Summary Statistics for One Quantitative Variable

Topic 1.6 described center and spread by eye. Summary statistics pin them down as numbers you can calculate, compare and defend: a statistic is a number computed from sample data. There are two families. Measures of center and position say where values sit on the number line; measures of variability say how spread out they are. Choosing which ones to report depends on the shape of the data, and that choice is itself something the exam asks you to justify.

Center: the mean and the median

The mean is the sum of the values divided by how many there are. For a sample of $n$ values $x_1, x_2, \ldots, x_n$,

The symbol $\bar{x}$ ("x-bar") is the sample mean, and $x_i$ is the $i$th value. Physically, the mean is the balance point: if each value were a weight on a ruler, the ruler would balance at $\bar{x}$. That is why one far-out value moves the mean a lot; a heavy weight at the end of a ruler shifts the balance point toward it.

The median is the middle value once the data are sorted from smallest to largest. With an odd number of values it is the single middle one, in position $\frac{n+1}{2}$. With an even number there are two middle values, and the median is their mean. For 2, 5, 9, 11 the median is $\frac{5 + 9}{2} = 7$, a number that isn't in the data at all, which is fine. The smallest value is the minimum and the largest is the maximum.

Both can be found by hand or with technology. On a TI-84, enter the data in a list and run 1-Var Stats: it reports $\bar{x}$, $s_x$, $n$, the minimum, $Q_1$, the median, $Q_3$ and the maximum.

Position: quartiles and percentiles

The median splits the sorted data into a lower half and an upper half. The first quartile, $Q_1$, is the median of the lower half, and the third quartile, $Q_3$, is the median of the upper half. When $n$ is odd, the median itself belongs to neither half. (The median is sometimes called $Q_2$.) About 25% of the values are at or below $Q_1$ and about 75% are at or below $Q_3$, so the middle half of the data lies between $Q_1$ and $Q_3$.

For example, the 9 sorted values 1, 3, 4, 6, 7, 9, 10, 12, 15 have median 7 (the 5th value). The lower half is 1, 3, 4, 6, so $Q_1 = \frac{3 + 4}{2} = 3.5$; the upper half is 9, 10, 12, 15, so $Q_3 = \frac{10 + 12}{2} = 11$. Software packages use slightly different quartile rules, so a spreadsheet can disagree with your calculator. The halves method shown here is the one the TI-84 uses; the AP framework doesn't prescribe a single quartile rule.

The $p$th percentile is the value with $p\%$ of the data at or below it. $Q_1$ is the 25th percentile, the median the 50th, $Q_3$ the 75th. A percentile is a position, not a score: a student at the 90th percentile on a test did as well as or better than 90% of test takers, whatever their actual score was.

Variability: range, IQR and standard deviation

The range is maximum minus minimum. It depends only on the two most extreme values, so one unusual observation can make it huge.

The interquartile range is

the width of the interval holding the middle 50% of the data. In the 9-value example, $\text{IQR} = 11 - 3.5 = 7.5$.

The standard deviation measures how far values typically are from the mean. The sample standard deviation is

Read it from the inside out. Each deviation $x_i - \bar{x}$ says how far one value is from the mean. The deviations always add to 0 (the mean is the balance point), so you can't average them directly; squaring makes them all positive. Adding the squares and dividing by $n - 1$ gives the sample variance, $s^2$, and the square root brings the units back to the units of the data. Dividing by $n - 1$ rather than $n$ makes $s^2$ a better estimate of the population's variance, because deviations measured from the sample's own mean come out a little too small on average. $s = 0$ only when every value is the same; otherwise $s > 0$.

What changing units does

Converting every value with the same rule changes the statistics in a predictable way.

• Adding a constant $a$ to every value shifts every measure of center and position (mean, median, quartiles, percentiles, minimum, maximum) by $a$, and leaves every measure of spread (range, IQR, standard deviation) unchanged. Sliding all the data along the number line doesn't spread it out.
• Multiplying every value by a positive constant $b$ multiplies every measure of center and position by $b$ and every measure of spread by $b$ as well. The variance is multiplied by $b^2$.

Converting Fahrenheit to Celsius uses both: $C = \frac{5}{9}(F - 32)$. Subtracting 32 moves the centers but not the spread; then multiplying by $\frac{5}{9}$ scales everything.

Outliers and resistance

Topic 1.6 called a value a "possible outlier" by eye. Two numerical rules make the call:

The two rules don't always agree, because they are built from different statistics. Name the rule you are using.

A statistic is resistant (or robust) if extreme values barely affect it. The median and IQR are resistant: making the largest value ten times larger leaves the middle of the sorted list where it was. The mean, standard deviation and range are not resistant: the mean is a balance point, and the standard deviation squares each deviation, so a single far-out value moves them a lot.

That gives the rule for choosing a summary. For a distribution that is skewed or has outliers, report the median and IQR, which describe the bulk of the data. For a roughly symmetric distribution without outliers, the mean and standard deviation are appropriate, and they are the statistics that later inference is built on.

Summary statistics also let you compare two independent samples (two separate groups, such as two classes or two companies): compare their centers, their variability and their outliers, and use the gap between mean and median as a hint about shape. Topic 1.8 explains why a mean well above the median signals right skew, and Topic 1.9 covers comparisons in full.

Worked examples

Where students lose points

On the exam

Expect to compute a mean, median or quartile by hand from a short list, and to use 1-Var Stats for anything longer; with a calculator, write down the statistics it gave you rather than just the final answer. An outlier question needs the computed fence or the value of $\bar{x} \pm 2s$, a comparison with the value, and the conclusion. When asked which statistic is more appropriate, the justification is resistance: name the skew or outlier and say that the median or IQR isn't strongly affected by it.