Compute and interpret measures of center (mean, median) and spread (range, IQR, standard deviation), and compare two distributions using center, spread, and shape (A.DSR, Data and Statistical Reasoning).

What this topic is asking

This Data and Statistical Reasoning (A.DSR) standard asks you to measure the center and spread of a data set and to compare two distributions fairly. The center measures are the mean and median; the spread measures are the range, the interquartile range (IQR), and the standard deviation. The key judgment the Georgia Milestones EOC tests is which measure to use: the mean and standard deviation for symmetric data, but the median and IQR when there are outliers or skew. Comparing distributions then means comparing center, spread, and shape together.

Measures of center

• Mean: add all values and divide by the count. It uses every value, so a single outlier shifts it noticeably.
• Median: the middle value when the data are ordered (the average of the two middle values for an even count). It is resistant to outliers.

For $4, 6, 7, 9, 24$: mean $= \frac{50}{5} = 10$, median $= 7$. The outlier 24 pulls the mean well above the median.

Measures of spread

• Range: maximum minus minimum. Simple, but sensitive to outliers.
• Interquartile range (IQR): $Q_3 - Q_1$, the spread of the middle 50 percent. Resistant to outliers.
• Standard deviation: a measure of the typical distance of values from the mean. A larger standard deviation means more spread-out data; a smaller one means data clustered near the mean.

Choosing the right measures

The decision rule is a frequent EOC item.

• Symmetric, no outliers: use the mean and standard deviation.
• Skewed or outliers present: use the median and IQR.

Comparing two distributions

To compare two groups (for example, two classes' test scores), address three things:

• Center: which group has the higher mean or median (typical value)?
• Spread: which group is more consistent (smaller IQR or standard deviation)?
• Shape: is either skewed, or are there outliers?

A complete comparison uses comparable measures for both groups (median and IQR for both, or mean and standard deviation for both) and states the comparison in context.

How the Milestones examines this topic

• Numeric entry. Compute the mean, median, range, or IQR of a data set.
• Multiple choice. Choose the better measure of center given skew or outliers, or compare two box plots.
• Constructed response. Compare two distributions by center, spread, and shape, in context.

Why outliers force the median

The reason an outlier pushes you toward the median is worth seeing concretely. The mean is computed from the sum of all values, so a single very large value adds a lot to the total and drags the mean toward itself, even though it is just one observation. In $4, 6, 7, 9, 24$, removing the 24 would drop the mean from 10 to about 6.5, a huge swing from one point, while the median would barely move. The median, by contrast, only cares about the position of the middle value, not how extreme the tails are, so it stays put. This is exactly why income data (which has a few very high values) is usually reported as a median, not a mean: the median reflects what a typical person experiences, while the mean is inflated by the extremes. Recognizing when one value is doing the talking tells you to switch to the resistant measures.

Standard deviation as consistency

Standard deviation is the spread measure for symmetric data, and on the EOC it is usually interpreted qualitatively rather than computed by hand. A smaller standard deviation means the values cluster tightly around the mean, so the data are more consistent or reliable; a larger standard deviation means the values are spread out, so the data are more variable. When comparing two symmetric data sets with similar means, the one with the smaller standard deviation is the more consistent, which is often the practically better outcome (a machine that fills bottles with less variation, a player with more consistent scores). Reading standard deviation as "how consistent" rather than as a formula is what the EOC comparison items expect.

Try this

Q1. Find the median of $3, 8, 8, 10, 15, 20$. [1 point]

• Cue. Even count; average the two middle values: $\frac{8 + 10}{2} = 9$.

Q2. Two classes have the same mean, but class A has a smaller standard deviation. Which is more consistent? [1 point]

• Cue. Class A, because a smaller standard deviation means less spread.