Comparing Data Sets by Center, Spread - Statistics and Probability

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    Median. Median resists the pull of outliers.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    Median. Long whisker and outlier indicate right skew.

28. What does standard deviation measure in a data set?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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