ACT 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 ACT 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: What is the mode of the data set $3,3,4,5,5,5,6$?

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

Flashcard 2: What is $\text{IQR}$ if $Q_1=12$ and $Q_3=20$?

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

Flashcard 3: Which measure of center is most resistant to outliers: mean or median?

Answer: Median. Median isn't affected by extreme values.

Flashcard 4: What does the term 'outlier' refer to in a data set?

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

Flashcard 5: How is the spread of a data set measured?

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

Flashcard 6: Which measure of spread is most resistant to outliers: range or IQR?

Answer: IQR. IQR isn't affected by extreme values.

Flashcard 7: What is the median of an ordered data set with an even number $n$ of values?

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

Flashcard 8: Identify the median of the data set: 11, 17, 19, 23, 27.

Answer: Median = 19. Middle value of five ordered numbers.

Flashcard 9: If every value increases by $5$, how does the mean change?

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

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

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

Flashcard 11: What is a box plot used for?

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

Flashcard 12: Identify the skewness of the data: 1, 2, 3, 4, 10.

Answer: Skewed right. Outlier 10 creates long right tail.

Flashcard 13: Calculate the mean of the data set: 10, 20, 30.

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

Flashcard 14: Identify the first quartile (Q1) of the data set: 10, 20, 30, 40, 50.

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

Flashcard 15: Which measure of central tendency represents the most frequent value?

Answer: Mode. Value appearing most often in dataset.

Flashcard 16: What is the median of an ordered data set with an odd number $n$ of values?

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

Flashcard 17: Calculate the range for the data: 9, 13, 21, 28.

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

Flashcard 18: Identify the median of the data set: 11, 17, 19, 23, 27.

Answer: Median = 19. Middle value of five ordered numbers.

Flashcard 19: Identify the first quartile (Q1) of the data set: 1, 3, 5, 7, 9.

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

Flashcard 20: What is a box plot used for?

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

Flashcard 21: What is the z-score formula for a value $x$ with mean $\mu$ and standard deviation $\sigma$?

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

Flashcard 22: State the effect of a constant addition on data mean.

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

Flashcard 23: Identify the skewness of the data: 1, 2, 3, 4, 10.

Answer: Skewed right. Outlier 10 creates long right tail.

Flashcard 24: Calculate the mean of the data set: 4, 5, 6, 8, 10.

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

Flashcard 25: State the formula for variance of a data set.

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

Flashcard 26: What does the range of a data set indicate?

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

Flashcard 27: Find the mode of the data set: 3, 3, 4, 5, 5, 5.

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

Flashcard 28: Find the standard deviation of the data set: 4, 4, 4, 4.

Answer: Standard deviation = 0. No variation means zero spread.

Flashcard 29: If every value is multiplied by $-2$, how does the IQR change?

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

Flashcard 30: State the formula for the range of a data set.

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

Flashcard 31: What does $Q_3$ represent in an ordered data set?

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

Flashcard 32: What measure is used to summarize the dispersion of a data set?

Answer: Standard deviation. Quantifies how spread out data values are.

Flashcard 33: Which distribution shape has mean $<$ median: left-skewed or right-skewed?

Answer: Left-skewed. Left tail pulls mean below median.

Flashcard 34: What is the formula for finding the z-score?

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

Flashcard 35: What is the mean absolute deviation?

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

Flashcard 36: State the formula for variance of a data set.

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

Flashcard 37: What is the mean of the data set: 7, 9, 11, 13, 15?

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

Flashcard 38: What is the median of the data set: 3, 5, 7, 9, 11?

Answer:

1. Middle value when arranged in order.

Flashcard 39: What is the mean of the data set $2,4,6,8$?

Answer: $5$. $(2+4+6+8)÷4 = 20÷4 = 5$.

Flashcard 40: What is the z-score of $x=86$ if $\mu=80$ and $\sigma=3$?

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

Flashcard 41: What is the formula for the standard deviation?

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

Flashcard 42: What measure is used to summarize the dispersion of a data set?

Answer: Standard deviation. Quantifies how spread out data values are.

Flashcard 43: How do you find the mode of a data set?

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

Flashcard 44: Calculate the range for the data: 9, 13, 21, 28.

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

Flashcard 45: How do you calculate the range of a data set?

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

Flashcard 46: What does a positively skewed distribution indicate?

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

Flashcard 47: What does $Q_1$ represent in an ordered data set?

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

Flashcard 48: Which measure of central tendency is least affected by outliers?

Answer: Median. Middle value remains stable with extreme values.

Flashcard 49: What is the mean of a data set with 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 count.

Flashcard 50: What is the first quartile (Q1) of the data set: 5, 7, 10, 15, 18?

Answer: Q1 = 7. 25th percentile of ordered data.

Flashcard 51: Find the range of the data set: 2, 4, 6, 8.

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

Flashcard 52: What does a negatively skewed distribution indicate?

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

Flashcard 53: State the formula for the range of a data set.

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

Flashcard 54: Calculate the mean of the data set: 4, 5, 6, 8, 10.

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

Flashcard 55: If every value is multiplied by $3$, how does the range change?

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

Flashcard 56: How is the shape of a data set described?

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

Flashcard 57: Identify the range of the data set: 4, 8, 15, 16, 23, 42.

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

Flashcard 58: What is the interquartile range (IQR)?

Answer: IQR = Q3 - Q1. Range of middle 50% of data values.

Flashcard 59: What is the standard deviation a measure of?

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

Flashcard 60: What is the median of the ordered data set $2,4,6,8$?

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

Flashcard 61: What is the formula for finding the z-score?

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

Flashcard 62: What is the five-number summary for a data set?

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

Flashcard 63: What is the outlier upper fence if $Q_1=4$ and $Q_3=12$?

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

Flashcard 64: What is the effect of an outlier on the mean?

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

Flashcard 65: How do you find the mode of a data set?

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

Flashcard 66: What is the median of the ordered data set $2,4,6,8$?

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

Flashcard 67: What is the median of the data set: 3, 5, 7, 9, 11?

Answer:

1. Middle value when arranged in order.

Flashcard 68: What is the mode of a data set?

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

Flashcard 69: What is the range of a data set?

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

Flashcard 70: What term describes a distribution skewed to the right?

Answer: Positively skewed. Distribution tail extends toward higher values.

Flashcard 71: How is the shape of a data set described?

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

Flashcard 72: Identify $Q_3$ for the ordered data set $2,4,6,8,10,12,14,16$.

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

Flashcard 73: What is the interquartile range (IQR) in terms of quartiles?

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

Flashcard 74: Which option best describes a symmetric distribution in terms of mean and median?

Answer: Mean equals median. Symmetric distributions have balanced tails.

Flashcard 75: Identify the median of the data set: 12, 15, 22, 24, 30, 31.

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

Flashcard 76: Identify the median in a set: 1, 2, 3, 4, 5, 6, 7, 8.

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

Flashcard 77: What does the standard deviation measure in a data set?

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

Flashcard 78: What does $Q_3$ represent in an ordered data set?

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

Flashcard 79: What is the standard deviation a measure of?

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

Flashcard 80: Identify the mode of the data set: 4, 5, 6, 6, 7.

Answer: Mode = 6. Most frequently occurring value appears twice.

Flashcard 81: What is the formula for the mean of a data set?

Answer: Mean = $\frac{\text{sum of all data points}}{\text{number of data points}}$. Formula divides total sum by count of values.

Flashcard 82: What is the range of a data set?

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

Flashcard 83: Which measure describes the variability of a data set?

Answer: Standard deviation. Measures how spread out data points are.

Flashcard 84: Identify $Q_1$ for the ordered data set $2,4,6,8,10,12,14,16$.

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

Flashcard 85: What is the formula for the standard deviation?

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

Flashcard 86: What is the purpose of a scatter plot?

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

Flashcard 87: What is the five-number summary for $1,2,3,4,5$?

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

Flashcard 88: What does it mean if a z-score is negative?

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

Flashcard 89: What is the first quartile (Q1) of the data set: 5, 7, 10, 15, 18?

Answer: Q1 = 7. 25th percentile of ordered data.

Flashcard 90: Identify the mode of the data set: 2, 2, 3, 4, 5.

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

Flashcard 91: What is the formula for the mean of a data set?

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

Flashcard 92: What is $\text{IQR}$ if $Q_1=12$ and $Q_3=20$?

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

Flashcard 93: What is the interquartile range (IQR)?

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

Flashcard 94: What is the shape of a uniform distribution?

Answer: Rectangular. All values occur with equal frequency.

Flashcard 95: What is the midrange of the data set with $\min=10$ and $\max=22$?

Answer: $16$. $(10+22)÷2 = 32÷2 = 16$.

Flashcard 96: Which distribution shape has mean $>$ median: left-skewed or right-skewed?

Answer: Right-skewed. Right tail pulls mean above median.

Flashcard 97: What does $Q_1$ represent in an ordered data set?

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

Flashcard 98: Which measure of central tendency is least affected by outliers?

Answer: Median. Outliers don't affect middle position value.

Flashcard 99: Which measure of center is best for skewed data?

Answer: Median. Resistant to extreme values' influence.

Flashcard 100: Identify the spread measure shown by the length of a box in a box-and-whisker plot.

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