PSAT Math Flashcards: Center Shape And Spread Of Data

What this deck covers

This deck focuses on Center Shape And Spread Of Data, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

Answer: Median. Outliers pull the mean but not the median.

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

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

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

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

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

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

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

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

Flashcard 7: What is the median of a data set?

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

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

Answer: $8$. $20-12=8$

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

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

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

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

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

Answer: $5$. $(2+4+6+8)÷4=20÷4=5$

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

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

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

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

Flashcard 14: What does $Q_1$ represent in a data set?

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

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

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

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

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

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

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

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

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

Flashcard 19: What is the median of a data set?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: Mean. High outliers pull the mean rightward.

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

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

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

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

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

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

Flashcard 36: What is the range of a data set?

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

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

Answer: IQR. Based on quartiles, it ignores extreme values.

Flashcard 38: What is the mean of a data set in terms of $n$ values $x_1, x_2, \dots, x_n$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 54: What does $Q_3$ represent in a data set?

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

Flashcard 55: What does $Q_1$ represent in a data set?

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

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

Answer: Mean. Sum all values and divide by the count.

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

Answer: Mean. Sum all values and divide by the count.

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

Answer: $5$. $(2+4+6+8)÷4=20÷4=5$.

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: IQR. Range includes outliers; IQR doesn't.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 78: What is the mode of a data set?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: 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.

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

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

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

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

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

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

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

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

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

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

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

Answer: IQR. IQR uses quartiles only; range uses extremes.

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

Answer: Mean. Average pulls toward extreme values.

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

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

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

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

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

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

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

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

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

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

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

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

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

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