Use statistics appropriate to the shape of a distribution to compare center (mean, median) and spread (range, IQR, standard deviation), and interpret differences in context (TN A1.S.ID.A.2, A1.S.ID.A.3).

What this topic is asking

Standards A1.S.ID.A.2 and A1.S.ID.A.3 ask you to compare two data sets by center and spread, and to choose statistics that fit the shape. Center is the mean or median; spread is the range, interquartile range (IQR), or standard deviation. The big idea is that skew and outliers decide which statistics are appropriate.

Center: mean versus median

• Mean: add all values, divide by the count. Sensitive to every value, including outliers.
• Median: the middle of the ordered data. Resistant to outliers, since only the position matters.

When the data is skewed or has an outlier, the mean is dragged toward the tail while the median stays put. The data set $4, 6, 6, 8, 100$ has median $6$ but mean $24.8$, the lone $100$ distorts the mean.

Spread: range, IQR, standard deviation

• Range $=$ max $-$ min: simple but driven entirely by the two extremes.
• IQR $= Q_3 - Q_1$: the middle 50 percent's width; resists outliers.
• Standard deviation: the typical distance of values from the mean; larger means more spread. Like the mean, it is sensitive to outliers.

Choosing statistics by shape

The rule (A1.S.ID.A.3): match the statistic to the shape.

• Symmetric, no outliers -> mean and standard deviation (they describe it well).
• Skewed or with outliers -> median and IQR (they resist the distortion).

How TNReady examines this topic

• Numeric response. Compute a mean, median, range, or IQR.
• Multiple choice. Compare two sets' centers or spreads, or choose the appropriate statistic for a shape.
• Multiple select. Choose all true comparisons of two displays.

A clarifying idea is that this topic pairs with representing data: you read the shape from a histogram or box plot, then that shape tells you whether the mean or the median is the trustworthy center.

Why outliers change the choice of statistic

An outlier is a value far from the rest, and its effect on each statistic is what drives the standard's advice. The mean uses every value's exact size, so a single large outlier can pull it far from the bulk of the data, as the $100$ did above. The standard deviation, which squares each distance from the mean, is even more sensitive, one extreme value can inflate it dramatically. By contrast, the median depends only on the position of the middle value, so adding an outlier shifts it by at most one place, and the IQR ignores the outer 25 percent on each end entirely. This is why a report on incomes or house prices, which are usually right-skewed by a few very large values, uses the median: it reflects the typical case, while the mean would be misleadingly high. On the EOC, when a data set or display shows an obvious outlier or a clear skew, the appropriate choices are the median and IQR, and recognizing this is often the whole point of the item.

Try this

Q1. Find the median of $3, 7, 9, 12, 15$. [1 point]

• Cue. Middle value of the ordered set: $9$.

Q2. Two sets have the same mean, but Set X has IQR $5$ and Set Y has IQR $20$. Which is more spread out? [1 point]

• Cue. Set Y (larger IQR means more variability in the middle half).