Mistake Master
Means follow totals, medians follow position
Every summary statistic compresses the same data a different way, and the SAT tests whether you know which one does what. The arithmetic is small. The points slip away when an outlier is expected to move the wrong measure, when the median skips its sort, and when a perfectly computed mean answers a median question. Know each recipe, then report the one named.
§1
What this topic is about
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The mean, median, mode, and range each compress a data set into one number, and each answers a different question. This topic's points ride on three skills: knowing how the mean and median react when the data is lopsided, running the median's recipe exactly, and reporting the measure the question names.
§2
The mean tracks totals; the median tracks position
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The mean is the total divided by the count, so every value tugs on it, and an extreme value tugs hard. The median is the middle of the ordered list, so extremes barely touch it.
- One huge value drags the mean toward it and leaves the median nearly alone.
- Skewed-high data: mean above median. Skewed-low: mean below.
- For a typical value in lopsided data (salaries, home prices), trust the median.
Worked example. A data set has mean $20$ and median $19$. The value $90$ joins the set. What happens?
The mean absorbs the $90$ into its total and jumps.
The median slides at most one slot in the ordered list; it barely moves. The mean is the outlier-sensitive one.
§3
The median's recipe, exactly
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Two steps, both mandatory: sort, then take the middle. An even count has two middles, and the median is their average.
- Always sort first; the list's written order means nothing.
- Odd count: the single middle value.
- Even count: the mean of the two middle values, even when they differ.
Worked example. Find the median of $6, 11, 3, 9, 14, 5$.
$3, 5, 6, 9, 11, 14$.
$$\dfrac{6 + 9}{2} = 7.5.$$ Taking $6$ or $9$ alone, or the middle of the unsorted list, are the two classic slips.
§4
Work through the total
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Mean questions that add or remove values are total questions in disguise: recover the total, adjust it, divide by the new count.
- Total $=$ mean $\times$ count, always recoverable.
- After adding or removing values, divide by the NEW count.
- Removing a below-average value raises the mean; removing an above-average one lowers it.
Worked example. Five scores average $82$; the lowest, a $62$, is dropped. New mean?
$5 \cdot 82 = 410$, minus $62$ leaves $348$.
$$\dfrac{348}{4} = 87.$$ Dividing by the old $5$, or expecting the mean to hold still, are the slips this pattern sells.
§5
Report the measure that was named
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One data set produces a mean, a median, a mode, a range, a maximum. All of them will appear as answer choices. The question names exactly one.
The mean follows totals and outliers; the median follows position and the sort-then-middle recipe; and the final answer is whichever measure the question actually named.
§6
Three patterns that cost real points
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Three patterns recur on center-and-spread questions. They are the same ones the diagnostic routes on.
The mean and median swap reactions.
An outlier is expected to move the median, or the mean is expected to hold still while the total changes. Skewed data gets summarized by the wrong center.
Fix. Ask which mechanism each measure uses: the mean divides the TOTAL, the median finds the POSITION. Extremes hammer totals and barely graze positions.
The median recipe gets shortcut.
The middle of the unsorted list gets reported, or one of an even count's two middles stands in for their average.
Fix. Sort first, every time. Count the values: odd takes the middle, even averages the middle two.
The wrong measure gets reported.
The mean answers a median question, the mode answers a median question, the maximum answers a range question. Every alternative is sitting in the choices.
Fix. Underline the measure named, compute exactly that, and remember the range is the difference between the extremes, not either one of them.
Ten quick checks across the patterns: outlier behavior, the median recipe, working through totals, and reporting the named measure. Pick or type your answer, then check. Progress is saved.