Compute and compare measures of center (mean, median) and spread (range, interquartile range, and informally standard deviation), and choose appropriate measures accounting for outliers (Ohio S-ID.2, S-ID.3).

What this topic is asking

Ohio standards S-ID.2 and S-ID.3 ask you to compute and compare measures of center (mean, median) and spread (range, interquartile range, and informally standard deviation), and to choose the right measure when data has outliers or skew. This is the analytical heart of the Statistics category, and it builds on the data displays.

Measures of center

Two numbers describe a "typical" value, and they can disagree.

When the mean is much larger than the median, a high outlier or right skew is pulling it; when the mean is much smaller, a low outlier or left skew is.

Measures of spread

Spread says how variable the data is around its center.

Standard deviation is the other spread measure: roughly the typical distance of values from the mean. Algebra I treats it informally, you compare "more spread" (larger standard deviation) versus "less spread," usually with technology, rather than computing it by hand.

Choosing the right measures

The key decision is whether the data is skewed or has outliers.

How Ohio examines this topic

• Numeric response. Compute a mean, median, range, or IQR.
• Multiple choice and multiple-select. Choose the better center or spread for a given distribution, or compare two data sets.
• Reasoning items. Explain why the mean and median differ, or why a measure is or is not resistant.

Why outliers pull the mean but not the median

The mean and median respond to an extreme value very differently, and the reason is in how each is built. The mean adds up every value, so a single very large (or very small) number contributes its full size to the total and drags the average toward itself, the more extreme the outlier, the bigger the pull. The median, by contrast, cares only about the position of the middle value: moving the largest number from $16$ to $1600$ does not change which value sits in the middle, so the median stays put. This is why the median is called resistant and the mean is not. When the two measures diverge, the gap itself is information: a mean well above the median points to a high outlier or right skew, and the median is the safer summary of a "typical" value.

Why the IQR is a more stable spread than the range

The range uses only the two most extreme values, the maximum and minimum, so it is entirely at the mercy of outliers: one unusually large value inflates the range even if every other value is tightly clustered. The interquartile range instead measures the spread of the middle 50% of the data, from $Q1$ to $Q3$, deliberately ignoring the top and bottom quarters where extremes live. That is why the IQR is resistant and gives a truer picture of how spread out the bulk of the data is. For skewed data or data with outliers, reporting the IQR alongside the median describes the distribution honestly, whereas the range and mean would both be distorted by the same extreme value.

Try this

Q1. Find the mean of $10, 12, 14, 16, 18$. [1 point]

• Cue. $\frac{70}{5} = 14$.

Q2. A data set is strongly skewed right. Should you report the mean or the median as the center? [1 point]

• Cue. The median (resistant to the skew).