Statistics Flashcards: Comparing Data Sets By Center Spread

What this deck covers

This deck focuses on Comparing Data Sets By Center Spread, giving you a quick way to review the definitions, rules, and examples that matter most for Statistics.

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: Identify the correct interpretation: if $z=2$, how many standard deviations above the mean is $x$?

Answer: $2$ standard deviations above the mean. Z-score directly tells distance in $\sigma$ units.

Flashcard 2: Which pair is most affected by outliers: mean and standard deviation, or median and IQR?

Answer: Mean and standard deviation. Outliers pull the mean and inflate the standard deviation.

Flashcard 3: Find the correct outlier cutoff: if $Q_1=10$ and $Q_3=18$, what is the lower fence?

Answer: $-2$. $Q_1-1.5(IQR)=10-1.5(8)=-2$.

Flashcard 4: State the formula for the outlier fences using the $1.5\times IQR$ rule.

Answer: Lower: $Q_1-1.5(IQR)$; Upper: $Q_3+1.5(IQR)$. Values beyond these bounds are potential outliers.

Flashcard 5: What is the best pair to report for a roughly symmetric distribution with no outliers?

Answer: Mean and standard deviation. Best describes center and spread for normal-like data.

Flashcard 6: Which comparison is more appropriate for skewed data: compare means and $s$, or compare medians and $IQR$?

Answer: Compare medians and $IQR$. Use resistant measures when distributions aren't symmetric.

Flashcard 7: Which option indicates more variability: $\sigma_A=4.1$ or $\sigma_B=3.8$?

Answer: Data set $A$. Larger standard deviation means more spread from mean.

Flashcard 8: Which data set is more consistent if $s_A=4.1$ and $s_B=6.0$ and both are symmetric?

Answer: Data set $A$. Smaller standard deviation means less variability.

Flashcard 9: What is the sample standard deviation formula in summation notation?

Answer: $s=\sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}$. Divides by $n-1$ for unbiased sample estimate.

Flashcard 10: Which option is more appropriate to compare typical values with an outlier present: mean or median?

Answer: Median. Median resists the pull of outliers.

Flashcard 11: Which comparison is more appropriate for symmetric data with no outliers: compare means and $s$, or compare medians and $IQR$?

Answer: Compare means and $s$. Use non-resistant measures for well-behaved distributions.

Flashcard 12: What is the usual outlier rule using $Q_1$, $Q_3$, and $IQR$ for high outliers?

Answer: High outlier if $x>Q_3+1.5(IQR)$. Values beyond this are unusually high.

Flashcard 13: Which statistics are most resistant to outliers: mean and standard deviation, or median and IQR?

Answer: Median and IQR. Outliers strongly affect mean and standard deviation.

Flashcard 14: What is the best measure of spread for a strongly skewed distribution?

Answer: Interquartile range (IQR). Resistant to outliers since it only uses middle 50% of data.

Flashcard 15: What is the best measure of center for an approximately symmetric distribution with no outliers?

Answer: Mean. Best represents typical value when data is balanced around center.

Flashcard 16: State the formula for the interquartile range in terms of quartiles.

Answer: $IQR=Q_3-Q_1$. Difference between third and first quartiles.

Flashcard 17: Which pair is resistant to outliers: mean and standard deviation, or median and IQR?

Answer: Median and IQR. Middle value and middle 50% range aren't pulled by extreme values.

Flashcard 18: Choose the correct conclusion: if two symmetric sets have equal means but $\sigma_A>\sigma_B$, which is more spread out?

Answer: Data set $A$ is more spread out. Larger $\sigma$ means values vary more from center.

Flashcard 19: Which option best compares centers when units differ: compare raw means, or compare z-scores?

Answer: Compare z-scores. Standardizes values to same scale for comparison.

Flashcard 20: Identify the correct comparison statement if both sets are right-skewed: use mean and $\sigma$, or median and $IQR$?

Answer: Use median and $IQR$. Skewed distributions require resistant measures.

Flashcard 21: Which data set has the larger spread if $IQR_A=12$ and $IQR_B=9$?

Answer: Data set $A$. Larger $IQR$ means more variability in middle 50%.

Flashcard 22: Choose the correct summary to compare two skewed data sets: report $(\bar{x},s)$ or $(\text{median},IQR)$?

Answer: $(\text{median},IQR)$. Use resistant measures for skewed distributions.

Flashcard 23: What is the usual outlier rule using $Q_1$, $Q_3$, and $IQR$ for low outliers?

Answer: Low outlier if $x<Q_1-1.5(IQR)$. Values below this are unusually low.

Flashcard 24: What is the best measure of center for a strongly skewed distribution?

Answer: Median. Resistant to outliers and extreme values in the tail.

Flashcard 25: What is the best measure of spread for an approximately symmetric distribution with no outliers?

Answer: Standard deviation. Captures variability well when data follows bell-shaped pattern.

Flashcard 26: Find $IQR$ if $Q_1=18$ and $Q_3=31$.

Answer: $IQR=13$. Apply formula: $31-18=13$.

Flashcard 27: Identify the correct center measure if a boxplot shows a long right whisker and a high outlier.

Answer: Median. Long whisker and outlier indicate right skew.

Flashcard 28: What does standard deviation measure in a data set?

Answer: Typical distance of values from the mean. Average deviation from center, in original units.

Flashcard 29: Identify the correct measure of center for a bimodal distribution when comparing typical values across groups.

Answer: Median (mean may be misleading). Mean pulled by modes; median finds middle value.

Flashcard 30: What does it mean if a distribution has mean less than median?

Answer: It is left-skewed. Long left tail pulls mean below median.

Flashcard 31: What does it mean if a distribution has mean greater than median?

Answer: It is right-skewed. Long right tail pulls mean above median.

Flashcard 32: Find the correct outlier cutoff: if $Q_1=10$ and $Q_3=18$, what is the upper fence?

Answer: $30$. $Q_3+1.5(IQR)=18+1.5(8)=30$.

Flashcard 33: Which data set has the larger center if $\text{median}_A=52$ and $\text{median}_B=49$?

Answer: Data set $A$. Higher median indicates higher center location.

Flashcard 34: Which option indicates a larger spread: $IQR_A=12$ or $IQR_B=9$?

Answer: Data set $A$. Larger IQR indicates greater spread in the middle 50%.

Flashcard 35: Identify the correct spread measure if a distribution is skewed left with several low outliers.

Answer: Interquartile range (IQR). Use resistant spread measure for skewed distributions.

Flashcard 36: State the z-score formula for a value $x$ using mean and standard deviation.

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