Use measures of center (mean, median) and spread (range, interquartile range) to describe and compare data sets, and account for the effect of outliers (LA A1: S-ID.A.2, S-ID.A.3).

What this topic is asking

Standards A1: S-ID.A.2 and S-ID.A.3 ask you to use measures of center (mean, median) and spread (range, interquartile range) to describe and compare data, and to account for the effect of outliers. On LEAP 2025 these are Type I and Type II items in the Additional and Supporting Content category. No statistics formula is on the reference sheet, so know these.

Measures of center

The mean ($10$) exceeds the median ($8$) here because the high value $18$ pulls the mean up.

Measures of spread

• Range = maximum $-$ minimum. A single big or small value can inflate it.
• Interquartile range (IQR) = Q3 $-$ Q1, the spread of the middle 50 percent. It ignores the extreme quarters, so it is resistant to outliers.

For $6, 8, 8, 10, 18$: range $= 18 - 6 = 12$; with Q1 $= 7$ and Q3 $= 14$, IQR $= 7$.

The effect of outliers

An outlier is a value far from the rest. It strongly affects the mean and the range (both use the actual values), but it barely affects the median and the IQR (both depend on position). This is why, for skewed data or data with outliers, the median and IQR are the more representative summary.

Comparing two data sets

To compare, pair a center with a spread. A set with a higher median is typically larger; a set with a smaller IQR or range is more consistent. State both: "Class A's median score is higher, and its IQR is smaller, so A scored higher and more consistently than B."

How LEAP examines this topic

• Equation response. Compute a mean, median, range, or IQR.
• Type II reasoning. Explain which measure is more appropriate, or compare two sets.
• Multiple choice. Identify the effect of an outlier, or pick the better center.

A clarifying idea: when the mean and median are close, the data are roughly symmetric; when the mean is well above the median, the data are right-skewed (a high tail), and below, left-skewed.

Why the median resists outliers but the mean does not

The contrast between the mean and the median comes down to what each one uses, and understanding it is the heart of S-ID.A.3. The mean is computed from the actual size of every value, so changing one value, especially to an extreme, changes the total and therefore the mean. Replace a $10$ with a $1000$ and the mean jumps, because that single large number now dominates the sum. The median, by contrast, is determined only by the position of the middle value once the data are ordered. Pushing the largest value even higher does not change which value sits in the middle, so the median stays put. This is why a few very high earners pull the mean income well above the median income, and why reports of "typical" values often use the median for skewed data like incomes or home prices. The same logic separates the range from the IQR: the range uses the two most extreme values, so an outlier inflates it, while the IQR measures only the middle 50 percent and ignores the extreme quarters. Choosing a resistant measure (median, IQR) or a sensitive one (mean, range) is therefore a judgment about whether extreme values should count fully, which is exactly the reasoning the standard wants you to make explicit.

Try this

Q1. Find the mean and median of $4, 4, 6, 8, 28$. [2 points]

• Cue. Mean $= \frac{50}{5} = 10$; median $= 6$. The outlier $28$ raises the mean.

Q2. Which spread measure is resistant to an outlier, the range or the IQR? [1 point]

• Cue. The IQR.