Center and Spread

Part V: Statistics and Probability

Part V of the ACT Math test asks what a set of data says and how far you can trust what it says. Most of these questions are short and reward careful reading: you compute one number, read one table, or decide which conclusion a study can honestly support. You will usually meet this material in three layers. First, how to summarize a single list of numbers (center and spread). Second, how the data was collected, which decides what you are allowed to conclude. Third, how two variables relate, using lines of best fit and correlation. This first chapter handles the first layer.

Center and Spread

The mean is the sum of all values divided by the count. The median is the middle value once the data is ordered, or the average of the two middle values if the count is even. The mode is the value that appears most frequently. A missing-value problem gives the mean and all but one value; multiplying the mean by the count recovers the total, and subtracting the known values isolates the missing one.

Spread describes how far the data is scattered, not where it is centered. The range is the maximum minus the minimum. The interquartile range, IQR, is $Q_3 - Q_1$, the spread of the middle 50% of the data. Standard deviation measures the typical distance of each value from the mean; a data set with values spread farther from the mean has a larger standard deviation, even if two data sets share the same mean.

Where this appears as a model: several groups combined into one average. Two class sections of 20 and 15 students with means of 78 and 85 combine to an overall mean of 81, found by weighting each mean by its class size instead of averaging 78 and 85 directly.

Quick reference

Missing-value problems

Worked Example. Four numbers have a mean of 15. Three of them are 9, 14, and 21. Find the fourth.

Recover the total by multiplying the mean by the count, then subtract the known values.

Total: $15 \times 4 = 60$

Fourth number: $60 - (9 + 14 + 21) = 16$

The same move works in reverse for questions that add or remove a value. The mean is always total divided by count, so write the total before and after, then divide by the new count. Never average two means unless the groups are the same size.

Combining groups: the weighted mean

When two groups are merged, each group's mean counts in proportion to its size. For the two class sections above, the totals are $20 \times 78 = 1560$ and $15 \times 85 = 1275$. Together that is $2835$ points among $35$ students, and $2835 \div 35 = 81$. The simple average of 78 and 85 would be 81.5, which wrongly treats the two sections as the same size. Because the larger section has the lower mean, the true combined mean is pulled toward 78.

Comparing spread across data sets

Two data sets can share the same mean while having very different spreads. A data set whose values sit far from the mean has a larger standard deviation than one whose values cluster tightly around it, regardless of what that shared mean actually is. On the ACT you almost never compute a standard deviation. You compare: which set is more spread out, or which set has values closer to its mean?

Spread also reacts to changes in the data in predictable ways. Adding the same number to every value shifts the mean and median by that number but leaves the range, IQR, and standard deviation unchanged. Multiplying every value by a positive number multiplies the mean, median, range, IQR, and standard deviation by that number.

Distribution shape and histograms

A distribution has shape as well as center and spread. A roughly symmetric distribution behaves similarly on both sides of its center. A right-skewed distribution has a longer tail toward larger values; those large values pull the mean to the right, so the mean is usually greater than the median. A left-skewed distribution has a longer tail toward smaller values, pulling the mean to the left, so the mean is usually less than the median.

A histogram displays how often values fall into each interval, so read its overall shape before calculating anything. Where is the center, how wide is the spread, is it symmetric or skewed, and are there isolated gaps or unusually extreme groups? When strong skew or outliers are present, the median and IQR are usually more resistant summaries of the data than the mean and standard deviation, which are both pulled toward extreme values.

Worked Example. A data set has mean 62 and median 51. What does this suggest about the distribution's shape?

Compare the mean and median directly. The mean (62) is greater than the median (51). This pattern indicates a right-skewed distribution, with a tail of unusually high values pulling the mean upward.

A handy memory aid: the mean chases the tail. If the tail points right, the mean sits to the right of the median; if the tail points left, the mean sits to the left.

Strategy checklist

• Sort the data before finding a median, a mode, or quartiles.
• For an even count, average the two middle values.
• For any mean puzzle, turn the mean into a total with $\text{mean} \times \text{count}$.
• For combined groups, weight by group size.
• For spread, ask how far values sit from the mean; do not look at the mean itself.
• For skew, compare mean and median and let the mean chase the tail.
• If an outlier appears, expect the mean, range, and standard deviation to change a lot, and the median and IQR to change little.