Analyzing Distributions

A track coach recorded the 100-meter sprint times for every student in a gym class. Most students finished between 14 and 18 seconds, but two students who were on the varsity team finished in under 12 seconds. When the coach calculated the average time, it came out lower than almost every student's actual time. The mean got pulled toward those two fast runners, and it stopped representing the group well. That pull — and how to recognize it — is what analyzing distributions is about.

A distribution is the pattern of how data values spread across a range. When you look at a dot plot or histogram, the first thing you read is the shape. Three shapes come up constantly on the Regents.

A symmetric distribution has a roughly mirror-image appearance. The left half looks like the right half. The data piles up in the middle, and both tails taper off at about the same rate. A classic bell curve is symmetric.

A skewed right distribution has most of its data bunched on the left, with a long tail stretching toward the right. Those extreme high values — outliers on the high end — pull the tail right. House prices in a neighborhood are often skewed right: most homes sell for similar amounts, but a few mansions push the tail far to the right.

A skewed left distribution is the mirror image: most data bunches on the right, with a long tail reaching left. Test scores on an easy exam often look like this — most students score high, but a few very low scores trail off to the left.

Here is a helpful visual of all three shapes:

Once you identify the shape, you choose measures of center and spread. The two measures of center are the mean and the median. The mean is the arithmetic average — add all values and divide by the count. The median is the middle value when data is sorted.

For a symmetric distribution, the mean and median are close to each other and both work well. Use either one.

For a skewed distribution, the mean gets dragged toward the tail. The median stays near the bulk of the data and represents the typical value better. This is why real estate reports use median home price instead of mean home price — a few expensive properties would inflate the mean and make the market look wealthier than it is.

Spread tells you how tightly or loosely the data clusters. Three measures matter here.

The range is simply the maximum value minus the minimum value.

The interquartile range, or IQR, is the range of the middle half of the data. You find the first quartile $Q_1$ (the median of the lower half) and the third quartile $Q_3$ (the median of the upper half), then subtract.

The IQR ignores the extreme values entirely, which makes it resistant to outliers. Pair it with the median when your distribution is skewed.

Standard deviation measures how far values typically stray from the mean. A small standard deviation means data clusters tightly around the mean. A large one means data is spread out. Pair it with the mean when your distribution is symmetric. You do not need to compute standard deviation by hand on the Algebra I Regents — your calculator handles it — but you need to interpret what it means.

The guiding rule: when the distribution is symmetric, report the mean and standard deviation. When the distribution is skewed or has outliers, report the median and IQR.

Here are two worked examples reading and comparing distributions.

A data set of weekly hours spent on homework by 9 students is: 2, 3, 3, 4, 5, 5, 6, 7, 14. Find the mean, median, and IQR, and decide which measures best describe the data.

Two classes took the same quiz. Class A had scores: 70, 72, 74, 75, 76, 78, 79, 80, 82. Class B had scores: 50, 55, 60, 72, 78, 82, 88, 92, 98. Both classes have the same median. Compare their spreads.