Week 3 · Chapter

Chapter 3 — Numerical Summaries: Center & Spread

Introduction to Statistics Free sample edition

The module's primary reading · ~35–45 minutes · Objective 2 (summarize and display a single quantitative or categorical variable with appropriate tables and graphs, and describe its distribution's shape, center, and spread with the right numerical summaries) · Read it first this week; the lecture, the tutorial, the quiz, and the Data Lab all build on it.

In plain language

Last week you drew pictures of 12 month lengths and 342 penguin body masses; this week you put numbers on them. Two questions drive the whole chapter: where is the middle, and how spread out are the values? You will meet three ways to name the middle, called measures of center (single numbers that say where the middle of the data sits). You will also meet ways to measure spread (how far the values tend to stray from that middle). You will also learn which number to trust when one odd value shows up in your data.

Why you already care about this

Three friends split a dinner. One ordered a $12 plate, one a $15 plate, and one an $18 plate. Together they spent $45 on food. If they agree that everyone pays the same share, each pays $15 toward the bill. That fair share is an average, and you have used it all your life without a name for it.

Now imagine a fourth friend joins, and that friend's plate cost $75. The fair share jumps, yet three of the four people still ate a modest meal. One number for the middle can hide that story. A second number, one that ignores how big the $75 is, would tell it better. A third number, one that says how far apart the plates were, would tell the rest.

That is this week. A nurse comparing patient wait times, a shop owner reading a month of daily sales, and a field biologist weighing penguins all ask the same two questions: what is typical, and how much do the values vary?

By the end of this chapter you will be able to:

compute the mean, the median, and the mode, and say which level of measurement each one needs;
explain why one wild value drags the mean but barely moves the median;
read the lean of a distribution by comparing its mean with its median;
compute the range, the variance, and the standard deviation, and say when to divide by one less than the count;
build a five-number summary, describe its boxplot, and flag unusual values with the 1.5 × IQR rule.

Carry this sentence with you: the median holds its ground; the mean follows the tail.

Where we are

Already covered: Week 1, Foundations & Types of Data, taught you to tell a population from a sample and to sort every variable by its level of measurement. Week 2, Summarizing Data with Tables & Graphs, taught you to picture one variable and to describe its shape in words.
This week: numbers for the middle (mean, median, mode), numbers for the spread (range, variance, standard deviation, IQR), the boxplot, and a rule for flagging unusual values.
How they connect: the level of measurement from Week 1 decides which center you may use, and the shape from Week 2 decides which center tells the truth.
Where we're going next: Week 4, Exploring Relationships Between Two Variables, moves from one variable to two and asks whether they move together.

Two ideas from earlier weeks do the heavy lifting today. From Week 1: a population is the whole group a question is about, and a sample is the part you actually measure. The letter n (the sample size) counts a sample, and the letter N (the population size) counts a whole population. From Week 2: the tail is the thin end of a distribution, and you name the skew by its tail. Last week you saw the penguin body masses lean right and the month lengths lean left. This week you prove both with two numbers.

1. Center: find the middle three ways

Picture a line of five customers at a food truck, each holding a receipt. You could ask three different questions about the middle. What would each pay if the total were shared equally? Who stands in the middle when you line them up by amount spent? Which amount shows up most often? Each question gives a different measure of center.

A measure of center is a single number that describes where the middle of a distribution sits; the mean, the median, and the mode are the three common ones.
The mean is the sum of all the values divided by how many values there are; it uses the size of every value and needs interval or ratio data.
The median is the middle value once the data are put in order, or the average of the two middle values when the count is even; it needs data that are at least ordinal.
The mode is the value or category that occurs most often; it is the only measure of center that works for nominal data.

The mean gets two symbols, because Week 1's population-versus-sample line runs right through it. The sample mean is the mean of the values in a sample, written x̄ and read "x bar"; it is a statistic. The population mean is the mean of every value in a whole population, written μ and read "mu"; it is a parameter. In a formula, Σ means "add them all up," and x stands for one data value, so the mean is Σx ⁄ n for a sample and Σx ⁄ N for a population.

Worked example — the center of a fair die. The six faces of a fair die are 1, 2, 3, 4, 5, and 6. They are every face there is, so they form a whole population, and N (the population size) is 6.

Add the faces. 1 + 2 + 3 + 4 + 5 + 6 = 21.
Divide by the count. μ = (1 + 2 + 3 + 4 + 5 + 6) ⁄ 6 = 21 ⁄ 6 = 3.5.
Find the median. The faces are already in order. With six values, there is no single middle one, so you average the two middle faces, 3 and 4: (3 + 4) ⁄ 2 = 3.5.
Look for a mode. Each face appears once, so no face is more common than another, and the die has no single mode.

In words: the balance point of a die is 3.5, even though you will never roll a 3.5. A mean does not have to be a value the data can take.

Worked example — the center of the month lengths. The twelve months of a common year have these lengths, in days: 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31. They are all twelve months, so they form a whole population with N = 12.

Add the lengths. The twelve lengths add to 365, so Σx = 365 days.
Divide by the count. The mean is 365 ⁄ 12 ≈ 30.4167 days, rounded to four decimals at the end.
Sort before you find the median. In order the lengths are one 28, four 30s, and seven 31s, and 1 + 4 + 7 = 12. Written out: 28, 30, 30, 30, 30, 31, 31, 31, 31, 31, 31, 31.
Find the median. With twelve values, the middle pair is the 6th and the 7th. Both are 31, so the median is 31.
Find the mode. The value 31 occurs 7 times, more than any other, so the mode is 31.

In words: a typical month has 31 days, and the average month has a little over 30 days. Section 3 explains why those two answers differ.

The level of measurement decides which center you may use

Week 1 sorted every variable into four levels: nominal, ordinal, interval, and ratio. Each center needs a certain level.

Nominal data are labels with no order, like penguin species. Only the mode works. Among the 344 penguins in the field log, 152 are Adelie, 68 are Chinstrap, and 124 are Gentoo, so the mode is Adelie. There is no "middle species" and no "average species."
Ordinal data have an order but no fixed gaps, like drink sizes. Five customers order small, small, medium, large, and large. Lined up, the middle order is medium, so the median size is medium. A mean would need you to add "small" and "large," which makes no sense.
Interval and ratio data are real numbers with equal gaps, like body mass in grams or days in a month. All three centers work.

Misconception check. Students often think that any column of numbers is fair game for an average. Suppose a spreadsheet codes the islands as Biscoe = 1, Dream = 2, and Torgersen = 3. The column now holds numbers, and a spreadsheet will happily compute their average, something like 1.7. But the codes are names wearing number costumes. "1.7" is not an island, and no penguin lives on it. The cure: before you average, ask what level the variable is. If the numbers are labels, report the mode instead.

In practice (your tools). Load the field log into Google Sheets with the same =IMPORTDATA(...) formula you used in Week 1, so body mass sits in column F. In an empty cell, =AVERAGE(F2:F345) returns the mean and =MEDIAN(F2:F345) returns the median. The two NA entries are text, so both functions skip them and use the 342 recorded masses. For the month lengths, type the twelve values in A1:A12 and use =AVERAGE(A1:A12), =MEDIAN(A1:A12), and =MODE(A1:A12). In Excel, the same three functions work with the same names.

2. Resistance: one wild value, two very different reactions

Picture a small street with five homes. Four sell for prices close to one another, and then a sports star buys the fifth, a mansion. The street's average sale price leaps. Yet the home in the middle of the price list has not changed at all. That is the whole idea of this section.

On that street, the middle price is the resistant number.

A resistant measure is a summary that changes little when one extreme or mistyped value is added or changed; the median is resistant and the mean is not.

The reason is in how each one is built, and the mansion shows it. The mean uses the size of every value, so one huge price pulls it hard. The median uses only the position of the middle home, so a huge price at the end of the line stays at the end of the line.

Worked example — a mistyped die face. Someone types the six faces of a die into a spreadsheet and hits the 0 key by accident, so the 6 becomes 60.

Recompute the mean. (1 + 2 + 3 + 4 + 5 + 60) ⁄ 6 = 75 ⁄ 6 = 12.5.
Recompute the median. In order the values are 1, 2, 3, 4, 5, 60. The two middle values are still 3 and 4, so the median stays (3 + 4) ⁄ 2 = 3.5.
Compare. One typo caused a more than 3-fold jump in the mean versus no change at all in the median. The mean moved from 3.5 to 12.5, and the median stayed at 3.5.

In words: a mean of 12.5 claims the typical face is bigger than any real face on the die. The median of 3.5 still describes the die. If you were checking a data file for typing errors, a mean that lands outside the range of sensible values is your first warning sign.

Misconception check. Students often hear "the median is resistant" and conclude that the median is always the better number. It is not. The median throws away information about how big the values are, and sometimes that information matters. A business owner who needs total sales for the month needs the mean, because mean × count = total. The cure: the median answers "what is a typical value?", and the mean answers "what is the fair share of the total?" Pick the question first, then the number.

Honesty note: resistance is about one or a few extreme values. If half the data change, the median moves too.

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

(a) Think of a number you have seen reported as an "average," such as a price, a salary, or a wait time; would one extreme value be likely in that data, and would you rather see the median?
(b) Think of a situation where you would want the mean even though an extreme value is present, such as splitting a bill or planning a budget; why does the total matter there?

3. Skew: let the mean and the median tell you the shape

Picture a quiz where most of the class scores near the top and a few students who missed a lesson score low. The few low scores drag the class average down, but the middle student still scored high. The average lands below the middle. That gap is a clue to the shape.

Last week you named skew by looking at a picture of the 12 months and the 342 penguins. This week two numbers back up your eye. Skew pulls the mean toward the long tail, because the mean feels the size of every value in that tail and the median does not.

Here is how the two numbers line up with the three shapes you met in Week 2.

In a left-skewed distribution, like the 12 month lengths, the long tail stretches toward the smaller values, and the mean usually sits below the median.
In a right-skewed distribution, like the 342 penguin masses, the long tail stretches toward the larger values, and the mean usually sits above the median.
In a symmetric distribution, the mean and the median sit close together.

Worked example — three datasets, three shapes. Compare each mean with its median.

The month lengths: left-skewed. The mean, 30.4167, sits below the median, 31. The lone 28 in the left tail pulls the mean down, toward February.
The penguin body masses: right-skewed. The 342 penguin body masses have mean x̄ (the sample mean) of 4,201.75 g, above the median of 4,050 g. The gap is 4,201.75 − 4,050 = 151.75 g, because a tail of heavy birds pulls the mean up.
The die faces: symmetric. The mean and the median are both 3.5, and the faces balance evenly around it.

Mean and median of the three week-3 datasets, with the shape each comparison suggests
Dataset	Mean	Median	Mean compared with median	Shape
Six die faces	3.5	3.5	equal	symmetric
Twelve month lengths (days)	30.4167	31	below	left-skewed
342 penguin body masses (g)	4,201.75	4,050	above	right-skewed

In words: a biologist who reports only the mean penguin mass describes a bird a little heavier than the typical one. A biologist who reports the median describes the bird in the middle of the line. For a skewed variable, the median is the safer "typical" value, and the mean is the one to use when the total matters.

Misconception check. Students often treat the mean-versus-median comparison as a law that always names the skew. It is a strong habit, not a law: in small or oddly shaped datasets the comparison can point the wrong way. The cure: use the comparison to back up the picture, never to replace it. Look at the dot plot or the histogram first, and then check whether the two numbers agree with what you see.

Caveat: in your own data, "the mean follows the tail" describes what usually happens. State the evidence (the picture and the two numbers) every time you name a skew.

In practice (your tools). In Google Sheets, put =AVERAGE(F2:F345) and =MEDIAN(F2:F345) side by side, then type =AVERAGE(F2:F345)-MEDIAN(F2:F345) in a third cell. A positive gap points to a right tail and a negative gap to a left tail. In Excel, the same three formulas work.

4. Spread: measure how far the values stray

Picture two coffee shops that each sell 100 cups on an average day. At the first shop, every day sells between 95 and 105 cups. At the second, some days sell 40 cups and some sell 160. The averages match, but only the first owner can order milk with confidence. The center is the same; the spread is not.

The range

The quickest measure of spread needs only 2 numbers from your list.

The range is the largest value minus the smallest value; it uses only the two most extreme values, so a single unusual value can stretch it.

The die faces have range 6 − 1 = 5, the month lengths 31 − 28 = 3 days, and the body masses 6,300 − 2,700 = 3,600 g. The range is quick, but it ignores every value in between, and the mistyped 60 from Section 2 would stretch the die's range from 5 to 59.

Deviations, variance, and standard deviation

A better measure of spread uses every value on your list. Start with the 6 faces of a die, and ask how far each face sits from the mean of 3.5.

Three new words carry this idea, and the die example right after them makes each one concrete.

A deviation is how far one value sits from the mean, found by subtracting the mean from the value; deviations above the mean are positive and those below are negative.
Variance is the average size of the squared deviations from the mean; it is measured in the square of the data's units.
The standard deviation is the square root of the variance; it describes the typical distance of a value from the mean, in the data's own units.

Why square the deviations? On the die, the plain deviations add to 0, because the faces above 3.5 exactly balance the faces below it. Squaring makes every deviation positive, so they can no longer cancel.

Worked example — the spread of a fair die, step by step. Use the six faces again, with mean 3.5.

Find each deviation. Each die face minus the mean 3.5 gives −2.5, −1.5, −0.5, 0.5, 1.5, 2.5, and these add to 0.
Square each deviation. (−2.5)² = 6.25, (−1.5)² = 2.25, (−0.5)² = 0.25, and the three positive deviations give the same squares in reverse.
Add the squares. 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5 square units.
Divide by the population size. The faces are the whole population, so you divide by N = 6. Dividing by N gives σ² = 17.5 ⁄ 6 ≈ 2.9167, rounded to four decimals.
Take the square root. σ = √2.9167 ≈ 1.71.

In words: a typical face sits about 1.71 pips from the balance point of 3.5. Here σ² (the population variance, read "sigma squared") is about 2.9167 square pips, and σ (the population standard deviation, read "sigma") is about 1.71 pips.

Sample or population: dividing by one less

Here is the twist that trips up many students, and it may trip you up too. Suppose the same six numbers came from six rolls of a die you were testing, not from a list of every face. Now they are a sample of the die's long-run behavior, and the formula changes.

The population variance, written σ², adds up the squared deviations from the population mean and divides by the population size.
The sample variance, written s², adds up the squared deviations from the sample mean and divides by one less than the sample size.
Bessel's correction is dividing a sample's sum of squared deviations by one less than the sample size instead of by the sample size, which keeps the sample variance from running too small on average.

Why one less? Think of 6 rolls of a die you are testing. Your sample's own mean sits closer to its own values than the true population mean does. So the squared deviations from x̄ come out a little too small, and dividing by a smaller number makes up the difference.

Worked example — the same six numbers as a sample. Treat 1, 2, 3, 4, 5, 6 as six rolls, so n (the sample size) is 6.

Reuse the sum of squares. The six squared deviations still add to 17.5 square units.
Divide by one less than the count. Dividing by 5 gives s² = 17.5 ⁄ 5 = 3.5.
Take the square root. The sample standard deviation s (the square root of s²) is √3.5 ≈ 1.87.

In words: the same six numbers give s ≈ 1.87 as a sample and σ ≈ 1.71 as a population. The sample version is larger by design, because it allows for the extra uncertainty of not seeing everything.

Variance and standard deviation of the numbers 1 through 6, computed as a population and as a sample
Treated as	Divide the 17.5 by	Variance	Standard deviation
A population (every face)	6	about 2.9167	about 1.71
A sample (six rolls)	5	3.5	about 1.87

Worked example — the month lengths and the penguins. The month lengths are a whole population, so σ ≈ 0.86 days. A spreadsheet's STDEV.S divides by 11 instead, and you get s ≈ 0.90 days. For the months, the population answer is the honest one, because you have every month. The 342 penguin masses are a different story. They are a sample of penguins, so you divide by 341, and you get s ≈ 801.95 g.

In words: a typical month is less than one day away from the mean month, and a typical penguin is roughly 800 g away from the mean penguin.

Misconception check. Students often mix up variance and standard deviation, or report a variance in the data's units. The penguin variance would be measured in square grams, a unit nobody can picture. The cure: the standard deviation is the one you report, because it is back in grams, days, or pips. A second slip is dividing by N for a sample. If your data are a sample, the formula divides by one less than the count, and STDEV.S is the spreadsheet function that does that.

In practice (your tools). In Google Sheets, =STDEV.S(F2:F345) returns the sample standard deviation of the penguin masses, about 801.95 g, and =VAR.S(F2:F345) returns the sample variance. For the twelve months in A1:A12, =STDEV.P(A1:A12) divides by the population size and returns about 0.86, while =STDEV.S(A1:A12) divides by 11 and returns about 0.90. The older name =STDEV is the sample version. In Excel, all four functions carry the same names. Pick the .S or the .P ending by asking one question: is this a sample or the whole population?

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

(a) Think of two things in your life with the same average but different spread, such as two commutes or two part-time jobs; which one would you choose, and why does the spread matter to you?
(b) Think of a product whose consistency matters more than its average, such as a medicine dose or a bag of coffee; who would care about its standard deviation, and why?

5. Five numbers: quartiles, the IQR, and the boxplot

Picture runners finishing a charity 5K. You could line them up by finishing time and stop the line at four spots: the first runner, the runner one quarter of the way along, the middle runner, the runner three quarters along, and the last. Those five people tell you a lot about the whole race. That is the five-number summary.

A quartile is one of the three values that cut ordered data into four parts with about the same number of values in each.
The first quartile, written Q1, is the median of the lower half of the ordered data; about a quarter of the values sit at or below it.
The third quartile, written Q3, is the median of the upper half of the ordered data; about three quarters of the values sit at or below it.
The five-number summary is the minimum, the first quartile, the median, the third quartile, and the maximum, listed in that order.
The interquartile range, written IQR, is the third quartile minus the first quartile; it is the spread of the middle half of the data and is resistant to extreme values.

Worked example — the five-number summary of penguin body mass. The 342 recorded masses, sorted from lightest to heaviest, give these five numbers.

Five-number summary of the 342 recorded penguin body masses, in grams
Minimum	Q1	Median	Q3	Maximum
2,700	3,550	4,050	4,750	6,300

Read the middle half. IQR of body mass: 4,750 − 3,550 = 1,200 g. The middle half of the penguins spans 1,200 g.
Notice the lean. The median sits 4,050 − 3,550 = 500 g above Q1 but 4,750 − 4,050 = 700 g below Q3. The upper half is more stretched out, which agrees with the right skew from Section 3.

Honesty note: different tools round the quartiles a little differently. Another common method puts this Q3 at 4,756.25 g instead of 4,750 g. The conclusions in this section come out the same under every common method.

The boxplot

A boxplot is a graph of the five-number summary: a box from Q1 to Q3 with a line at the median, whiskers to the most extreme unflagged values, and flagged values drawn as dots.

You can describe the penguin boxplot in words. A box runs from 3,550 g to 4,750 g along a number line, with a line inside it at 4,050 g. The line sits closer to the left end of the box. A whisker runs left to 2,700 g and a longer whisker runs right to 6,300 g. No dots sit beyond the whiskers. The longer right whisker and the off-center median both say "right-skewed."

The 1.5 × IQR rule

Week 2 left you with an open question about February: when is a value far enough away to call an outlier? Here is the rule the course uses.

The 1.5 × IQR rule is a rule of thumb that flags a value as a potential outlier when it lies more than one and a half interquartile ranges below the first quartile or above the third quartile.
A fence is one of the two cut-off values in the 1.5 × IQR rule: the first quartile minus 1.5 times the IQR, and the third quartile plus 1.5 times the IQR.

Worked example — fences for the penguin masses. You already have Q1, Q3, and the IQR, so you need three short steps.

Find the step. 1.5 × 1,200 = 1,800 g.
Build the fences. 3,550 − 1,800 = 1,750 g and 4,750 + 1,800 = 6,550 g.
Check the extremes. The minimum, 2,700 g, sits above 1,750 g. The maximum, 6,300 g, sits below 6,550 g. Both sit inside your fences, so no value is flagged.

Worked example — fences for the month lengths. Now try the rule on a small dataset you can check by hand.

Find the quartiles. The sorted lengths are 28, 30, 30, 30, 30, 31, 31, 31, 31, 31, 31, 31. The lower six end in 30 and 30 at their middle, so Q1 = 30. The upper six are all 31, so Q3 = 31.
Find the IQR and the step. The IQR is 31 − 30 = 1 day, and 1.5 × 1 = 1.5.
Build the fences. 30 − 1.5 = 28.5 and 31 + 1.5 = 32.5.
Check the extremes. February's 28 sits below 28.5, so it is flagged as a potential outlier. Nothing sits above 32.5.

In words: by the course's rule, February is unusual among the months. It is also a real, correct value, so it stays in the data. Its boxplot shows a short box from 30 to 31 with the median line at the right edge, 31, a whisker that ends at 30, and one separate dot at 28.

Misconception check. Students often treat "flagged" as "delete." A flagged value is a value to look at, not a value to throw away. February is flagged, and it is a fact of the calendar. A penguin recorded at 60,000 g would be flagged too, and that one is almost surely a typing error. The cure: when the rule flags a value, ask whether it is a mistake or a real case. Fix a mistake; keep and report a real case.

In practice (your tools). In Google Sheets, =MIN(F2:F345), =QUARTILE.INC(F2:F345,1), =MEDIAN(F2:F345), =QUARTILE.INC(F2:F345,3), and =MAX(F2:F345) return the five numbers 2,700, 3,550, 4,050, 4,750, and 6,300. In two more cells, =QUARTILE.INC(F2:F345,1)-1.5*(QUARTILE.INC(F2:F345,3)-QUARTILE.INC(F2:F345,1)) returns the lower fence, 1,750, and the matching formula with a plus sign on Q3 returns the upper fence, 6,550. Google Sheets has no one-click boxplot, so sketch it by hand from the five numbers. In Excel, select the column and use Insert, then Insert Statistic Chart, then Box and Whisker; then right-click the box, choose Format Data Series, and set Quartile Calculation to Inclusive median so it matches QUARTILE.INC.

Limitation: the 1.5 × IQR rule is a working rule of thumb, not a law of nature. It gives you and every classmate the same yardstick, but the final call on a flagged value is yours.

Key terms

measure of center — A single number that describes where the middle of a distribution sits; the mean, the median, and the mode are the three common ones. (first used: 1. Center: find the middle three ways)
mean — The sum of all the values divided by how many values there are; it uses the size of every value and needs interval or ratio data. (first used: 1. Center: find the middle three ways)
median — The middle value once the data are put in order, or the average of the two middle values when the count is even; it needs data that are at least ordinal. (first used: 1. Center: find the middle three ways)
mode — The value or category that occurs most often; it is the only measure of center that works for nominal data. (first used: 1. Center: find the middle three ways)
sample mean — The mean of the values in a sample, written x̄ and read "x bar"; it is a statistic. (first used: 1. Center: find the middle three ways)
population mean — The mean of every value in a whole population, written μ and read "mu"; it is a parameter. (first used: 1. Center: find the middle three ways)
resistant measure — A summary that changes little when one extreme or mistyped value is added or changed; the median is resistant and the mean is not. (first used: 2. Resistance: one wild value, two very different reactions)
range — The largest value minus the smallest value; it uses only the two most extreme values, so a single unusual value can stretch it. (first used: 4. Spread: measure how far the values stray)
deviation — How far one value sits from the mean, found by subtracting the mean from the value; deviations above the mean are positive and those below are negative. (first used: 4. Spread: measure how far the values stray)
variance — The average size of the squared deviations from the mean; it is measured in the square of the data's units. (first used: 4. Spread: measure how far the values stray)
standard deviation — The square root of the variance; it describes the typical distance of a value from the mean, in the data's own units. (first used: 4. Spread: measure how far the values stray)
population variance — The squared deviations from the population mean added up and divided by the population size, written σ². (first used: 4. Spread: measure how far the values stray)
sample variance — The squared deviations from the sample mean added up and divided by one less than the sample size, written s². (first used: 4. Spread: measure how far the values stray)
Bessel's correction — Dividing a sample's sum of squared deviations by one less than the sample size instead of by the sample size, which keeps the sample variance from running too small on average. (first used: 4. Spread: measure how far the values stray)
quartile — One of the three values that cut ordered data into four parts with about the same number of values in each. (first used: 5. Five numbers: quartiles, the IQR, and the boxplot)
first quartile — The median of the lower half of the ordered data, written Q1; about a quarter of the values sit at or below it. (first used: 5. Five numbers: quartiles, the IQR, and the boxplot)
third quartile — The median of the upper half of the ordered data, written Q3; about three quarters of the values sit at or below it. (first used: 5. Five numbers: quartiles, the IQR, and the boxplot)
five-number summary — The minimum, the first quartile, the median, the third quartile, and the maximum, listed in that order. (first used: 5. Five numbers: quartiles, the IQR, and the boxplot)
interquartile range — The third quartile minus the first quartile, written IQR; it is the spread of the middle half of the data and is resistant to extreme values. (first used: 5. Five numbers: quartiles, the IQR, and the boxplot)
boxplot — A graph of the five-number summary: a box from Q1 to Q3 with a line at the median, whiskers to the most extreme unflagged values, and flagged values drawn as dots. (first used: 5. Five numbers: quartiles, the IQR, and the boxplot)
1.5 × IQR rule — A rule of thumb that flags a value as a potential outlier when it lies more than one and a half interquartile ranges below the first quartile or above the third quartile. (first used: 5. Five numbers: quartiles, the IQR, and the boxplot)
fence — One of the two cut-off values in the 1.5 × IQR rule: the first quartile minus 1.5 times the IQR, and the third quartile plus 1.5 times the IQR. (first used: 5. Five numbers: quartiles, the IQR, and the boxplot)

Summary

The mean is the fair share of the total, what each friend pays when the food bill is split evenly; the median is the middle of the ordered list; the mode is the most common value.
Use the mode for nominal data, the median for ordinal or higher, and the mean only for interval or ratio data.
The median resists an extreme value; the mean does not.
Skew pulls the mean toward the long tail, so a mean below the median suggests left skew and a mean above it suggests right skew.
The range uses two values; the variance and the standard deviation use every value.
Divide by one less than the sample size for a sample and by the population size for a population.
Report the standard deviation, which is in the data's units, rather than the variance, which is in squared units.
The five-number summary and its boxplot show the center, the spread, and the lean at a glance.
The 1.5 × IQR rule flags a value for a closer look; a flagged real value stays in the data.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

A health survey records three things about each respondent: favorite fruit, a satisfaction rating (very unhappy, unhappy, neutral, happy, very happy), and minutes spent waiting at the clinic. Which measures of center may you use for each variable, and why?
Five apartments on one block rent for $1,150, $1,200, $1,250, $1,300, and $3,600 a month. Find the mean and the median rent. Which better describes a typical apartment on the block, and what shape does the comparison suggest?
A book-club member logs the pages she read on five days: 4, 6, 8, 10, and 12. Treat the five days as a sample. Find the sample variance and the sample standard deviation, and say what the standard deviation means.
A bakery times eleven deliveries, in minutes: 18, 20, 22, 22, 23, 24, 25, 27, 27, 29, 45. Find the five-number summary and the fences, and say whether any delivery is flagged.

Answers (worked reasoning — read after attempting)

Favorite fruit is nominal, so only the mode works: you can say which fruit is named most often, but fruit has no order and no average. The satisfaction rating is ordinal, so the mode and the median both work; a median of "happy" makes sense, but averaging the words does not. Waiting minutes are ratio data, so the mode, the median, and the mean all work. This item checks the misconception of averaging labels.
Mean: $1,150 + $1,200 + $1,250 + $1,300 + $3,600 = $8,500, and $8,500 ⁄ 5 = $1,700. Median: the rents are already in order, and the middle (3rd) one is $1,250. The mean is pulled up by the one $3,600 apartment, so it sits above four of the five rents. The median, $1,250, better describes a typical apartment. The mean above the median suggests a right skew, with the $3,600 rent in the right tail.
Mean: (4 + 6 + 8 + 10 + 12) ⁄ 5 = 40 ⁄ 5 = 8 pages. Deviations: −4, −2, 0, 2, 4, which add to 0. Squares: 16, 4, 0, 4, 16, which add to 40. This is a sample, so divide by one less than the count: s² = 40 ⁄ 4 = 10 square pages. Then s = √10 ≈ 3.16 pages. In words: on a typical day she read about 3 pages more or fewer than her average of 8. Dividing by 5 instead would give 8 and about 2.83, the population version, which is too small for a sample.
The eleven times are in order, so the median is the 6th, 24 minutes. The lower five are 18, 20, 22, 22, 23, so Q1 = 22. The upper five are 25, 27, 27, 29, 45, so Q3 = 27. The five-number summary is 18, 22, 24, 27, 45. The IQR is 27 − 22 = 5, and 1.5 × 5 = 7.5. The fences are 22 − 7.5 = 14.5 and 27 + 7.5 = 34.5. The 45-minute delivery sits above 34.5, so it is flagged; the 18-minute one sits inside. Before dropping the 45, the bakery should ask what happened: a wrong address is a real delivery, and it stays.

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, and the quiz — and next week the correlation coefficient measures how two variables move together.

Built with The Course Maker · samples.thecoursemaker.com