Compute and interpret measures of center (mean, median) and spread (range, interquartile range), read box plots and histograms, and describe the shape of a distribution.

What this topic is asking

The Statistics and Probability category opens with summarizing one-variable data (the S-ID standards). On the Grade 10 MCAS you compute and interpret center (mean, median) and spread (range, interquartile range), read box plots and histograms, and describe a distribution's shape. A recurring test point is which measure of center is appropriate when the data has outliers, so interpretation matters as much as calculation.

Measures of center

Two summaries describe the "typical" value:

• The mean (average) is the sum of all values divided by how many there are. It uses every value, so a single extreme value moves it.
• The median is the middle value when the data is ordered. With an odd count it is the single middle value; with an even count it is the average of the two middle values.

For $4, 7, 9, 12, 15, 20$ (six values), the median is $\frac{9 + 12}{2} = 10.5$, while the mean is $\frac{67}{6} \approx 11.2$. They differ because the data is slightly uneven.

Measures of spread

Spread describes how scattered the data is:

• The range is the simplest: maximum minus minimum. It uses only the two extremes, so an outlier inflates it.
• The interquartile range (IQR) is $Q_3 - Q_1$, the range of the middle 50% of the data. $Q_1$ is the median of the lower half and $Q_3$ the median of the upper half. The IQR ignores the extremes, so it resists outliers.

The IQR is the spread companion to the median: both describe the middle of the data and are robust to extreme values.

Box plots and the five-number summary

A box plot displays the five-number summary: minimum, $Q_1$, median, $Q_3$, and maximum. The box spans $Q_1$ to $Q_3$ (the IQR), a line inside marks the median, and the whiskers reach the minimum and maximum. The MCAS asks you to read these values off a box plot or to build one from data.

Shape of a distribution

A histogram groups data into intervals and shows their frequencies, revealing the shape:

• Symmetric: the left and right halves mirror each other; the mean and median are close.
• Skewed right (a long tail to the right): a few high values pull the mean above the median.
• Skewed left (a long tail to the left): a few low values pull the mean below the median.

The relationship between mean and median signals the skew, which is a frequent MCAS reasoning point: in right-skewed data, mean $>$ median.

Comparing two distributions

The MCAS often shows two data sets (two box plots, or two histograms) and asks which has a higher center or a larger spread. Compare the medians to judge typical value and the IQRs (or ranges) to judge consistency. A team whose scores have a higher median is typically better; a team with a smaller IQR is more consistent, even if its median is similar. So a full comparison names both a center and a spread, not just one.

A subtle point: two data sets can have the same mean but very different spreads. Scores of $50, 50, 50$ and $20, 50, 80$ both average 50, but the second is far more variable. This is why a center alone does not describe a distribution; the spread completes the picture, and the MCAS rewards mentioning both.

Effect of changing a value

If every value in a data set has a constant added, the mean and median rise by that constant, but the spread (range and IQR) is unchanged, because the whole distribution shifts without stretching. If every value is multiplied by a constant, both the center and the spread scale by that constant. Reasoning about these effects, without recomputing from scratch, is a quick MCAS skill: adding 5 to every test score raises the mean by 5 but leaves the IQR the same.

Try this

Q1. Find the mean of $6, 8, 10, 16$.

• Cue. $\frac{40}{4} = 10$.

Q2. A data set has $Q_1 = 12$ and $Q_3 = 27$. What is the IQR?

• Cue. $27 - 12 = 15$.