Center, Shape, & Spread of Data - ACT Math

1. What is the mode of the data set $3,3,4,5,5,5,6$?

   $5$. 5 appears three times, most frequent.

2. What does the term 'outlier' refer to in a data set?

   A data point significantly different from others. Extreme value far from typical range.

3. Which measure of spread is most resistant to outliers: range or IQR?

   IQR. IQR isn't affected by extreme values.

4. What is the median of an ordered data set with an even number $n$ of values?

   Average of positions $\frac{n}{2}$ and $\frac{n}{2}+1$. For even $n$, average the two middle values.

5. Identify the median of the data set: 11, 17, 19, 23, 27.

   Median = 19. Middle value of five ordered numbers.

6. Calculate the mean of the data set: 10, 20, 30.

   Mean = 20. Sum 60 divided by 3 data points.

7. Identify the first quartile (Q1) of the data set: 10, 20, 30, 40, 50.

   Q1 = 20. 25th percentile of the ordered dataset.

8. Which measure of central tendency represents the most frequent value?

   Mode. Value appearing most often in dataset.

9. Calculate the range for the data: 9, 13, 21, 28.

   Range = 28 - 9 = 19. Largest minus smallest value.

10. Identify the first quartile (Q1) of the data set: 1, 3, 5, 7, 9.

    Q1 = 3. 25th percentile value in ordered data set.

11. What is a box plot used for?

    Visualizing the distribution, center, and spread of data. Shows five-number summary graphically.

12. What is the z-score formula for a value $x$ with mean $\mu$ and standard deviation $\sigma$?

    $z=\frac{x-\mu}{\sigma}$. Standardizes values by mean and standard deviation.

13. State the effect of a constant addition on data mean.

    Adding a constant increases the mean by that constant. Shifts entire distribution by constant.

14. Identify the skewness of the data: 1, 2, 3, 4, 10.

    Skewed right. Outlier 10 creates long right tail.

15. Calculate the mean of the data set: 4, 5, 6, 8, 10.

    Mean = 6.6. Sum 33 divided by 5 data points.

16. State the formula for variance of a data set.

    Variance = $\frac{\text{sum of squared deviations from mean}}{n}$. Measures average squared distance from mean.

17. What does the range of a data set indicate?

    Spread of the data. Shows variability from minimum to maximum.

18. Find the mode of the data set: 3, 3, 4, 5, 5, 5.

    Mode = 5. Value 5 appears three times, most frequent.

19. What measure is used to summarize the dispersion of a data set?

    Standard deviation. Quantifies how spread out data values are.

20. What is the mean of the data set: 7, 9, 11, 13, 15?

    Mean = 11. Sum 55 divided by 5 data points.

21. What is the mean of the data set $2,4,6,8$?

    $5$. $(2+4+6+8)÷4 = 20÷4 = 5$.

22. What is the mean of a data set with values $x_1, x_2, \dots, x_n$?

    $\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}$. Sum all values and divide by count.

23. What is the first quartile (Q1) of the data set: 5, 7, 10, 15, 18?

    Q1 = 7. 25th percentile of ordered data.

24. State the formula for the range of a data set.

    Range = $\text{max value} - \text{min value}$. Difference between highest and lowest values.

25. How is the shape of a data set described?

    Using terms like symmetric, skewed left, skewed right. Describes distribution pattern and tail direction.

26. What is the formula for finding the z-score?

    z = $\frac{x - \text{mean}}{\text{standard deviation}}$. Standardizes values relative to distribution.

27. What is the five-number summary for a data set?

    $\min,\ Q_1,\ \text{median},\ Q_3,\ \max$. Five key values that summarize distribution.

28. How do you find the mode of a data set?

    Mode is the number that appears most frequently. Look for the value with highest frequency.

29. What is the median of the ordered data set $2,4,6,8$?

    $5$. Average of 4 and 6: $(4+6)÷2 = 5$.

30. What is the median of the data set: 3, 5, 7, 9, 11?

    7. Middle value when arranged in order.

31. What is the mode of a data set?

    The most frequently occurring value. Value that appears most often in the data.

32. What term describes a distribution skewed to the right?

    Positively skewed. Distribution tail extends toward higher values.

33. What is the interquartile range (IQR) in terms of quartiles?

    $\text{IQR}=Q_3-Q_1$. Spread of the middle 50% of data.

34. Identify the median of the data set: 12, 15, 22, 24, 30, 31.

    Median = (22 + 24) / 2 = 23. Average of two middle values in even-sized set.

35. Identify the median in a set: 1, 2, 3, 4, 5, 6, 7, 8.

    Median = (4 + 5) / 2 = 4.5. Average of 4th and 5th values.

36. What does the standard deviation measure in a data set?

    It measures the spread of data around the mean. Shows how far values deviate from average.

37. What does $Q_3$ represent in an ordered data set?

    The median of the upper half (the $75\%$ percentile). Third quartile divides top 25% from rest.

38. What is the standard deviation a measure of?

    Spread of data points around the mean. Measures variability or dispersion of data values.

39. What is the range of a data set?

    $\text{range}=\max-\min$. Difference between largest and smallest values.

40. Which measure describes the variability of a data set?

    Standard deviation. Measures how spread out data points are.

41. What is the formula for the standard deviation?

    Standard deviation = $\sqrt{\text{variance}}$. Square root of the variance value.

42. What does it mean if a z-score is negative?

    The value is below the mean. Negative z-scores indicate below-average values.

43. Identify the mode of the data set: 2, 2, 3, 4, 5.

    Mode = 2. Value 2 appears twice, most frequent.

44. What is the formula for the mean of a data set?

    Mean = $\frac{\text{sum of all data points}}{\text{number of data points}}$. Add all values and divide by count.

45. What is $\text{IQR}$ if $Q_1=12$ and $Q_3=20$?

    $8$. $20-12 = 8$.

46. What is the interquartile range (IQR)?

    IQR = Q3 - Q1, difference between third and first quartiles. Measures spread of middle 50% of data.

47. What is the midrange of the data set with $\min=10$ and $\max=22$?

    $16$. $(10+22)÷2 = 32÷2 = 16$.

48. Which distribution shape has mean $>$ median: left-skewed or right-skewed?

    Right-skewed. Right tail pulls mean above median.

49. What does $Q_1$ represent in an ordered data set?

    The median of the lower half (the $25\%$ percentile). First quartile divides bottom 25% from rest.

50. Identify the spread measure shown by the length of a box in a box-and-whisker plot.

    $\text{IQR}=Q_3-Q_1$. Box length represents interquartile range.

51. Calculate the range of the data set: 15, 22, 29.

    Range = 14. Maximum 29 minus minimum 15 equals 14.

52. Identify $Q_3$ for the ordered data set $2,4,6,8,10,12,14,16$.

    $13$. Median of upper half: 10,12,14,16.

53. If every value increases by $5$, how does the median change?

    Median increases by $5$. Adding constant shifts median by same amount.

54. What is the shape of a uniform distribution?

    Rectangular. All values occur with equal frequency.

55. What characteristic does the median measure?

    Central tendency. Describes typical or average value position.

56. Find the range of the data set: 2, 4, 6, 8.

    Range = 6. Maximum 8 minus minimum 2 equals 6.

57. What is a characteristic of a symmetric distribution?

    Mean and median are approximately equal. No skew means balanced distribution.

58. What is the outlier lower fence if $Q_1=4$ and $Q_3=12$?

    $-8$. $Q_1 - 1.5×IQR = 4 - 1.5×8 = -8$.

59. If $1,2,3,4$ is changed to $101,102,103,104$, which measure stays the same: range or mean?

    Range. Adding constant doesn't change spread measures.

60. Calculate the mean for the data: 5, 10, 15.

    Mean = 10. Sum (30) divided by count (3).

61. Which measure of central tendency is least affected by outliers?

    Median. Outliers don't affect middle position value.

62. If every value is multiplied by $-2$, how does the IQR change?

    IQR is multiplied by $2$. Absolute value of multiplier scales IQR.

63. Which option best describes a symmetric distribution in terms of mean and median?

    Mean equals median. Symmetric distributions have balanced tails.

64. What is the midrange of a data set in terms of min and max?

    $\text{midrange}=\frac{\min+\max}{2}$. Average of minimum and maximum values.

65. What is the median of an ordered data set with an odd number $n$ of values?

    The middle value, position $\frac{n+1}{2}$. For odd $n$, median is at position $\frac{n+1}{2}$.

66. What does a positively skewed distribution indicate?

    Tail on the right side is longer; mean > median. Right-skewed with outliers pulling mean up.

67. Identify the mode of the data set: 3, 3, 6, 9, 9, 12.

    Modes = 3 and 9. Both 3 and 9 appear twice.

68. What is the range of the data set $-2,0,3,7$?

    $9$. $7-(-2) = 9$.

69. If one extreme outlier is added, which changes less: mean or median?

    Median. Median is resistant to outliers, mean isn't.

70. What is the mean absolute deviation?

    Average of absolute deviations from the mean. Measures typical distance from the mean.

71. What is the standard deviation used to measure in a data set?

    Typical distance of values from the mean. Measures variability or spread around the mean.

72. What is the formula for variance?

    Variance = $\frac{\text{sum of squared deviations from mean}}{n}$. Sum of squared deviations divided by count.

73. What is the outlier cutoff rule using $Q_1$, $Q_3$, and $\text{IQR}$?

    Outliers $<Q_1-1.5\,\text{IQR}$ or $>Q_3+1.5\,\text{IQR}$. Values beyond 1.5 IQR from quartiles are outliers.

74. Determine the range of the data set: 6, 12, 18, 24.

    Range = 18. Maximum 24 minus minimum 6 equals 18.

75. What type of data does a dot plot represent?

    Frequency of individual data points. Shows individual values with stacked dots.

76. Which measure of center is most resistant to outliers: mean or median?

    Median. Median isn't affected by extreme values.

77. What is the median of the ordered data set $1,3,5,7,9$?

    $5$. Middle value of five ordered values.

78. Which distribution shape has mean $<$ median: left-skewed or right-skewed?

    Left-skewed. Left tail pulls mean below median.

79. Identify the third quartile (Q3) of the data set: 4, 5, 6, 7, 8, 9, 10.

    Q3 = 9. 75th percentile of ordered data values.

80. What is the purpose of a scatter plot?

    To show relationships between two variables. Displays correlation patterns between two variables.

81. What is the mean of the data set: 2, 4, 6, 8, 10?

    Mean = 6. Sum (30) divided by count (5).

82. What is a characteristic of a bimodal distribution?

    Two distinct peaks. Distribution has two separate high points.

83. Which measure of center is best for skewed data?

    Median. Resistant to extreme values' influence.

84. What is the outlier upper fence if $Q_1=4$ and $Q_3=12$?

    $24$. $Q_3 + 1.5×IQR = 12 + 1.5×8 = 24$.

85. If every value increases by $5$, how does the mean change?

    Mean increases by $5$. Adding constant shifts mean by same amount.

86. If every value is multiplied by $k>0$, how does the standard deviation change?

    Standard deviation is multiplied by $k$. Multiplying scales standard deviation by $|k|$.

87. What is the effect of an outlier on the mean?

    An outlier can significantly affect the mean. Extreme values pull the average away.

88. Identify $Q_1$ for the ordered data set $2,4,6,8,10,12,14,16$.

    $5$. Median of lower half: 2,4,6,8.

89. Identify the mode of the data set: 4, 5, 6, 6, 7.

    Mode = 6. Most frequently occurring value appears twice.

90. What is the shape of a normal distribution curve?

    Bell-shaped. Symmetric curve resembling a bell.

91. What does a histogram display?

    Frequency distribution of a data set. Shows how often values occur in intervals.

92. What does a negatively skewed distribution indicate?

    Tail on the left side is longer; mean < median. Left-skewed with outliers pulling mean down.

93. Calculate the median of the data set: 5, 8, 12, 14, 20.

    Median = 12. Middle value of five ordered data points.

94. What is the median of the data set: 3, 7, 9?

    Median = 7. Middle value when ordered: 3, 7, 9.

95. If every value is multiplied by $3$, how does the range change?

    Range is multiplied by $3$. Multiplying scales all spread measures.

96. How do you calculate the range of a data set?

    Subtract the smallest value from the largest value. Difference between maximum and minimum values.

97. Find the standard deviation of the data set: 4, 4, 4, 4.

    Standard deviation = 0. No variation means zero spread.

98. How is the spread of a data set measured?

    Using range, interquartile range, and standard deviation. Various measures quantify data variability.

99. What is the z-score of $x=86$ if $\mu=80$ and $\sigma=3$?

    $2$. $z = \frac{86-80}{3} = \frac{6}{3} = 2$.

100. Identify the range of the data set: 4, 8, 15, 16, 23, 42.

     Range = 42 - 4 = 38. Subtract minimum from maximum value.

101. What is the five-number summary for $1,2,3,4,5$?

     $1,\ 2,\ 3,\ 4,\ 5$. Min=1, $Q_1$=2, median=3, $Q_3$=4, max=5.

102. If $z=-1.5$, $\mu=50$, and $\sigma=4$, what is the value $x$?

     $44$. $x = 50 + (-1.5)(4) = 50 - 6 = 44$.

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