Comparing Data Sets by Center, Spread - Statistics
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State the formula for the interquartile range in terms of quartiles.
State the formula for the interquartile range in terms of quartiles.
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$IQR=Q_3-Q_1$. Difference between third and first quartiles.
$IQR=Q_3-Q_1$. Difference between third and first quartiles.
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State the formula for the outlier fences using the $1.5\times IQR$ rule.
State the formula for the outlier fences using the $1.5\times IQR$ rule.
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Lower: $Q_1-1.5(IQR)$; Upper: $Q_3+1.5(IQR)$. Values beyond these bounds are potential outliers.
Lower: $Q_1-1.5(IQR)$; Upper: $Q_3+1.5(IQR)$. Values beyond these bounds are potential outliers.
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What is the sample standard deviation formula in summation notation?
What is the sample standard deviation formula in summation notation?
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$s=\sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}$. Divides by $n-1$ for unbiased sample estimate.
$s=\sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}$. Divides by $n-1$ for unbiased sample estimate.
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What does standard deviation measure in a data set?
What does standard deviation measure in a data set?
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Typical distance of values from the mean. Average deviation from center, in original units.
Typical distance of values from the mean. Average deviation from center, in original units.
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Which comparison is more appropriate for skewed data: compare means and $s$, or compare medians and $IQR$?
Which comparison is more appropriate for skewed data: compare means and $s$, or compare medians and $IQR$?
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Compare medians and $IQR$. Use resistant measures when distributions aren't symmetric.
Compare medians and $IQR$. Use resistant measures when distributions aren't symmetric.
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Which comparison is more appropriate for symmetric data with no outliers: compare means and $s$, or compare medians and $IQR$?
Which comparison is more appropriate for symmetric data with no outliers: compare means and $s$, or compare medians and $IQR$?
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Compare means and $s$. Use non-resistant measures for well-behaved distributions.
Compare means and $s$. Use non-resistant measures for well-behaved distributions.
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What is the best measure of center for an approximately symmetric distribution with no outliers?
What is the best measure of center for an approximately symmetric distribution with no outliers?
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Mean. Best represents typical value when data is balanced around center.
Mean. Best represents typical value when data is balanced around center.
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What is the best measure of spread for a strongly skewed distribution?
What is the best measure of spread for a strongly skewed distribution?
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Interquartile range (IQR). Resistant to outliers since it only uses middle 50% of data.
Interquartile range (IQR). Resistant to outliers since it only uses middle 50% of data.
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What is the best measure of center for a strongly skewed distribution?
What is the best measure of center for a strongly skewed distribution?
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Median. Resistant to outliers and extreme values in the tail.
Median. Resistant to outliers and extreme values in the tail.
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Choose the correct summary to compare two skewed data sets: report $(\bar{x},s)$ or $(\text{median},IQR)$?
Choose the correct summary to compare two skewed data sets: report $(\bar{x},s)$ or $(\text{median},IQR)$?
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$(\text{median},IQR)$. Use resistant measures for skewed distributions.
$(\text{median},IQR)$. Use resistant measures for skewed distributions.
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Find $IQR$ if $Q_1=18$ and $Q_3=31$.
Find $IQR$ if $Q_1=18$ and $Q_3=31$.
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$IQR=13$. Apply formula: $31-18=13$.
$IQR=13$. Apply formula: $31-18=13$.
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Which data set is more consistent if $s_A=4.1$ and $s_B=6.0$ and both are symmetric?
Which data set is more consistent if $s_A=4.1$ and $s_B=6.0$ and both are symmetric?
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Data set $A$. Smaller standard deviation means less variability.
Data set $A$. Smaller standard deviation means less variability.
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Which data set has the larger spread if $IQR_A=12$ and $IQR_B=9$?
Which data set has the larger spread if $IQR_A=12$ and $IQR_B=9$?
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Data set $A$. Larger $IQR$ means more variability in middle 50%.
Data set $A$. Larger $IQR$ means more variability in middle 50%.
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Which data set has the larger center if $\text{median}_A=52$ and $\text{median}_B=49$?
Which data set has the larger center if $\text{median}_A=52$ and $\text{median}_B=49$?
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Data set $A$. Higher median indicates higher center location.
Data set $A$. Higher median indicates higher center location.
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Identify the correct spread measure if a distribution is skewed left with several low outliers.
Identify the correct spread measure if a distribution is skewed left with several low outliers.
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Interquartile range (IQR). Use resistant spread measure for skewed distributions.
Interquartile range (IQR). Use resistant spread measure for skewed distributions.
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Identify the correct center measure if a boxplot shows a long right whisker and a high outlier.
Identify the correct center measure if a boxplot shows a long right whisker and a high outlier.
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Median. Long whisker and outlier indicate right skew.
Median. Long whisker and outlier indicate right skew.
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What is the best measure of spread for an approximately symmetric distribution with no outliers?
What is the best measure of spread for an approximately symmetric distribution with no outliers?
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Standard deviation. Captures variability well when data follows bell-shaped pattern.
Standard deviation. Captures variability well when data follows bell-shaped pattern.
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Which pair is resistant to outliers: mean and standard deviation, or median and IQR?
Which pair is resistant to outliers: mean and standard deviation, or median and IQR?
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Median and IQR. Middle value and middle 50% range aren't pulled by extreme values.
Median and IQR. Middle value and middle 50% range aren't pulled by extreme values.
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Which pair is most affected by outliers: mean and standard deviation, or median and IQR?
Which pair is most affected by outliers: mean and standard deviation, or median and IQR?
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Mean and standard deviation. Outliers pull the mean and inflate the standard deviation.
Mean and standard deviation. Outliers pull the mean and inflate the standard deviation.
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Find the correct outlier cutoff: if $Q_1=10$ and $Q_3=18$, what is the lower fence?
Find the correct outlier cutoff: if $Q_1=10$ and $Q_3=18$, what is the lower fence?
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$-2$. $Q_1-1.5(IQR)=10-1.5(8)=-2$.
$-2$. $Q_1-1.5(IQR)=10-1.5(8)=-2$.
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What does it mean if a distribution has mean less than median?
What does it mean if a distribution has mean less than median?
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It is left-skewed. Long left tail pulls mean below median.
It is left-skewed. Long left tail pulls mean below median.
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What does it mean if a distribution has mean greater than median?
What does it mean if a distribution has mean greater than median?
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It is right-skewed. Long right tail pulls mean above median.
It is right-skewed. Long right tail pulls mean above median.
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Identify the correct measure of center for a bimodal distribution when comparing typical values across groups.
Identify the correct measure of center for a bimodal distribution when comparing typical values across groups.
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Median (mean may be misleading). Mean pulled by modes; median finds middle value.
Median (mean may be misleading). Mean pulled by modes; median finds middle value.
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Choose the correct conclusion: if two symmetric sets have equal means but $\sigma_A>\sigma_B$, which is more spread out?
Choose the correct conclusion: if two symmetric sets have equal means but $\sigma_A>\sigma_B$, which is more spread out?
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Data set $A$ is more spread out. Larger $\sigma$ means values vary more from center.
Data set $A$ is more spread out. Larger $\sigma$ means values vary more from center.
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Which option is more appropriate to compare typical values with an outlier present: mean or median?
Which option is more appropriate to compare typical values with an outlier present: mean or median?
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Median. Median resists the pull of outliers.
Median. Median resists the pull of outliers.
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What is the usual outlier rule using $Q_1$, $Q_3$, and $IQR$ for low outliers?
What is the usual outlier rule using $Q_1$, $Q_3$, and $IQR$ for low outliers?
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Low outlier if $x<Q_1-1.5(IQR)$. Values below this are unusually low.
Low outlier if $x<Q_1-1.5(IQR)$. Values below this are unusually low.
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Which option best compares centers when units differ: compare raw means, or compare z-scores?
Which option best compares centers when units differ: compare raw means, or compare z-scores?
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Compare z-scores. Standardizes values to same scale for comparison.
Compare z-scores. Standardizes values to same scale for comparison.
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What is the usual outlier rule using $Q_1$, $Q_3$, and $IQR$ for high outliers?
What is the usual outlier rule using $Q_1$, $Q_3$, and $IQR$ for high outliers?
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High outlier if $x>Q_3+1.5(IQR)$. Values beyond this are unusually high.
High outlier if $x>Q_3+1.5(IQR)$. Values beyond this are unusually high.
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Identify the correct comparison statement if both sets are right-skewed: use mean and $\sigma$, or median and $IQR$?
Identify the correct comparison statement if both sets are right-skewed: use mean and $\sigma$, or median and $IQR$?
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Use median and $IQR$. Skewed distributions require resistant measures.
Use median and $IQR$. Skewed distributions require resistant measures.
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Identify the correct interpretation: if $z=2$, how many standard deviations above the mean is $x$?
Identify the correct interpretation: if $z=2$, how many standard deviations above the mean is $x$?
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$2$ standard deviations above the mean. Z-score directly tells distance in $\sigma$ units.
$2$ standard deviations above the mean. Z-score directly tells distance in $\sigma$ units.
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