    Neither, as speed is not related to MAD.

    Set Y, because the delivery times are typically closer to the average time.

    Set X, because a larger MAD means faster delivery.

20. If a data set has a Mean Absolute Deviation (MAD) of $0$, what must be true about all the data points in that set?

    All the numbers in the set are negative.

    The data set contains only two numbers.

    The mean of the set must be $0$.

    All data points in the set are identical.

21. The absolute deviations calculated for a small data set are $2, 4, 1,$ and $3$. What is the final step calculation for the Mean Absolute Deviation (MAD)?

    $(2+4+1+3) / 4 = 2.5$

    $2 \times 4 \times 1 \times 3 = 24$

    $(2+4+1+3) / 5 = 2$

    $4 - 1 = 3$

22. A baker records the temperature deviations for four ovens: $1.5^\circ, 2.5^\circ, 0.5^\circ,$ and $3.5^\circ$. Calculate the Mean Absolute Deviation (MAD) for the ovens.

    $1.5$

    $8.5$

    $2.5$

    $2.0$

23. After finding all the absolute deviations of a data set, what is the next and final step required to calculate the Mean Absolute Deviation (MAD)?

    Find the largest deviation.

    Find the difference between the largest and smallest deviation.

    Average the list of absolute deviations.

    Multiply the sum of the deviations by the number of data points.

24. The MAD for the weekly sales of a coffee shop is calculated to be $\$15$. What does this low MAD value suggest about the sales data?

    The range of weekly sales is exactly $\$15$.

    The weekly sales figures are very consistent and close to the mean.

    The coffee shop generally has very low weekly sales.

    The average weekly sales value is $\$15$.

25. Calculate the Mean Absolute Deviation (MAD) for the number of pets owned by five students: $\{1, 2, 3, 4, 5\}$.

    1.5

    6

    1

    1.2

26. A car mechanic recorded the time (in minutes) taken for five oil changes: $\{10, 10, 15, 20, 20\}$. The mean time is $15$ minutes. What is the MAD?

    0 minutes

    10 minutes

    4 minutes

    5 minutes

27. In a real-world context, what does a very high Mean Absolute Deviation (MAD) value indicate about the data being analyzed?

    The data points are spread widely away from the central mean.

    The data points are perfectly consistent.

    The median and the mean are equal.

    The average of the data is also very high.

28. The list of absolute deviations for a set of scores is $\{0.8, 1.2, 0.8, 1.2\}$. Calculate the MAD.

    4.0

    1.2

    1.0

    0.8

29. Data Set P has a MAD of $2.1$, and Data Set Q has a MAD of $5.9$. Which data set exhibits greater variability?

    Data Set Q, because the typical distance from its mean is larger.

    Both data sets show the same level of variability.

    Variability cannot be determined by MAD.

    Data Set P, because its data is less spread out.

30. If a data set measures the weight of apples in grams, what are the appropriate units for the Mean Absolute Deviation (MAD)?

    Kilograms

    Grams$^2$

    Number of apples

    Grams

31. Two investment portfolios, R and S, both have an average return of $7\%$. Portfolio R has a MAD of $0.5\%$, and Portfolio S has a MAD of $3.0\%$. Which portfolio is more volatile (has less predictable returns)?

    Portfolio S, because its returns typically deviate $3.0\%$ from the average.

    Portfolio R, because its MAD is smaller.

    Portfolio R, because its returns are more consistent.

    Neither, because the average returns are identical.

32. Which calculation is performed when calculating the MAD using the list of absolute deviations?

    Finding the standard deviation.

    Squaring the sum of the deviations.

    Finding the arithmetic mean of the deviations.

    Finding the median of the deviations.

33. Calculate the MAD for the data set $\{10, 10, 10, 10\}$.

    40

    10

    0

    1

34. The mean height of a team is $160$ cm, and the MAD is $4$ cm. How should this MAD value be interpreted?

    On average, the players' heights differ from the mean height by $4$ cm.

    The range of heights is $4$ cm.

    The tallest player is $164$ cm.

    All players are between $156$ cm and $164$ cm tall.

35. A data set lists the daily high temperatures in two cities. City A has a MAD of $15^\circ F$, and City B has a MAD of $3^\circ F$. Which city has more consistent daily temperatures?

    Both cities are equally consistent.

    Neither city, as temperature consistency is not related to MAD.

    City B, because its temperatures are generally closer to the mean temperature.

    City A, because its MAD is higher.

36. Calculate the MAD for the data set $\{8, 9, 10, 11, 12\}$.

    1.2

    0.8

    10

    1

37. If the Mean Absolute Deviation (MAD) for a set of data is $12$ pounds, what does this tell us about the data?

    A typical data point is 12 pounds away from the mean.

    The mean weight is 12 pounds.

    The heaviest data point is 12 pounds.

    The data is clustered closely around the center.

38. Why is the process of calculating MAD often useful in quality control or manufacturing?

    It determines the minimum acceptable level of quality.

    It quantifies the consistency and uniformity of the products.

    It ensures that the mean product size meets specifications.

    It sets the price based on average cost.

39. Calculate the MAD for the data set $\{2, 4, 6, 8, 10\}$. The mean is $6$.

    6

    2.5

    2.4

    3

40. Which statement best summarizes the comparative role of MAD?

    A smaller MAD indicates that a data set is less variable than one with a larger MAD.

    The data set with the larger MAD always has a larger mean.

    MAD compares the mean of two different data sets.

    MAD measures the absolute size of the largest outlier.

41. A student calculated the test scores for five quizzes as $85, 92, 78, 90,$ and $85$. The mean score is $86$. After finding the mean absolute deviation (MAD), the student obtained a value of $4.0$. Which of the following statements provides the most accurate interpretation of this MAD value in the context of the quiz scores?

    The student can expect their future quiz scores to always fall exactly between $82$ and $90$.

    The student's score of $78$ deviates from the mean by $8$, proving the MAD calculation is incorrect.

    On average, the student's individual quiz scores differ from the mean score by $4.0$ points.

    The average quiz score was $86$, and $4.0$ represents the total range of scores.

42. Two companies, $\text{X}$ and $\text{Y}$, both report an average employee salary of $\$45,000$. Company $\text{X}$ has a Mean Absolute Deviation (MAD) of $\$3,200$, while Company $\text{Y}$ has a MAD of $\$16,000$. Which conclusion about the distribution of salaries is best supported by the MAD values?

    Company $\text{X}$ has salaries that are less spread out and more consistent than Company $\text{Y}$.

    The median salary for Company $\text{Y}$ must be much lower than $\$45,000$.

    Company $\text{X}$ must employ more people than Company $\text{Y}$ to achieve a lower MAD.

    The highest and lowest salaries are likely identical in both companies since the mean is the same.

43. Five saplings were measured for height (in inches): $60, 62, 65, 67,$ and $68$. Calculate the Mean Absolute Deviation (MAD) for this data set, recognizing that the mean is $64.4$ inches.

    $3.2$ inches

    $2.4$ inches

    $2.8$ inches

    $2.72$ inches

44. Consider a data set of seven temperatures recorded in Celsius. If one temperature reading, which was a significant outlier (far above the mean), is removed from the data set, how would the Mean Absolute Deviation (MAD) typically change, and why?

    The MAD would usually decrease because the removal of an outlier makes the data cluster more tightly around the new mean.

    The MAD would usually increase because the mean would decrease, increasing the deviation of the remaining lower values.

    The MAD would usually remain the same, because MAD is focused on the central tendency, not extreme values.

    The MAD would usually decrease because the range of the data expands dramatically upon removal of an outlier.

45. A data set consists of four scores: $10, 12, 16,$ and $x$. If the Mean Absolute Deviation (MAD) of this set is $2.5$, what is the value of the missing score, $x$?

    $8$

    $11$

    $14$

    $10$

46. A factory produces standardized bolts. Machine A produces bolts with a mean length of $5.0$ cm and a MAD of $0.05$ cm. Machine B produces bolts with a mean length of $5.0$ cm but a MAD of $0.15$ cm. If quality control requires bolts to be highly consistent (low variability), which statement is true?

    Machine B is superior because the larger MAD indicates a higher potential for extreme maximum lengths.

    Both machines are equally reliable since they both achieve the target mean length of $5.0$ cm.

    Machine B's output is less reliable because, on average, the bolts deviate further from the required mean length.

    Machine A is producing larger bolts than Machine B since its MAD is smaller.

47. A scientist measured the growth (in cm) of four bacteria cultures: $4.0, 5.5, 6.0,$ and $6.5$. Calculate the Mean Absolute Deviation (MAD) of the growth data.

    $0.8$ cm

    $0.75$ cm

    $0.6$ cm

    $0.9$ cm

48. Compared to the Range, why is the Mean Absolute Deviation (MAD) generally considered a superior measure of the typical variability within a dataset for statistical comparison?

    The Range only considers the two extreme data points, making it highly sensitive to single outliers.

    The MAD calculation is simpler and less prone to mathematical error than calculating the Range.

    The Range is influenced by every single data point in the set, unlike the MAD.

    The MAD only utilizes the median of the data set, ensuring $50\%$ of the data is accounted for.

49. The average wait time at a popular amusement park ride is $45$ minutes, and the calculated Mean Absolute Deviation (MAD) is $5$ minutes. If you were to interpret this value, what does the MAD tell you about the consistency of the wait times?

    A typical wait time for this ride is expected to deviate from the mean of $45$ minutes by about $5$ minutes.

    The variability is low, meaning wait times rarely exceed $50$ minutes or fall below $40$ minutes.

    $5$ minutes is the total difference between the longest and shortest recorded wait times.

    This indicates that $50\%$ of all recorded wait times fall within $5$ minutes of the average time.

50. Two basketball teams, Team A and Team B, each played four games and scored the following points. Team A: $8, 12, 15, 25$. Team B: $10, 14, 16, 20$. Both teams have the same mean score of $15$ points. Determine the difference between the Mean Absolute Deviations ($MAD_A - MAD_B$) and interpret the result regarding team consistency.

    $2.0$; Team B is the more consistent team, deviating less from the average.

    $5.0$; Team A is more consistent because its minimum score was lower, indicating better variance control.

    $8.0$; Team A is less consistent because the total deviation sum for Team A is $20$ points higher than Team B.

    $2.0$; Team A is the more consistent team, deviating less from the average.

51. A scientist recorded the masses (in grams) of five minerals: $\{15, 18, 20, 21, 26\}$. After correctly calculating the mean mass ($\bar{x}$) to be $20$ grams, they found the list of absolute deviations. If the final step is to calculate the Mean Absolute Deviation (MAD) by averaging these deviations, what is the resulting MAD value, and what does the value conceptually represent?

    $3.2$ grams; it represents the average squared difference between the observations and the mean.

    $2.8$ grams; it represents the total spread of the data set from minimum to maximum value, divided by the number of samples.

    $3.5$ grams; it represents the median distance between any data point and the lowest mass recorded.

    $2.8$ grams; it represents the typical distance each mineral's mass deviates from the average mass of $20$ grams.

52. A small company tracks the daily call volume for six employees. If the calculated Mean Absolute Deviation (MAD) is $4.5$ calls, this implies the total sum of all six absolute deviations must be $27$. If five of the absolute deviations found were $3, 7, 1, 9,$ and $2$, and the overall mean call volume was $50$, what is one possible value for the sixth employee's actual call volume?

    $52$ calls

    $48$ calls

    $45$ calls

    $55.5$ calls

53. Two Grade 7 classes took the same statistics quiz. Class R had a mean score of $85$ and a Mean Absolute Deviation (MAD) of $5.2$ points. Class S also had a mean score of $85$, but its MAD was $1.8$ points. What is the most accurate interpretation of these results regarding the variability of the scores?

    Class R performed better overall because its scores were more spread out, indicating greater potential for high achievement.

    The scores in Class S are clustered more tightly around the mean of $85$, indicating significantly less variability and higher consistency in performance compared to Class R.

    The range of scores must be identical for both classes, as they share the same mean.

    The students in Class R were more consistent in their scoring since $5.2$ is a larger deviation, indicating a wider range of high and low scores.

54. A chef is testing the temperature stability of a new oven, recording five distinct temperature fluctuations (in degrees Celsius) after reaching a stable point: $\{10, 11, 12, 12, 13\}$. Calculate the Mean Absolute Deviation (MAD) for this data set, ensuring precision despite the resulting non-integer mean.

    $0.85^\circ$ C

    $0.92^\circ$ C

    $1.0^\circ$ C

    $0.88^\circ$ C

55. Which statement provides the most precise and complete interpretation of the Mean Absolute Deviation (MAD) value within the framework of data variability?

    MAD is the measure of central tendency that identifies the exact middle point of the data set, ensuring $50\%$ of the data lies above and below it.

    MAD serves as an average measure of variability, detailing how far, on average, individual data points are dispersed or spread out from the arithmetic mean of the data set.

    MAD indicates the specific position of the mode within the distribution relative to the overall range.

    MAD is a measure that quantifies the total spread of the data set, calculated by taking the sum of the differences between consecutive data points.

56. A small data set initially contains 4 values and has a calculated Mean Absolute Deviation (MAD) of $5$ units. The analyst considers adding a new, fifth data point that is exactly equal to the current mean of the set. How does adding this single data point affect the calculated MAD, and why?

    The MAD will increase slightly (e.g., to $5.25$) because the increase in the total number of data points causes the divisor in the final calculation to be larger.

    The MAD will increase to $6.25$, because the inclusion of the new point shifts the mean, causing all previous deviations to increase slightly.

    The MAD will decrease to $4$, because the new point contributes a deviation of zero, lowering the average dispersion relative to the increased count of data points.

    The MAD will remain exactly $5$, because a point that is exactly at the mean contributes zero deviation, having no impact on the existing spread.

57. A coach recorded the weightlifting capacity (in pounds) of two athletes, P and Q, across four trials. Set P: $\{2, 4, 6, 8\}$. Set Q: $\{4, 4, 6, 6\}$. Both athletes achieved the same mean capacity of $5$ pounds. Calculate the MAD for both sets and determine which athlete shows significantly less variability in their performance.

    MAD(P) is $1.5$ lbs and MAD(Q) is $1.0$ lbs; Athlete P is less variable.

    MAD(P) is $1.0$ lbs and MAD(Q) is $2.0$ lbs; Athlete Q is less variable.

    MAD(P) is $2.0$ lbs and MAD(Q) is $1.0$ lbs; Athlete Q shows significantly less variability.

    MAD(P) is $2.0$ lbs and MAD(Q) is $1.5$ lbs; Athlete Q is less variable.

58. A food factory calculates the Mean Absolute Deviation (MAD) for the weight of cookie packets to be $0.5$ ounces. The mean weight is $16.0$ ounces. If a consumer affairs group mandates that the typical deviation (MAD) must be less than $0.4$ ounces, what operational conclusion must the factory draw based solely on the MAD value?

    The MAD value is irrelevant; the factory must instead check that the median weight meets the required $16.0$ ounces.

    The factory's weighing system is adequate because $0.5$ ounces is very close to $0.4$ ounces, demonstrating high precision.

    The factory must redesign its packaging, as the MAD of $0.5$ ounces indicates the existence of significant outliers far exceeding the maximum weight.

    The factory is failing to meet the requirement because their packets typically deviate $0.5$ ounces from the mean, exceeding the mandated maximum typical deviation of $0.4$ ounces.

59. A student is calculating the Mean Absolute Deviation (MAD) for the dataset $\{30, 32, 38\}$. They correctly calculate the mean as $33.33...$ and find the absolute deviations to be $3.33..., 1.33...,$ and $4.67...$. Which of the following describes the necessary final step, according to the definition of MAD, and the resulting calculation?

    Averaging the first and last absolute deviations ($3.33...$ and $4.67...$) and ignoring the middle deviation.

    Calculating the difference between the largest deviation ($4.67...$) and the smallest deviation ($1.33...$) to find the range of deviations.

    Summing the absolute deviations ($3.33... + 1.33... + 4.67... = 9.33...$) and dividing the total by the count of data points ($3$), resulting in a MAD of $3.11...$.

    Squaring the sum of the deviations to emphasize the larger differences from the mean, resulting in a measure called variance.

60. A data analyst records three measurements in a new experiment: $\{x, 8, 16\}$. What integer value must $x$ take in order for the calculated Mean Absolute Deviation (MAD) of the set to be exactly $4$?

    $x=6$

    $x=4$

    $x=2$

    $x=10$