Calculate and interpret measures of center (mean, median) and spread (range, interquartile range, standard deviation), and choose appropriate measures based on the shape of the distribution and the presence of outliers (MA.912.DP.1.2, MA.912.DP.1.3).

What this topic is asking

MA.912.DP.1 asks you to compute and compare measures of center (mean, median) and spread (range, interquartile range, standard deviation) and to choose the appropriate one based on the distribution's shape and outliers. The B.E.S.T. Algebra 1 EOC tests both the calculation and the judgment of which measure fits.

Measures of center

• Mean. Add all values and divide by the count. It uses every value, so a single extreme value shifts it.
• Median. Order the data and take the middle value (or the average of the two middle values). It depends only on position, so extreme values do not move it.

Measures of spread

• Range. Maximum minus minimum. Simple but sensitive to a single extreme.
• Interquartile range (IQR). $Q_3 - Q_1$, the width of the middle 50 percent. Resistant to outliers.
• Standard deviation. The typical distance of values from the mean. Larger means more spread. Like the mean, it is affected by outliers.

Choosing the right measure

Match the measure to the shape:

• Symmetric, no outliers: the mean (center) and standard deviation (spread) are appropriate.
• Skewed or has outliers: the median (center) and IQR (spread) are appropriate, because they resist extreme values.

How the B.E.S.T. EOC examines this topic

• Number entry and equation editor. Compute a mean, median, range, or IQR.
• Multiple choice. Choose the better measure for a skewed or outlier-laden set.
• Comparison items. Compare the center or spread of two groups from their box plots or summaries.

A clarifying idea: center answers "what is typical?" and spread answers "how varied?". A complete description needs one of each, and for skewed data the resistant pair (median and IQR) tells the more honest story than the mean and standard deviation.

Why the median resists outliers but the mean does not

The contrast comes from how each measure uses the data. The mean is a balance point computed by adding every value, so each value contributes its full size; replacing an ordinary value with a huge one adds that whole excess to the total and drags the average toward it. The median, by contrast, depends only on the value's rank, not its magnitude. Turning the largest data point from $8$ into $800$ leaves it still the largest, so the middle position does not move and the median is unchanged. This is why a single mansion in a list of house prices inflates the mean but leaves the median near a typical home: the mean feels the dollar amount, the median only feels the ordering. The same logic makes the IQR (built from quartile positions) resistant while the range and standard deviation (built from extreme values and distances) are not.

Try this

Q1. Find the IQR of a data set with Q1 $= 12$ and Q3 $= 28$. [1 point]

• Cue. $28 - 12 = 16$.

Q2. A test-score set has one very low score. Which center is more representative? [1 point]

• Cue. The median, because it resists the low outlier.