Summary Statistics for One Quantitative Variable

Measures of center

The mean is the sum of the values divided by the number of values. For a sample of $n$ values,

where $x_i$ stands for the $i$th value in the sample. 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 value. With an even number it is the mean of the two middle values. The smallest value is the minimum and the largest is the maximum.

Find these by hand for a small data set. For a large one, use a calculator's one-variable statistics.

Quartiles and percentiles

The first quartile, $Q_1$, is the median of the lower half of the sorted data, the values below the median's position. The third quartile, $Q_3$, is the median of the upper half. When $n$ is odd, this course leaves the median out of both halves, as the TI-84 does. About 25% of the values are at or below $Q_1$ and about 75% are at or below $Q_3$, so the two quartiles mark off the middle 50% of the data. The median is the second quartile, $Q_2$.

The $p$th percentile is the value with $p$% of the data at or below it. $Q_1$ is the 25th percentile and $Q_3$ is the 75th. A test score at the 90th percentile is at or above 90% of all the scores. That is a statement about position, not about getting 90% of the questions right.

Measures of variability

Three numbers measure how spread out a distribution is.

• The range is the maximum minus the minimum.
• The interquartile range is $\mathrm{IQR}=Q_3-Q_1$, the width of the middle half of the data.
• The standard deviation measures a typical distance between the data values and their mean.

For a sample, the standard deviation is

In words: find each value's deviation from the mean, square the deviations, add them up, divide by $n-1$, and take the square root. The quantity under the square root, $s^2$, is the sample variance. A calculator labels the sample standard deviation $s_x$. The ${\sigma }_x$ beside it divides by $n$ instead, so it is not the one to report for a sample.

The deviations always add to 0, which is why they are squared before they are added. Dividing by $n-1=4$ gives $s^2=\frac{178}{4}=44.5$, so $s=\sqrt{44.5}\approx 6.67$ minutes. A typical commute is about 6.7 minutes away from the mean of 14 minutes.

Two rules for outliers

There are many ways to decide that a value is unusually far from the rest. Two are used all the time.

The $1.5\times \mathrm{IQR}$ rule. A value is an outlier if it is more than $1.5\times \mathrm{IQR}$ above $Q_3$ or more than $1.5\times \mathrm{IQR}$ below $Q_1$. For the getting-ready times, $1.5\times 12=18$, so the cutoffs, often called fences, are $18-18=0$ and $30+18=48$ minutes. The 62-minute student is an outlier.

The two standard deviation rule. A value is an outlier if it is more than 2 standard deviations above or below the mean. Here $26.5+2(13.5)=53.5$ and $26.5-2(13.5)=-0.5$, so 62 is an outlier by this rule as well.

The two rules do not always agree. An extreme value inflates the standard deviation, which pushes the two standard deviation cutoffs farther out, so that rule can miss a value the IQR rule catches. When a question names a rule, use that rule.

Resistant and nonresistant statistics

A statistic is resistant if outliers do not change it much, if at all. The median and the IQR are resistant, because they depend on the middle of the sorted data and not on how far out the extremes sit. The mean, the standard deviation and the range are nonresistant. Every value feeds into the mean and the standard deviation, and the range is built from the two most extreme values, so one extreme value can move all three a long way.

Nine students in a club list their follower counts on a photo app: 210, 340, 380, 415, 450, 520, 610, 700 and 48,000. The mean is about 5,736 followers, more than eight times what any of the other eight students has. The median, 450, describes a typical member. Drop the 48,000 and the mean falls to about 453, while the median only moves to 432.5.

That gives a working rule:

• For a distribution that is roughly symmetric with no outliers, report the mean and standard deviation.
• For a distribution that is strongly skewed or has outliers, report the median and IQR.

Comparing samples with summary statistics

Summary statistics also compare two or more independent samples on center, variability, shape and outliers. Random samples of 40 students from two lunch periods recorded how many minutes they waited in the cafeteria line.

Fourth period's waits are longer, with a median of 9 minutes against 6, and far more variable, with an IQR of 7 minutes against 3. Its mean sits above its median and its maximum is 26 minutes, both hints of a long right tail. A student who wants a short, predictable wait should pick fifth period.

Changing units

Changing the units of measurement changes the summary statistics, in a predictable way.

Multiplying every value by a constant, as when you convert minutes to seconds, multiplies every measure of center, position and variability by that constant. In seconds, the getting-ready times have a mean of about 1,593 seconds, an IQR of $12\times 60=720$ seconds and a standard deviation of about 811 seconds.

Adding a constant to every value shifts the mean, the median and the quartiles by that constant but leaves the range, the IQR and the standard deviation exactly where they were. Every value moves, and the distances between values do not.

Temperature conversion does both. Fahrenheit is 1.8 times Celsius, plus 32. If a city's daily highs for two weeks have a mean of 18 degrees Celsius and a standard deviation of 4 degrees, then in Fahrenheit the mean is $1.8(18)+32=64.4$ degrees and the standard deviation is $1.8(4)=7.2$ degrees. The 32 shifts the center but adds nothing to the spread.