Use statistics appropriate to the shape of the distribution to compare center and spread of two or more data sets (NC.M1.S-ID.2).

What this topic is asking

NC.M1.S-ID.2 asks you to compare the center and spread of two or more data sets, using statistics appropriate to the shape of the distribution. The key judgement is choosing the mean and standard deviation for symmetric data, or the median and IQR for skewed data or data with outliers.

Measures of center

Two common centers, with different sensitivities.

Measures of spread

Spread describes how variable the data is.

• Range: maximum minus minimum. Easily distorted by one extreme value.
• Interquartile range (IQR): $Q_3 - Q_1$, the spread of the middle half. Resistant to outliers.
• A larger spread means the data is more variable; a smaller spread means it clusters tightly.

Choosing the right measure

Comparing two data sets

To compare two groups, compare a center and a spread using the same measures for both. "Class A has a higher median test score but a larger IQR than Class B" says A typically scored higher but with more variability. The comparison should always name both center and spread.

A common EOC item shows two box plots stacked on the same axis and asks you to compare them. Read the median lines to compare center, and read the box widths (the IQRs) to compare spread. For example, if Class A's box plot has its median line at $82$ and Class B's at $75$, Class A's typical score is higher; if Class A's box runs from $78$ to $88$ (IQR $10$) while Class B's runs from $65$ to $90$ (IQR $25$), Class A's scores are also far more consistent. The box plot makes both comparisons visual: a box shifted right means a higher center, and a narrower box means a smaller spread. Always state the comparison in context, not just the numbers.

How the NC Math 1 EOC examines this topic

• Gridded response. Compute a mean, median, range, or IQR.
• Multiple choice. Choose the better measure of center, or compare two data sets.
• Technology-enhanced. Match summaries to box plots, or select all true comparison statements.

This builds on representing distributions, since shape decides the measure, and the idea of a resistant statistic recurs throughout statistics.

Why outliers force a choice of statistic

The mean treats every value equally, so a single extreme value can drag it far from the bulk of the data, while the median only cares about position, so an outlier barely moves it. The same split holds for spread: the range depends entirely on the two most extreme values, but the IQR ignores the outer quarters. This is why S-ID.2 ties the choice of statistic to the shape: in skewed or outlier-heavy data, the median and IQR tell the honest story, while the mean and range can mislead. Recognizing when an outlier is present, and switching to resistant measures, is the judgment the EOC is testing, not just the arithmetic of computing a mean.

Try this

Q1. Find the median of $2, 5, 9, 11$. [1 point]

• Cue. Even count: average the two middle values, $\frac{5 + 9}{2} = 7$.

Q2. A data set is strongly skewed right. Which center is more representative? [1 point]

• Cue. The median (the mean is pulled toward the high tail).