Comparing Data Sets - ACT Science

1. Define the range in a data set.

   Difference between the maximum and minimum values. Measures the spread of the data set.

2. Define standard deviation in the context of data sets.

   Standard deviation measures data spread around the mean. It quantifies how much individual data points deviate from the average.

3. What is the definition of an outlier when comparing two distributions or series?

   A value far from the overall pattern of the data. An unusual data point that doesn't fit the pattern.

4. Calculate the mean of the data set: 2, 4, 6, 8, 10.

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

5. What does it mean if one data set is a constant multiple of another across all $x$?

   They differ by a multiplicative factor (scaling). One is proportionally larger at all points.

6. Identify range: 1, 3, 5, 7, 10.

   9. Maximum (10) minus minimum (1).

7. Define frequency distribution in data analysis.

   A summary of how often values occur. It counts how many times each value appears.

8. In a data set, what is an outlier?

   A value significantly higher or lower than most others. Extreme value that deviates from the pattern.

9. What is the definition of a dependent variable when comparing two data sets?

   The measured response, often on the $y$-axis. The output variable that responds to changes.

10. What is a bimodal distribution?

    A distribution with two modes. Two values occur with the highest frequency.

11. Calculate the percent difference between $20$ and $24$ using \frac{|A-B|}{\frac{A+B}{2}}\times 100\text{%}.

    18.2\text{%}. $\frac{|20-24|}{22} \times 100\% = 18.2\%$.

12. What is the mode of the data set: 5, 5, 9, 14, 14, 14?

    14. 14 appears most frequently (3 times).

13. Identify the data set with greater variability: A: 3, 3, 3, 3, 3; B: 1, 5, 9, 13, 17.

    Set B has greater variability. Set A has no variation while Set B spans a wide range.

14. What is the meaning of a higher slope for data set A than data set B on the same axes?

    A changes faster per unit $x$ than B. Set A increases more rapidly than set B.

15. What is an outlier in a data set?

    A value significantly different from others. An extreme value that falls far from the typical range.

16. Define 'median' in statistics.

    Middle value when data is ordered. Central position in ordered data.

17. Identify the range of the data set: 4, 8, 10, 15, 22.

    18. Maximum (22) minus minimum (4).

18. Identify the correct conclusion if two data sets share a correlation with $x$ but differ in magnitude.

    They show similar trends but different levels (different $y$-values). Similar patterns but different scales or intensities.

19. What is the range of the data set: 10, 15, 20, 25, 30?

    The range is 20. Subtract the minimum (10) from the maximum (30).

20. What is the most direct way to compare rates of change of two data sets on a line graph?

    Compare slopes over the same $x$-interval. Calculate how steep each line is.

21. Identify the correct comparison if two bars represent different sample sizes $n$.

    Compare proportions or rates, not raw counts. Use ratios to account for different group sizes.

22. What is the correct interpretation if error bars do not overlap for two means?

    The means likely differ more than the displayed uncertainty. Non-overlapping bars suggest a clearer difference.

23. Calculate the mode: 3, 3, 6, 9.

    3. 3 appears twice, most frequent.

24. In a normal distribution, what percentage of data falls within one standard deviation of the mean?

    Approximately 68%. This follows the empirical rule for normal distributions.

25. Which option best describes a fair comparison when units differ (e.g., mg vs g)?

    Convert to the same units before comparing values. Use consistent units to make meaningful comparisons.

26. How do you find the median?

    Arrange data in order, pick the middle value. Standard method for finding median.

27. What is the definition of variability in a data set?

    How spread out the values are around a typical value. Measures how much the data points differ from each other.

28. Identify the median in the data set: 2, 4, 6, 8, 10.

    The median is 6. The middle value when data is arranged in order.

29. Find the slope between $(2,5)$ and $(6,13)$ using $\frac{y_2-y_1}{x_2-x_1}$.

    $2$. $\frac{13-5}{6-2} = \frac{8}{4} = 2$.

30. Identify which comparison is most appropriate for exponential growth across data sets.

    Compare multiplicative changes or slopes on a log scale. Log scales better show exponential growth patterns.

31. How is variance related to standard deviation?

    Variance is the square of the standard deviation. Variance equals standard deviation squared, or $σ^2$.

32. What is the mode of 1, 2, 2, 3, 4?

    2. 2 appears twice, most frequent.

33. What type of graph shows the cumulative frequency of data?

    Ogive or cumulative frequency graph. Also called a cumulative frequency curve.

34. What does it mean if two data sets have the same mean but different spread?

    They have similar centers but different variability. Central tendency is similar but consistency differs.

35. What is the mean of the data set: 3, 5, 7, 9, 11?

    The mean is 7. Sum all values and divide by the count of values.

36. What is the definition of a data set in an ACT Science table or graph?

    A collection of measured or observed values for one or more variables. Information gathered from measurements or observations.

37. What is a key characteristic of a normal distribution?

    Symmetry around the mean. The distribution is bell-shaped with equal sides around the center.

38. Which measure is resistant to outliers: mean, median, or mode?

    Median is resistant to outliers. Median doesn't change much when extreme values are present.

39. What is the effect of outliers on the mean of a data set?

    Outliers can skew the mean. Outliers pull the mean toward their extreme values.

40. What is the formula for percent change from an initial value $I$ to a final value $F$?

    $\frac{F-I}{I}\times 100\%$. Shows relative change from the starting value.

41. What is the formula for the range?

    Maximum value - Minimum value. Basic range calculation formula.

42. What is the definition of a trend when comparing data sets across an $x$-variable?

    A consistent pattern of increase, decrease, or no change. A pattern showing how values change consistently.

43. What does it mean if data is negatively skewed?

    Tail on the left side; mean < median. The distribution has a longer tail on the left side.

44. What measure is least affected by outliers?

    Median. Resistant to extreme values.

45. Calculate the IQR for the data set: 1, 3, 5, 7, 9.

    The IQR is 4. $Q^1 = 3$, $Q^3 = 7$, so $IQR = 7 - 3 = 4$.

46. In a positively skewed distribution, which is greater: mean or median?

    Mean is greater. Outliers in the right tail pull the mean higher.

47. How do you identify an outlier in a box plot?

    Points outside the whiskers. Whiskers extend to the most extreme non-outlier values.

48. What does a flat histogram indicate about data distribution?

    Uniform distribution. All values occur with approximately equal frequency.

49. What is the interquartile range (IQR) of the data set: 1, 3, 5, 7, 9?

    4. Third quartile (7) minus first quartile (3).

50. Identify the mean of: $1, 3, 5, 7$.

    4. Sum ($16$) divided by count ($4$).

51. What is the mode of 2, 4, 4, 6, 8?

    4. 4 appears twice, most frequent.

52. What does it mean if two data sets have the same shape but one is shifted upward?

    They differ mainly by an additive offset in $y$. Same pattern with a constant vertical displacement.

53. What is a negative correlation?

    As one variable increases, the other decreases. Both variables move in opposite directions to each other.

54. What does IQR stand for?

    Interquartile Range. Statistical abbreviation for quartile spread.

55. What is the median of the data set: 3, 7, 9, 15, 18?

    9. Middle value of the ordered set.

56. What is the most direct way to compare two data sets at the same $x$-value?

    Compare their $y$-values at that shared $x$. Look at the heights at the same horizontal position.

57. Find the absolute difference between $18$ and $11$ using $|A-B|$.

    $7$. $|18-11| = 7$.

58. What is the correct interpretation of overlapping error bars for two means?

    The difference may not be statistically clear from the graph alone. Overlap suggests the difference might not be significant.

59. Define correlation in the context of data analysis.

    A measure of the strength and direction of a relationship. Ranges from -1 to +1, indicating linear association strength.

60. How is the interquartile range calculated?

    Subtract the first quartile from the third quartile. Formula: $Q_3 - Q_1$.

61. What is the formula for the ratio of data set A to data set B at the same $x$?

    $\frac{A}{B}$. Divides one value by the other for comparison.

62. What is the key caution when comparing two data sets measured under different conditions?

    Differences may be due to confounding variables. Other factors may influence the observed differences.

63. Interpret a correlation coefficient of -0.8.

    Strong negative correlation. The variables have a strong inverse relationship.

64. What statistical measure is best for skewed data: mean, median, or mode?

    Median is best for skewed data. Median is less affected by extreme values than the mean.

65. What is the median of: 8, 10, 12, 14?

    11. Average of middle two values (10+12)/2.

66. What does a logarithmic axis imply when comparing equal vertical steps on a graph?

    Equal steps represent equal ratios, not equal differences. Spacing shows multiplicative rather than additive changes.

67. Which measure of central tendency is most affected by outliers?

    Mean. Outliers pull the mean away from the center.

68. What is the interquartile range (IQR)?

    IQR = Q3 - Q1. The range of the middle 50% of the data.

69. Find the mode: 7, 8, 9, 9, 10.

    9. 9 appears twice, most frequent.

70. At the same $x$, data set A has $y=12$ and B has $y=9$; what is $A-B$?

    $3$. $12 - 9 = 3$.

71. What is the definition of a control group or baseline data set in an experiment?

    The reference condition used for comparison. The standard used to compare against other groups.

72. What is the formula for calculating the interquartile range?

    Q3 - Q1. Standard IQR calculation notation.

73. What is the formula for percent difference between two values $A$ and $B$?

    $\frac{|A-B|}{\frac{A+B}{2}}\times 100\text{\%}$. Compares values relative to their average.

74. Find the mean: 4, 6, 8, 10.

    7. Sum (28) divided by count (4).

75. What is the purpose of a line graph?

    To show trends over time. It displays how data changes across time intervals.

76. What is the interquartile range for: 5, 10, 15, 20, 25?

    10. Third quartile (20) minus first quartile (10).

77. Calculate the mean: 5, 10, 15.

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

78. Calculate the percent change from $40$ to $50$ using \frac{F-I}{I}\times 100\text{%}.

    25\text{%}. $\frac{50-40}{40} \times 100\% = 25\%$.

79. Identify the mean of: 1, 3, 5, 7.

    4. Sum (16) divided by count (4).

80. What graphical representation is best for categorical data?

    Bar chart. Shows frequency of different categories or groups.

81. What is a positive correlation?

    As one variable increases, the other also increases. Both variables move in the same direction together.

82. Find the range: 3, 6, 8, 12, 20.

    17. Maximum (20) minus minimum (3).

83. What does a box plot visually represent?

    Distribution, median, quartiles, and outliers. Shows the five-number summary of a data set graphically.

84. Data set A increases from $6$ to $15$ while B increases from $4$ to $10$; which has larger absolute increase?

    Data set A. A increases by $9$, B increases by $6$.

85. What is the definition of an independent variable when comparing two data sets?

    The variable controlled or placed on the $x$-axis. The input variable that's manipulated or changed.

86. How is range calculated in a data set?

    Range = Maximum value - Minimum value. This formula measures the spread between the highest and lowest values.

87. Which is more robust to outliers: mean or median?

    Median. Median is less influenced by extreme values.

88. What does it mean if two data set curves intersect at an $x$-value?

    They have equal $y$-values at that $x$. The data sets have the same value at that point.

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

    Positively skewed. The tail extends toward larger values; mean > median.

90. What is the formula for the difference (A minus B) at the same $x$ when comparing data sets?

    $A-B$. Subtracts one value from the other.

91. What is the range of 12, 15, 18, 21, 24?

    12. Maximum (24) minus minimum (12).

92. Find the mode of this data set: 1, 2, 2, 3, 4.

    The mode is 2. The value 2 appears twice, more than any other value.

93. What is the mode of the data set: 4, 4, 5, 6, 7?

    The mode is 4. The value that appears most frequently in the data set.

94. What is the range of: 9, 15, 22, 25?

    16. Maximum (25) minus minimum (9).

95. Find the mean of 5, 10, 15, 20.

    12.5. Sum (50) divided by count (4).

96. What is the purpose of a stem-and-leaf plot?

    To display quantitative data distribution. It organizes data by stems (tens) and leaves (units).

97. What does it mean if two lines on a graph are parallel over an interval?

    They have the same slope (same rate of change). Both lines change at the same rate.

98. Calculate IQR for: 10, 20, 30, 40, 50.

    20. Third quartile (40) minus first quartile (20).

99. What does a high standard deviation indicate about a data set?

    High spread or variability from the mean. Data points are widely scattered from the average value.

100. What is the interquartile range of 2, 4, 6, 8, 10?

     4. Third quartile (8) minus first quartile (4).

101. What is the median of the ordered set $[2,4,7,9,12]$?

     $7$. The middle value when data is arranged in order.

102. What is a scatter plot used for?

     To display relationships between two variables. It visualizes how two quantitative variables relate to each other.

103. What is the formula for absolute difference between two values $A$ and $B$?

     $|A-B|$. Distance between values, always positive.

104. Define 'mode' in statistics.

     Most frequently occurring value in a data set. Value with highest frequency.

105. Identify the median of 6, 8, 10, 12.

     9. Average of middle two values (8+10)/2.

106. What is the best first step before comparing two graphs with different $y$-axis scales?

     Check axis labels, units, and scale intervals. Ensures accurate comparison of data values.

107. Calculate the mean of 4, 8, 10, 12, 14.

     The mean is 9.6. Sum the values (48) and divide by count (5).

108. What is the mean of the values $3,5,10,12$?

     $7.5$. Sum divided by count: $(3+5+10+12) ÷ 4 = 7.5$.

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

     Identify the most frequently occurring value. Count frequency of each value.

110. What is the formula for slope between points $ ($x_1$,$y_1$) $ and $ ($x_2$,$y_2$) $?

     $\frac{y_2-y_1}{x_2-x_1}$. Rise over run between two points.

111. What does a correlation coefficient of 0 indicate?

     No linear relationship. The variables have no linear association with each other.

112. What is the formula for calculating the mean?

     $\frac{\text{sum of values}}{\text{number of values}}$. Standard formula for arithmetic mean.

113. Identify the median of the data set: 11, 13, 15, 17, 19.

     15. Middle value of 5 ordered numbers.

114. How does a histogram differ from a bar chart?

     Histograms display frequency of numerical data. Bar charts show categories; histograms show continuous data ranges.

115. What is the median of 3, 7, 8, 12, 14?

     8. Middle value of 5 ordered numbers.

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