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Week 3 · Chapter

Chapter 3 — Numerical Summaries: Center & Spread

Introduction to Statistics Generic evergreen edition

Course: Introduction to Statistics (18-week generic edition)
The module's primary reading · ~25–35 minutes · Objective 2 · Read this before (or alongside) the week's lecture — everything else in the module builds on it.

Why you already care about this

You made decisions this week based on averages you never questioned. Your maps app told you a route "averages" a certain time. A listing site told you what apartments "average" in your area. A syllabus told you the class average on something. An "average" is the most quoted statistic in the world — and the easiest one to be misled by, because the word hides a genuine choice: which average, and what is it not telling you?

Last week you learned to turn a pile of numbers into a picture and describe its shape. A picture is honest but bulky — you can't text someone a histogram and expect a decision back. This week we compress: a whole dataset squeezed into two or three numbers — one for center (where's the middle of this?) and one for spread (how much do the values wander?). Compression always loses information. The skill of the week is knowing exactly what got lost.

By the end of this chapter you will be able to: compute and choose among the mean, median, and mode; explain resistance — why one wild value drags the mean but barely touches the median; compute a standard deviation by hand and say what it means in words; build a five-number summary, find the IQR, and apply the 1.5×IQR rule that formally flags outliers; and use a z-score to compare values from completely different scales. One line to carry through the week: center without spread is half a story.

Where we are

Week 1 gave you the machinery of trust: populations vs. samples, parameters vs. statistics, and how data get collected. Week 2 turned raw numbers into pictures — histograms and their shapes, including skew (a long tail to one side) and the idea of values that sit suspiciously far from the rest. This week those shapes get measured. You'll see that choosing the right summary depends on the shape — which is exactly why the pictures came first.

1. Three centers: mean, median, mode

Start with the plain ideas — all three are honest attempts to answer "what's a typical value?"

  • The mean is the everyday "average": add everything, divide by how many. Picture it as the balance point — if the values were weights on a seesaw, the mean is where it balances.
  • The median is the middle value once the data are sorted: half the values sit below it, half above. With an even count, average the two middle values. Sorting first is not optional — the median is a position, and positions only exist in sorted data.
  • The mode is the most frequent value. It's the humblest of the three, but it's the only center that works for categorical data: "the most common apartment type is a one-bedroom" is a mode statement.

Worked example. Five one-way commute times, in minutes: 15, 20, 20, 25, 30.

  • Mean: 15 + 20 + 20 + 25 + 30 = 110, and 110 ÷ 5 = 22 minutes.
  • Median: the list is already sorted; the middle (3rd) value is 20 minutes.
  • Mode: 20 appears twice, everything else once, so the mode is 20 minutes.

Three defensible "centers," two different numbers. The mean sits a little above the median because the 30 tugs the balance point upward while the median only counts positions. That small gap is about to become the entire plot of this chapter.

Once the ideas are solid, the notation is quick — and it rhymes with Week 1. The sample mean is written x̄ ("x-bar"); the population mean is μ ("mu"). Samples get Latin letters; populations get Greek. The statistic x̄ is what you compute from your data; the parameter μ is the true value you're estimating — the same have-versus-want split as p̂ and p.

Misconception check. "The mode has to be unique, and it's basically useless." Neither. Ties are legal — a dataset with two modes (bimodal) is often a clue that two different groups got mixed together, a shape idea straight from Week 2. And for categorical data, the mode isn't just useful; it's the only center that exists.

2. Resistance: the day the mean lied

Here is the most important experiment in the chapter, small enough to do in your head.

Worked example. A rideshare driver logs five airport runs: 12, 18, 20, 22, 28 minutes.

  • Sum = 12 + 18 + 20 + 22 + 28 = 100, so the mean = 100 ÷ 5 = 20 minutes.
  • Sorted, the middle value gives a median of 20 minutes. Perfect agreement.

Now the twist. One day a crash closes the highway, and the 28-minute run becomes 88 minutes. The data are now 12, 18, 20, 22, 88.

  • New sum = 12 + 18 + 20 + 22 + 88 = 160, so the new mean = 160 ÷ 5 = 32 minutes.
  • New median: sorted, the list is 12, 18, 20, 22, 88 — the middle value is still 20 minutes.

One bad day moved the mean twelve minutes. It moved the median zero. The mean of 32 now describes a commute that basically never happens.

This behavior has a name. A summary is resistant if extreme values (outliers) barely budge it. The median is resistant; the mean is not — every value tugs on the balance point, and a far-away value tugs hardest.

Resistance connects directly to Week 2's shapes. In a right-skewed distribution — a long tail toward high values, the classic shape of incomes, home prices, and rents — the tail's extreme values drag the mean up past the median. Left skew drags it below. The memory hook: the mean chases the tail. That is exactly why rent and income reports lead with the median: one penthouse shouldn't get to redefine "typical."

So which center should you report?

  • Roughly symmetric data with no wild values → the mean (it uses every value's actual size, which is a feature when nothing abuses it).
  • Skewed or outlier-prone data → the median (and report the mean too if you like — the gap between them is itself a diagnostic).
  • Categorical data → the mode is your only option.

Misconception check. "If a value looks extreme, just delete it and move on." Never silently. First ask: typo or truth? A 9,999-gram penguin is a data-entry error; an 88-minute commute really happened. The professional move is to investigate, report results with and without the value, and use a formal rule (Section 4) — not a feeling — to decide what counts as an outlier in the first place.

3. Spread: from the range to the standard deviation

Two grocery stores. Your last five weekly bills at Store A: 90, 95, 100, 105, 110 dollars. At Store B: 70, 85, 100, 115, 130. Both average exactly $100 — check A: 90 + 95 + 100 + 105 + 110 = 500, and 500 ÷ 5 = 100; B: 70 + 85 + 100 + 115 + 130 = 500, same mean. Same center. Completely different experience: Store B is a budgeting rollercoaster. Center without spread is half a story.

The quickest spread measure is the range: maximum − minimum. Store A: 110 − 90 = $20. Store B: 130 − 70 = $60. Fast and intuitive — but the range uses only two values, so a single outlier owns it completely, and it's blind to everything in between.

The professional tool is built from deviations: each value's gap from the mean, (value − x̄), with its sign. Deviations have one famous quirk: they always sum to zero. That's not a bug — it's the balance point doing its job — but it means we can't just average them to measure spread. The fix: square them first.

Worked example — the standard deviation recipe. Five quiz scores (out of 10): 6, 6, 8, 10, 10.

  1. Mean: 6 + 6 + 8 + 10 + 10 = 40, and 40 ÷ 5 = 8.
  2. Deviations: 6 − 8 = −2 · 6 − 8 = −2 · 8 − 8 = 0 · 10 − 8 = 2 · 10 − 8 = 2. (They sum to 0 — good, the arithmetic checks.)
  3. Square each: 4, 4, 0, 4, 4.
  4. Add: 4 + 4 + 0 + 4 + 4 = 16.
  5. Divide by n − 1 = 4: 16 ÷ 4 = 4. This is the sample variance, s² = 4 — in squared units (points²).
  6. Square root: √4 = 2. The sample standard deviation is s = 2 points.

Now say it in words, because this is the sentence that matters: a typical quiz score sits about 2 points from the mean of 8. That is what the standard deviation is — the typical distance from the mean. Back at the grocery stores, technology gives Store A s ≈ 7.9 and Store B s ≈ 23.7: same center, three times the wobble.

The vocabulary, locked down:

  • Variance (s²) is the average squared deviation (with the n − 1 divisor). Its units are squared — points², dollars² — which is why nobody speaks variance out loud; it's a stepping stone.
  • Standard deviation (s) is the square root of the variance, back in the data's own units. SD is never negative, and s = 0 means every value is identical.
  • Why divide by n − 1? One line at this level: a sample's deviations are measured from x̄, which came from the same sample, so they run slightly small — dividing by n − 1 instead of n corrects the books. Your spreadsheet and calculator do this automatically for sample data. The population versions use n and get Greek letters: σ (population SD) and σ² (population variance).
  • Like the mean it's built on, the SD is not resistant — the crash-day 88 would inflate it dramatically.

Misconception check. "The variance is 4, so the spread is 4 points." Units betray you here: that 4 is in points squared, which measures nothing you can picture. Take the square root and report s = 2 points. Variance and SD are related, never interchangeable — and on a quiz, "s² = 4" and "s = 4" are very different claims.

In practice (your tools). The whole toolkit is five functions, identical in Google Sheets and Excel: =AVERAGE(), =MEDIAN(), =MODE(), =STDEV(), and =VAR(). One trap worth naming now: =STDEV() uses the sample divisor (n − 1) — that's the one this course means. Its lookalike =STDEVP() divides by n (the population version); on 6, 6, 8, 10, 10 the wrong function returns ≈1.79 instead of 2. If your SD looks slightly small, check your divisor — you'll run this exact trap on purpose in the data lab.

4. Five numbers, one picture: quartiles, IQR, and the outlier fence

The median splits the data in half. Do it twice more and you've mapped the whole dataset:

  • Q1 (first quartile) = the median of the lower half — about 25% of values sit below it.
  • Q3 (third quartile) = the median of the upper half — about 75% sit below it.
  • The five-number summary — minimum, Q1, median, Q3, maximum — is a five-word biography of any dataset.

Worked example. Monthly rents for eight studio apartments, already sorted (dollars): 950, 1000, 1100, 1150, 1250, 1300, 1400, 2400.

  • Median: even count, so average the 4th and 5th values: (1150 + 1250) ÷ 2 = 1200.
  • Q1: median of the lower half (950, 1000, 1100, 1150) = (1000 + 1100) ÷ 2 = 1050.
  • Q3: median of the upper half (1250, 1300, 1400, 2400) = (1300 + 1400) ÷ 2 = 1350.
  • Five-number summary: 950 · 1050 · 1200 · 1350 · 2400.

From the quartiles comes the resistant spread measure: the interquartile range, IQR = Q3 − Q1 = 1350 − 1050 = 300 — the width of the middle 50% of the data. Notice the $2,400 luxury listing never touched it: the IQR, built from positions, is resistant.

And from the IQR comes the course's formal outlier rule. The 1.5×IQR rule: compute 1.5 × IQR = 1.5 × 300 = 450, then set fences at

  • lower fence = Q1 − 450 = 1050 − 450 = 600,
  • upper fence = Q3 + 450 = 1350 + 450 = 1800.

Any value outside the fences is flagged as an outlier. Here, 2400 > 1800 — formally flagged (some landlord is dreaming). Flagged means investigate — typo or truth? — never silently delete.

For the record, the mean of these rents is 10,550 ÷ 8 = 1318.75, well above the median of 1200 — the luxury listing drags the balance point, the median shrugs. Section 2's lesson, now with a dollar sign.

The boxplot (from Week 2's plot family) is the five-number summary drawn: a box from Q1 to Q3 holding the middle 50%, a line at the median — not the mean — whiskers out to the most extreme values still inside the fences, and individual dots beyond the fences for flagged outliers. A lopsided box or one long whisker is skew you can see, and side-by-side boxplots are the fastest honest comparison of groups you'll meet in this course.

Finally, the pairing rule that organizes everything so far:

The mean travels with the SD. The median travels with the IQR. Report the resistant pair (median + IQR) for skewed or outlier-prone data; the classic pair (mean + SD) for well-behaved data.

Misconception check. "The line in the middle of a boxplot is the mean." It's the median. A boxplot is built entirely from the five-number summary — resistant parts only. The mean isn't in the picture at all, which is precisely why boxplots stay honest around outliers.

In practice (your tools). =QUARTILE(range, 1) and =QUARTILE(range, 3) return the quartiles; subtract for the IQR; =MIN() and =MAX() finish the five-number summary. One honest footnote: software computes quartiles by interpolation, so =QUARTILE can differ slightly from our split-the-halves hand method. Neither is wrong — they're different conventions. Say which method you used, and don't panic when the spreadsheet disagrees with your pencil by a few dollars.

5. z-scores: how many SDs from home?

A raw number means little without context. Is a 44-minute commute bad? Depends on what's typical for that route and how much it usually wobbles. The z-score folds both into one universal number:

z = (value − mean) ÷ SD — how many standard deviations from the mean.

  • z = 0 → exactly average. Positive → above the mean; negative → below.
  • Sign is direction; size is rarity. |z| around 2 or beyond is starting to be genuinely unusual (a precise version of "unusual" arrives with the normal model later in the course).

Worked example — comparing across different scales. You scored 85 on a statistics exam where the class mean was 75 with SD 5. You scored 80 on a chemistry exam with class mean 70 and SD 10. Which performance is more impressive?

  • Statistics: z = (85 − 75) ÷ 5 = 10 ÷ 5 = 2.0 — two standard deviations above the class.
  • Chemistry: z = (80 − 70) ÷ 10 = 10 ÷ 10 = 1.0 — one standard deviation above.

Both scores beat their class average by the same 10 raw points — but the statistics performance is rarer, because in a tighter distribution the same gap is a bigger feat. z-scores make different scales comparable: exams, commute times, penguin masses — anything with a mean and an SD.

Misconception check. Two in one here. "z = 2 means top 2%" — no: z is a distance in SD units, not a percentile; converting z to a percentage needs the normal model (coming later in the course). And "a negative z-score means something went wrong" — no: negative just means below the mean, which for commute times or grocery bills is the direction you want. Read the sign as direction, never as judgment.

Key terms

  • Mean (x̄ sample, μ population) — the balance point: sum of values ÷ count. Not resistant.
  • Median — the middle value of the sorted data (average the middle two for an even count). Resistant.
  • Mode — the most frequent value; the only center available for categorical data.
  • Resistant measure — a summary that extreme values barely budge (median, IQR — yes; mean, range, SD — no).
  • Range — maximum − minimum; quick but owned entirely by the two end values.
  • Deviation — (value − mean), with sign; deviations always sum to zero.
  • Variance (s² sample, σ² population) — the average squared deviation (n − 1 divisor for samples); units are squared.
  • Standard deviation (s sample, σ population) — √variance; the typical distance from the mean, in the data's own units; never negative.
  • Five-number summary — min, Q1, median, Q3, max.
  • Quartiles (Q1, Q3) — medians of the lower and upper halves; about 25% and 75% of data sit below them.
  • IQR (interquartile range) — Q3 − Q1, the width of the middle 50%; the resistant spread measure.
  • 1.5×IQR rule — fences at Q1 − 1.5·IQR and Q3 + 1.5·IQR; values beyond the fences are formally flagged as outliers.
  • Boxplot — the five-number summary drawn: box Q1→Q3, line at the median, whiskers to the last values inside the fences, dots beyond.
  • z-score — (value − mean) ÷ SD: how many SDs a value sits from the mean; compares values across different scales.

Summary

  • Three centers: mean (balance point), median (middle of the sorted list), mode (most common; the only one for categories). Samples write x̄ and s; populations write μ and σ — Latin for samples, Greek for populations.
  • Resistance is the deciding factor: one crash-day 88 moved a mean from 20 to 32 and the median not at all. The mean chases the tail — so skewed data (rents, incomes) report the median.
  • Spread: range = max − min (fragile); SD = the typical distance from the mean, via deviations → square → add → ÷(n−1) → root. Variance is SD² in squared units. On 6, 6, 8, 10, 10: s = 2 points.
  • Five-number summary (min, Q1, median, Q3, max) → IQR = Q3 − Q1 → fences at 1.5×IQR beyond the quartiles → values outside are flagged outliers, to be investigated, never silently deleted. The boxplot draws it all, with the line at the median.
  • Pairing rule: mean travels with SD; median travels with IQR.
  • z = (value − mean) ÷ SD measures relative standing in SD units — direction from the sign, rarity from the size, and no percentages until the normal model.

Check your understanding

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

  1. Five used textbooks cost 38, 42, 45, 47, 88 dollars. Compute the mean and the median, state which is larger, explain why in one sentence — and say which one you'd quote as "the typical price."
  2. Five study sessions last week ran 28, 28, 30, 32, 32 minutes. Compute the sample standard deviation by the full recipe (mean → deviations → squares → sum → ÷(n−1) → root), then say the result in words.
  3. Eight monthly water bills, sorted (dollars): 50, 60, 70, 80, 90, 100, 110, 170. Build the five-number summary, compute the IQR, set the 1.5×IQR fences, and say whether any bill is formally an outlier.
  4. One podcast's episodes average 46 minutes with SD 8; this week's episode ran 58. Another show's episodes average 30 minutes with SD 4; its latest ran 40. Which episode is more unusual for its own show, and what did the z-scores do that raw minutes couldn't?

Answers (worked reasoning — read after attempting)

  1. Sum = 38 + 42 + 45 + 47 + 88 = 260, so the mean = 260 ÷ 5 = $52. Sorted, the middle value gives median = $45. The mean is larger because the $88 book drags the balance point upward while the median only counts positions (the mean chases the tail). Quote the median ($45) as typical — four of the five books cost less than the mean, so $52 describes almost nobody's purchase.
  2. Mean = (28 + 28 + 30 + 32 + 32) ÷ 5 = 150 ÷ 5 = 30. Deviations: −2, −2, 0, 2, 2 (sum = 0 ✓). Squares: 4, 4, 0, 4, 4. Sum = 16. Divide by n − 1 = 4: variance s² = 4 (minutes²). Square root: s = 2 minutes. In words: a typical study session ran about 2 minutes from the mean of 30 — a very consistent week.
  3. Median = (80 + 90) ÷ 2 = 85. Q1 = median of (50, 60, 70, 80) = (60 + 70) ÷ 2 = 65. Q3 = median of (90, 100, 110, 170) = (100 + 110) ÷ 2 = 105. Five-number summary: 50 · 65 · 85 · 105 · 170. IQR = 105 − 65 = 40; 1.5 × 40 = 60; fences at 65 − 60 = 5 and 105 + 60 = 165. The $170 bill exceeds the upper fence → formally flagged as an outlier — investigate before deleting (a leak is truth, a typo is not).
  4. First show: z = (58 − 46) ÷ 8 = 12 ÷ 8 = 1.5. Second show: z = (40 − 30) ÷ 4 = 10 ÷ 4 = 2.5. The second show's episode is more unusual — it sits 2.5 SDs above its own average, versus 1.5 for the first — even though its raw excess (10 minutes) is smaller than the first show's (12 minutes). The z-scores put both episodes on the same ruler — SD units — which raw minutes can't do because the two shows wobble by different amounts.

Next up in this module: the slide deck if you want the lecture's visual walk-through, then the Readings & Resources page for optional deeper dives — and when you're ready to practice, Lecture Tutorial 3 turns your chatbot into your personal Week 3 tutor.