Center, Shape, & Spread of Data - PSAT Math

1. Which measure of center is typically larger for a right-skewed distribution: mean or median?

   Mean. Right tail pulls the mean higher than the median.

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

   Median. Outliers pull the mean but not the median.

3. Find $\text{IQR}$ if $Q_1=12$ and $Q_3=20$.

   $20-12=8$. Subtract first quartile from third quartile.

4. What is the median of the data set $3,7,9,10,12$?

   $9$. The middle value of the ordered set is $9$.

5. Find the range of the data set $\{3,3,9,12\}$.

   $9$. Range = max - min = $12-3=9$

6. Find the median of the data set $1,2,5,9,10,13$.

   $\frac{5+9}{2}=7$. Average the 3rd and 4th values: $\frac{5+9}{2}=7$.

7. What is the median of a data set?

   The middle value when the data are ordered. For odd count, it's the center value; for even, average the two middle values.

8. Given $Q_1=12$ and $Q_3=20$, what is the value of $\text{IQR}$?

   $8$. $20-12=8$

9. What is the IQR for the sorted data $1,2,3,4,5,6,7,8$?

   $4$. $Q_3-Q_1=6.5-2.5=4$

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

    $z=\frac{x-\mu}{\sigma}$. Measures how many standard deviations from the mean.

11. Find the mean of the data set $\{2,4,6,8\}$.

    $5$. $(2+4+6+8)÷4=20÷4=5$

12. Find the range of the data set $\{12,5,9,20\}$.

    $15$. $\text{max}=20$, $\text{min}=5$, so $20-5=15$

13. Find the median of $3,8,2,9,10$.

    $8$. Sort: $2,3,8,9,10$; middle is $8$.

14. What does $Q_1$ represent in a data set?

    The $25$th percentile (median of the lower half). 25% of data falls below this value.

15. What is the median of a sorted data set with an even number of values?

    Average of the two middle values. When even count, no single middle exists.

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

    Min, $Q_1$, median, $Q_3$, max. These five values summarize the distribution's spread.

17. If every value in a data set increases by $5$, how does the mean change?

    The mean increases by $5$. Adding a constant to all values shifts the mean by that constant.

18. Find the mean of $4,6,10$.

    $\bar{x}=\frac{20}{3}$. $\frac{4+6+10}{3}=\frac{20}{3}$

19. What is the median of a data set?

    The middle value when the data are ordered. For even counts, average the two middle values.

20. Identify $\text{IQR}$ if $Q_1=12$ and $Q_3=27$.

    $27-12=15$. Subtract first quartile from third quartile.

21. Which shape best describes a distribution with a long left tail?

    Left-skewed (negatively skewed). Tail extends left with more low values.

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

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

23. Find the median of the ordered data set $2, 4, 7, 9, 11, 15$.

    $8$. Average of 3rd and 4th values: $(7+9)/2=8$.

24. Find the median of the ordered data set $\{1,3,7,9\}$.

    $5$. Average of middle two: $(3+7)÷2=5$

25. What is the effect on the mean if you multiply every data value by a constant $k$?

    The mean is multiplied by $k$. Scaling all values scales their average.

26. Identify the new IQR if every value in a data set is multiplied by $3$.

    IQR is multiplied by $3$. Multiplying all values scales spread measures by same factor.

27. In a box plot, what does the length of the box (from $Q_1$ to $Q_3$) represent?

    The interquartile range, $\text{IQR}$. Shows the spread of the middle 50% of data.

28. What is the interquartile range (IQR) in terms of $Q_1$ and $Q_3$?

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

29. Identify the skew: if the mean is greater than the median, what is the distribution shape?

    Right-skewed (positively skewed). Tail extends right, pulling mean higher.

30. What is the median of the data set $3,7,8,12,20$?

    $8$. Already sorted; middle of 5 values is the 3rd.

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

    Typical distance of values from the mean. Larger SD means more spread out data.

32. In a right-skewed distribution, which is typically larger: mean or median?

    Mean. High outliers pull the mean rightward.

33. Identify the range of the data set $-2,0,3,11$.

    $11-(-2)=13$. Subtract minimum from maximum: $11-(-2)$.

34. State the $z$-score formula for a value $x$ with mean $\mu$ and standard deviation $\sigma$.

    $z=\frac{x-\mu}{\sigma}$. Measures how many standard deviations from mean.

35. What is the effect on the mean if you add the same constant $c$ to every data value?

    The mean increases by $c$. Adding constant shifts all values equally.

36. What is the range of a data set?

    $\text{max}-\text{min}$. Subtract the smallest value from the largest.

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

    IQR. Based on quartiles, it ignores extreme values.

38. What is the mean of a data set in terms of $n$ 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 the count.

39. Which measure of spread is more resistant to outliers: range or IQR?

    IQR. Outliers don't affect the middle 50% of data.

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

    $5$. $\frac{2+4+6+8}{4}=\frac{20}{4}=5$

41. What does it mean if a distribution is skewed left?

    Long tail to the left (toward smaller values). Most data clusters on the right with fewer low values.

42. Identify the skew: a distribution has a long tail to the right.

    Right-skewed (positively skewed). More data on left, tail extends right.

43. Identify the shape: in a right-skewed distribution, which is larger, mean or median?

    Mean is larger than median. Right tail pulls mean higher than median.

44. If every value in a data set is multiplied by $3$, how does the standard deviation change?

    The standard deviation is multiplied by $3$. Multiplying all values scales the spread by the same factor.

45. Which measure of spread is more resistant to outliers: range or $\text{IQR}$?

    $\text{IQR}$. IQR ignores extremes, while range includes them.

46. What are the five-number summary values used for a box plot?

    Minimum, $Q_1$, median, $Q_3$, maximum. These five values divide the data into four equal parts.

47. What do $Q_1$ and $Q_3$ represent in a data set?

    $Q_1$: $25$th percentile; $Q_3$: $75$th percentile. Quartiles divide sorted data into four equal parts.

48. Which measure is more resistant to extreme values: mean or median?

    Median. Outliers affect mean but not median's position.

49. What is the relationship between mean and median in a left-skewed distribution?

    Mean $<$ median. Left tail pulls mean below median.

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

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

51. What is the effect on the $\text{IQR}$ if you multiply every data value by a constant $k$?

    $\text{IQR}$ is multiplied by $k$. Scaling all values scales their spread.

52. Given $Q_1=10$ and $Q_3=18$, what is the upper outlier fence $Q_3+1.5(IQR)$?

    $30$. $18+1.5(8)=18+12=30$.

53. Given $Q_1=10$ and $Q_3=18$, what are the $1.5\,\text{IQR}$ fences?

    Lower $=-2$, upper $=30$. $Q_1-1.5(8)=10-12=-2$; $Q_3+1.5(8)=18+12=30$

54. What does $Q_3$ represent in a data set?

    The $75$th percentile (median of the upper half). 75% of data falls below this value.

55. What does $Q_1$ represent in a data set?

    The $25$th percentile (lower quartile). 25% of data falls below this value.

56. Which measure of center is the arithmetic average of a data set?

    Mean. Sum all values and divide by the count.

57. Which measure of center is the arithmetic average of all data values?

    Mean. Sum all values and divide by the count.

58. Find the mean of the data set $2,4,6,8$.

    $5$. $(2+4+6+8)÷4=20÷4=5$.

59. What is the outlier rule using $IQR$ for the upper fence?

    Upper fence $=Q_3+1.5(IQR)$. Values above this are potential outliers.

60. Find the IQR for $Q_1=12$ and $Q_3=20$.

    $8$. $IQR=Q_3-Q_1=20-12=8$

61. What is typically true about mean vs. median in a left-skewed distribution?

    Mean $<$ median. Left tail pulls mean below median.

62. What is the outlier fence rule using $Q_1$, $Q_3$, and IQR?

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

63. Find the range of $[3,3,9,12,15]$.

    $15-3=12$. Subtract smallest value from largest value.

64. Identify the median of the ordered data set $3,7,9,12,20$.

    $9$. With 5 values, the 3rd position holds the median.

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

    IQR. Range includes outliers; IQR doesn't.

66. What is the interquartile range (IQR) in terms of quartiles $Q_1$ and $Q_3$?

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

67. What is the mean of a data set in terms of a sum and the number of values $n$?

    $\text{mean}=\frac{\text{sum of values}}{n}$. Divide the total of all values by the count of values.

68. Identify the skew: if the mean is less than the median, what is the distribution shape?

    Left-skewed (negatively skewed). Tail extends left, pulling mean lower.

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

    Median. Extreme values don't affect the middle position.

70. Compute the $z$-score for $x=18$ when $\mu=12$ and $\sigma=3$.

    $\frac{18-12}{3}=2$. Value is 2 standard deviations above mean.

71. Identify the median of $[2,5,7,10,13]$.

    $7$. Middle value of 5 ordered values is the 3rd one.

72. What does $Q_3$ represent in a sorted data set?

    The median of the upper half (the $75$th percentile). Splits the top 25% from the rest.

73. Find $\text{IQR}$ given $Q_1=12$ and $Q_3=20$.

    $8$. Apply IQR formula: $20-12=8$

74. What is the median of an ordered data set with an odd number of values?

    The single middle value in the ordered list. With odd count, the middle value stands alone.

75. Find the median of the data set $3,7,9,10,12$.

    $9$. Middle value of ordered list is $9$.

76. What is the mean of a data set in terms of the sum and number of values?

    $\text{mean}=\frac{\text{sum of values}}{\text{number of values}}$. Divide the total by how many values there are.

77. Using $Q_1=10$, $Q_3=18$, what are the outlier fences?

    Lower $=-2$, upper $=30$. $Q_1-1.5(8)=-2$, $Q_3+1.5(8)=30$ where IQR$=8$.

78. What is the mode of a data set?

    The value that occurs most frequently. The value with the highest frequency.

79. Identify the median of $[1,4,6,8]$.

    $\frac{6+4}{2}=5$. Even count: average the 2nd and 3rd values.

80. Identify the shape: if the right tail is longer, is the distribution left- or right-skewed?

    Right-skewed (positively skewed). Longer right tail indicates positive skew.

81. Find the median of the ordered data set $3,5,7,9,11,13$.

    $8$. Average of 3rd and 4th values: $\frac{7+9}{2}=8$

82. What does it mean if a distribution is skewed right?

    Long tail to the right (toward larger values). Right skew means more extreme values on the high end.

83. What does a right-skewed distribution look like in terms of tail direction?

    Long tail to the right. Most data clustered on left with outliers extending right.

84. What does it mean if a distribution is skewed left?

    Long tail to the left (toward smaller values). Left skew means more extreme values on the low end.

85. Identify the skew if a distribution has a long tail to the left.

    Left-skewed (negatively skewed). Mean pulled toward the tail.

86. What is the outlier rule using IQR (lower and upper fences)?

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

87. Identify the center line in a box plot: which statistic does it represent?

    Median. The box plot's center line shows the middle value.

88. Which measure of spread is more resistant to outliers: range or $IQR$?

    $IQR$. IQR ignores extreme values, unlike range.

89. Identify the median of the ordered data set $1,5,6,10$.

    $\frac{5+6}{2}=5.5$. Average the 2nd and 3rd values: $\frac{5+6}{2}$.

90. Given $Q_1=12$ and $Q_3=20$, what is the interquartile range?

    $8$. $Q_3-Q_1=20-12=8$.

91. Find $\text{IQR}$ for the sorted data $1,2,3,4,5,6,7,8$ using median-of-halves quartiles.

    $4$. $Q_1=2.5$, $Q_3=6.5$; $6.5-2.5=4$.

92. Which spread measure is more resistant to outliers: range or IQR?

    IQR. IQR uses quartiles only; range uses extremes.

93. Which measure is more affected by extreme values: mean or median?

    Mean. Average pulls toward extreme values.

94. Using $Q_1=10$, $Q_3=22$, identify the upper outlier cutoff value.

    $40$. $Q_3+1.5(Q_3-Q_1)=22+1.5(12)=40$

95. What is the standard deviation conceptually (what does it measure)?

    Typical distance of values from the mean. Measures how spread out data points are from their average.

96. What are the five-number summary values for a box plot?

    Minimum, $Q_1$, median, $Q_3$, maximum. These five values divide data into four equal parts.

97. Given $Q_1=10$ and $Q_3=18$, what is $IQR$?

    $8$. $IQR=Q_3-Q_1=18-10=8$.

98. What does it mean if a distribution is left-skewed?

    It has a longer tail to the left. Most data clustered on right, few low values.

99. What is the median of a data set with an even number of values?

    Average of the two middle sorted values. Order the data and find the mean of the two center values.

100. What is the median of a data set with an even number of values?

     The average of the two middle sorted values. No single middle value, so average the two center ones.

101. Identify the mean of the data set $2,4,6,8$.

     $5$. $\frac{2+4+6+8}{4}=\frac{20}{4}=5$

102. Which measure of spread is most resistant to outliers: range or $\text{IQR}$?

     $\text{IQR}$. Based on quartiles, unaffected by extreme values.

103. What is the outlier 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 these bounds are unusual.

104. What is the outlier rule using $\text{IQR}$ (lower and upper fences)?

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

105. Identify $Q_3$ for the ordered data $[2,4,6,8,10,12]$ using the median-of-halves method.

     $10$. Median of upper half: $[8,10,12]$, so $Q_3=10$.

106. Given $Q_1=10$ and $\text{IQR}=6$, what is the lower outlier fence?

     $1$. $Q_1-1.5(\text{IQR})=10-1.5(6)=10-9=1$

107. What does standard deviation measure about a data set?

     Typical distance of values from the mean. Measures how spread out data points are.

108. State the formula for the mean of values $x_1, x_2, \dots, x_n$.

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

109. Given $Q_1=10$ and $Q_3=18$, what is the interquartile range?

     $8$. IQR = $Q_3-Q_1=18-10=8$

110. Find the median of the ordered data set $3, 5, 8, 10, 12$.

     $8$. Middle value of 5 ordered values is the 3rd one.

111. What does it mean if a distribution is symmetric?

     Left and right sides are roughly mirror images. Symmetric means balanced distribution around the center.

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

     $Q_3-Q_1$. Third quartile minus first quartile.

113. What does $Q_1$ represent in a sorted data set?

     The median of the lower half (the $25$th percentile). Splits the bottom 25% from the rest.

114. Identify $Q_1$ for the ordered data $[2,4,6,8,10,12]$ using the median-of-halves method.

     $4$. Median of lower half: $[2,4,6]$, so $Q_1=4$.

115. In a left-skewed distribution, which is typically larger: mean or median?

     Median. Low outliers pull the mean leftward.

116. What is the outlier rule using $IQR$ for the lower fence?

     Lower fence $=Q_1-1.5(IQR)$. Values below this are potential outliers.

117. What is the median of an ordered data set with an even number of values?

     Average of the two middle values. With even count, average the two center values.

118. Identify the shape: in a left-skewed distribution, which is larger, mean or median?

     Median is larger than mean. Left tail pulls mean lower than median.

119. Identify $Q_1$ and $Q_3$ for the sorted data $1,2,3,4,5,6,7,8$.

     $Q_1=2.5,\ Q_3=6.5$. $Q_1$ is between positions 2-3, $Q_3$ is between 6-7.

120. What does it mean if a distribution is skewed right?

     Long tail to the right (toward larger values). Most data clusters on the left with fewer high values.

121. Which measure of center is more resistant to outliers: mean or median?

     Median. Outliers affect mean more than median.

122. Which measure is more resistant to outliers: mean or median?

     Median. Median ignores extreme values, mean includes them.

123. What is the outlier rule using $\text{IQR}$?

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

124. Find the IQR for the ordered data $[2,4,6,8,10,12]$ using the median-of-halves method.

     $10-4=6$. Subtract first quartile from third quartile.

125. Find the range of $-2,5,7,7,9$.

     $11$. $9-(-2)=11$

126. Which measure is more resistant to outliers: range or IQR?

     IQR. IQR uses only middle 50%, unaffected by extreme values.

127. What is the interquartile range (IQR) in terms of quartiles $Q_1$ and $Q_3$?

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

128. What is the median of a data set with an odd number of values?

     The middle value after sorting. Sort first, then select the center value.

129. Find the median of the ordered data set $\{1,3,7,10\}$.

     $5$. Average the two middle values: $\frac{3+7}{2}=5$

130. What is true about mean and median in a symmetric distribution?

     Mean equals median. In symmetric data, center measures align.

131. What is the interquartile range (IQR) in terms of $Q_1$ and $Q_3$?

     $\text{IQR}=Q_3-Q_1$. Difference between third and first quartiles.

132. What is the effect on the median if you add the same constant $c$ to every data value?

     The median increases by $c$. Adding constant shifts all values equally.

133. Identify the skew: a distribution has a long tail to the left.

     Left-skewed (negatively skewed). More data on right, tail extends left.

134. What does a left-skewed distribution look like in terms of tail direction?

     Long tail to the left. Most data clustered on right with outliers extending left.

135. State the outlier rule using the $1.5\times\text{IQR}$ fences.

     Outlier if $<Q_1-1.5\,\text{IQR}$ or $>Q_3+1.5\,\text{IQR}$. Values beyond these fences are considered unusual.

136. Identify the skew for a distribution where mean $<$ median $<$ mode.

     Skewed left (negatively skewed). Tail extends left, pulling mean below median.

137. What is the mean of the data set $2,4,4,10$?

     $5$. $\frac{2+4+4+10}{4}=\frac{20}{4}=5$

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

     $\text{range}=\text{max}-\text{min}$. Measures total spread from smallest to largest value.

139. What is the outlier 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 IQRs from quartiles are outliers.

140. What is the median of the data set $1,5,6,10$?

     $\frac{11}{2}$. Average of 5 and 6: $\frac{5+6}{2}=\frac{11}{2}$

141. What does $Q_3$ represent in a data set?

     The $75$th percentile (upper quartile). 75% of data falls below this value.

142. What is the five-number summary of a data set?

     $\text{min},\ Q_1,\ \text{median},\ Q_3,\ \text{max}$. Lists the key values that divide the data into quarters.

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

     $\text{min},\ Q_1,\ \text{median},\ Q_3,\ \text{max}$. Lists key positions: minimum, first quartile, median, third quartile, maximum.

144. State the formula for the mean of $n$ values $x_1,\dots,x_n$.

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

145. Which shape best describes a distribution with a long right tail?

     Right-skewed (positively skewed). Tail extends right with more high values.

146. What does it mean if a distribution is right-skewed?

     It has a longer tail to the right. Most data clustered on left, few high values.

147. Find the mean of the data set $\{2,4,6,8\}$.

     $5$. $\bar{x}=\frac{2+4+6+8}{4}=\frac{20}{4}=5$

148. Identify the shape: mean $<$ median $<$ mode. What is the skew?

     Left-skewed (negatively skewed). Tail extends left; extreme low values pull mean down.

149. What is the outlier rule using $\text{IQR}$ for low and high outliers?

     Low $<Q_1-1.5\text{IQR}$; High $>Q_3+1.5\text{IQR}$. Values beyond 1.5 IQRs from quartiles are outliers.

150. A data set multiplies every value by $3$. How do the range and IQR change?

     Range $\times 3$; IQR $\times 3$. Multiplying scales spread measures by same factor.

151. What is the median of the data set $1,2,5,9$?

     $\frac{7}{2}$. Average of middle two: $\frac{2+5}{2}=\frac{7}{2}$

152. Identify the skew for a distribution where mean $>$ median $>$ mode.

     Skewed right (positively skewed). Tail extends right, pulling mean above median.

153. What is the mean of a data set in terms of the sum of values and $n$?

     Mean $=\frac{\text{sum of values}}{n}$. Divide the total of all values by the count of values.

154. Which measure is more resistant to outliers: mean or median?

     Median. Middle value unaffected by extreme values.

155. What do $Q_1$ and $Q_3$ represent in a data set?

     $Q_1$ is $25\%$; $Q_3$ is $75\%$ percentile. First and third quartiles divide data into quarters.

156. Find the mean of the data set $4, 6, 8, 12$.

     $7.5$. $(4+6+8+12)/4=30/4=7.5$.

157. What is the interquartile range (IQR) in terms of quartiles $Q_1$ and $Q_3$?

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

158. What is a resistant measure of center: mean or median?

     Median. Median isn't affected by extreme values.

159. Identify the shape: mean $>$ median $>$ mode. What is the skew?

     Right-skewed (positively skewed). Tail extends right; extreme high values pull mean up.

160. Identify whether $35$ is a high outlier if $Q_1=10$, $Q_3=18$, and the value is $35$.

     Yes, since $35>18+1.5(8)=30$. $IQR=18-10=8$; outlier if $>18+1.5(8)=30$

161. What is the range of the data set $-2,0,5,9$?

     $11$. $9-(-2)=11$

162. Find the range of the data set $-2,0,5,11$.

     $11-(-2)=13$. Subtract minimum from maximum: $11-(-2)=13$.

163. What is the relationship between mean and median in a right-skewed distribution?

     Mean $>$ median. Right tail pulls mean above median.

164. Find $\text{IQR}$ for the five-number summary: min $2$, $Q_1=5$, median $7$, $Q_3=12$, max $20$.

     $7$. $\text{IQR}=Q_3-Q_1=12-5=7$

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

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

166. A distribution is right-skewed. Which is larger: mean or median?

     Mean. The tail pulls the mean toward higher values.

167. What is typically true about mean vs. median in a right-skewed distribution?

     Mean $>$ median. Right tail pulls mean above median.

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

     Typical distance of values from the mean. Measures how spread out data points are.

169. Identify the skew if a distribution has a long tail to the right.

     Right-skewed (positively skewed). Tail direction indicates skew direction.

170. Identify the new mean if every value in a data set increases by $5$.

     Mean increases by $5$. Adding constant to all values shifts mean by same constant.

171. What is the effect on the range if you add the same constant $c$ to every data value?

     The range stays the same. Differences between values remain unchanged.

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

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

173. What is the median of a data set with an odd number of values?

     The middle value after sorting the data. Sort first, then pick the one in the middle.

174. Find the median of the data set $1,2,6,9$.

     $4$. Average of middle two: $(2+6)÷2=4$.

175. Find the range of the data set $-2, 0, 5, 9, 9$.

     $11$. $9-(-2)=11$.

176. Which measure of spread is more resistant to outliers: range or $\text{IQR}$?

     $\text{IQR}$. IQR ignores extremes, focusing on middle 50%.

177. What is a resistant measure of spread: range or IQR?

     IQR. IQR ignores extreme values, unlike range.

178. A data set adds $5$ to every value. How do the mean and median change?

     Mean $+5$; median $+5$. Adding constant shifts center measures equally.

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

     Median. Median uses middle position, unaffected by extreme values.

180. What is the median of a data set with an odd number of values?

     The middle value after sorting. Order the data and select the center value.

181. Find the range of the data set $-2,5,11,4$.

     $13$. $\text{max}(11)-\text{min}(-2)=11-(-2)=13$.

182. Identify the shape: if the left tail is longer, is the distribution left- or right-skewed?

     Left-skewed (negatively skewed). Longer left tail indicates negative skew.

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