Center, spread, and resistance

The sample mean is the arithmetic balance point: $\bar{x} = \frac{\sum x_i}{n}$. Every value participates, so every value can move it. The median is positional: order the data and take the middle value (or the average of the two middle values when $n$ is even). Only the values' order matters, not how far the extremes sit.

Watch the two react to a single change. For the data 10, 20, 30, 40, 50 both the mean and the median are 30. Replace the 50 with 250 and the mean becomes $\frac{350}{5} = 70$, while the median is still 30. The median is called a resistant measure of center for exactly this reason: outliers and long tails barely move it. The mean is nonresistant, and in a skewed distribution it is dragged toward the tail, so right skew usually leaves the mean above the median.

Neither is "the right answer" in general. The mean feeds later machinery (standard deviation, regression, inference); the median tells you what a typical case looks like when the data are lopsided. The choice is a judgment you are expected to justify from the shape of the data.

Three measures of variability appear in this course: the range (maximum minus minimum), the interquartile range, and the standard deviation. The sample standard deviation is

$$s = \sqrt{\frac{1}{n-1}\sum (x_i - \bar{x})^2},$$

and its plain-language reading is: roughly how far the values typically sit from their mean, in the same units as the data. To compute it by hand:

1. Find $\bar{x}$ and each deviation $x_i - \bar{x}$.
2. Square the deviations and add them up.
3. Divide by $n - 1$ (this quotient is the variance, $s^2$).
4. Take the square root to return to the data's units.

Three facts guard against misreadings. $s$ is never negative, because it is built from squares. $s = 0$ only when every value is identical. And $s$ is typical distance, not a boundary: individual values routinely sit more than one standard deviation from the mean. Whether an $s$ of 10 is "large" depends entirely on the units and the context: 10 grams of variation in newborn weights is nothing, 10 points of variation on a 10-point quiz is everything.

The first quartile $Q_1$ is the median of the lower half of the ordered data (from the minimum up to the position of the median); the third quartile $Q_3$ is the median of the upper half. About 25% of the data sit at or below $Q_1$ and about 75% at or below $Q_3$, so $Q_1$ and $Q_3$ bracket the middle 50%. The five-number summary is minimum, $Q_1$, median, $Q_3$, maximum.

For the seven values 6, 9, 11, 14, 17, 20, 44: the median is 14, the lower half 6, 9, 11 gives $Q_1 = 9$, and the upper half 17, 20, 44 gives $Q_3 = 20$.

The interquartile range is $\text{IQR} = Q_3 - Q_1 = 20 - 9 = 11$. Note what kind of thing that is: a single distance, in the data's units. "The IQR is from 9 to 20" is not an IQR; it is the location of the box. The IQR is the width of that middle-50% span, and because it is built from quartiles it shares the median's resistance: the 44 could have been 4,400 and the IQR would still be 11.

Two standard rules flag potential outliers. The 1.5 IQR rule: a value is an outlier if it falls more than $1.5 \times \text{IQR}$ above $Q_3$ or below $Q_1$. For the data above, the fences are $9 - 1.5(11) = -7.5$ and $20 + 1.5(11) = 36.5$, so 44 is flagged and nothing else is. The multiplication happens first: the fence is $Q_3 + 1.5 \times \text{IQR}$, never $Q_3 + 1.5$. The 2 SD rule: a value more than two standard deviations from the mean, outside $\bar{x} \pm 2s$, is flagged.

Outliers and skew are also what settle the choice of summary:

1. Roughly symmetric, no outliers: report the mean with the standard deviation.
2. Skewed or outlier-laden: report the median with the IQR, the resistant pair. In the data above the 44 drags the mean to about 17.3 while the median sits at 14.
3. Either way, state the units. Changing units changes the statistics: adding a constant shifts the mean, median, and quartiles but leaves every measure of spread alone, while multiplying (say, converting minutes to seconds) multiplies center and spread alike.

A flagged value is a candidate for investigation, not deletion. The rule points a finger; it does not pull a trigger.