Introduction to Statistics · Chapter 2 · Descriptive Statistics
A lab, not a reading: survey your own class, build a histogram and a box plot by hand, and argue about what the numbers and the pictures agree on.
bookSHelf · Introduction to Statistics · §2.8 · a self-paced section
Learning objectives — by the end of this section you will be able to
§2.8.1 — Stats Lab: Descriptive Statistics
This section is a lab, not a reading. Every other section of this chapter handed you a data set and asked you to describe it. Here you go get the data yourself, and then run the whole chapter over it: a histogram from §2.2, quartiles and the outlier test from §2.3, a box plot from §2.4, the mean from §2.5, the shape language from §2.6, and the standard deviation from §2.7.
The question is one you can answer without any equipment: how many pairs of shoes do you own? Count yours before you read on.
Context Pause — you are the measuring instrument here
Every number here arrives through a person
Every number in this lab arrives through a person answering a question about themselves, so the usual data problems show up at once — someone counts flip-flops, someone else does not, someone rounds thirty-something down to thirty. That does not make the data unusable. It makes the definition part of the result.
A data set collected under thirty slightly different definitions has a spread that is partly real and partly an artifact of the question — and nothing computed afterward can separate the two.
§2.8.1 — fix the rule before anyone answers
Fix the counting rule with your group before the first person answers. Does a pair of slippers count? Cleats you have not worn since high school? A single shoe whose partner is missing? There is no right answer — but the statistics care very much whether all thirty people used the same one.
Write the rule at the top of your data sheet in one sentence, and read it to each person you survey rather than paraphrasing it from memory — a rule restated thirty times from memory is thirty slightly different rules.
Your turn — same mean, different spread
Try It Now 2.8.1 — Pao Xiong's group vs. a second group
Pao Xiong's group and a second group run this lab in the same classroom. Pao tells their classmates to count every pair of shoes they own; the second group tells its classmates to count only the pairs they have worn in the last year. Both report a mean near 12, but Pao's standard deviation is nearly twice the other group's. Give a reason for the difference that has nothing to do with sampling error.
Step 1 — compare what the two questions admit. The second group's rule caps what any one person can report — there are only so many pairs anybody actually wears in a year. Pao's rule has no such cap.
Step 2 — connect that to spread. Pao's question lets the high end run much further from the center, so their deviations are larger even when the two groups survey the same people.
Answer: the two groups measured different variables under the same name. Pao's larger standard deviation is a property of their wider definition, not evidence of a worse sample — "describe how you collected the data" is not busywork.
A demonstration data set
| Position | Values |
|---|---|
| 1–10 | 2, 3, 4, 4, 5, 5, 6, 6, 6, 7 |
| 11–20 | 7, 8, 8, 8, 9, 9, 10, 10, 11, 12 |
| 21–30 | 12, 13, 14, 15, 16, 18, 20, 22, 25, 32 |
Table 2.8.1: A previous class's 30 responses to "how many pairs of shoes do you own?", sorted from smallest to largest.
Sort first, always. The median and both quartiles are read straight off a sorted list by counting positions — on an unsorted list every one of them turns into a hunt, and a hunt is where arithmetic mistakes come from.
Your turn — find the two middle positions
Try It Now 2.8.2 — Alondra Reyes checks the count
Using Table 2.8.1, confirm the count is 30, then name the value in position 15 and the value in position 16. Why are those two positions the ones worth finding first?
Step 1 — count. Three rows of ten values each gives 3×10=30 responses.
Step 2 — read positions 15 and 16. The second row holds positions 11–20; position 15 is the fifth entry, 9, and position 16 is the sixth, also 9.
Answer: n=30, and both positions hold 9. With an even count there is no single middle value, so the median is the mean of these two — finding them first pins the center before you compute anything.
Insight Note — sorting in one move
Open rāSHio, paste your class's 30 shoe counts into the spreadsheet with File → Delimited List… (it accepts commas, semicolons, tabs, or one value per line), then choose Transform → Sort to put them in ascending order — every position count in this lab is read straight off that sorted column.
Figure 2.8.1
Transform → Sort
Select the range of values and choose Transform → Sort to put them in ascending order — every position count in this lab is read straight off that sorted column.
Figure 2.8.1: Putting the lab's values in ascending order in rāSHio: Transform → Sort.
§2.8.2 — collect the data
Survey 30 classmates, build a frequency table and a histogram by hand, then calculate the mean and standard deviation — and check the ends of your data against a formal outlier test rather than a guess.
§2.8.2 — a formal test, not a feeling
Definition 2.8.1 — Potential Outlier (the 1.5 × IQR rule)
A data value is a potential outlier when it falls outside the two fences built from the quartiles:
lower fence=Q1−1.5(IQR)upper fence=Q3+1.5(IQR)where IQR=Q3−Q1. A value below the lower fence or above the upper fence is flagged as a potential outlier; a value inside both fences is not, however far from the mean it may feel.
The word potential is load-bearing — the rule flags a value for inspection, it does not convict it. Deleting a real value because a formula raised its hand makes your sample less representative, not more.
§2.8.2 — step 1: survey your class
| Pairs of shoes owned — 30 responses | ||||
|---|---|---|---|---|
| ⋮ | ||||
Table 2.8.2: Survey results — the number of pairs of shoes owned by each of your 30 classmates.
Randomly survey 30 classmates. Count your own pairs first, and read the counting rule you fixed earlier to each person rather than paraphrasing it from memory.
§2.8.2 — step 2: construct the histogram
Five to six bars is wide enough to show a shape and narrow enough that each bar still holds several people — with thirty values, ten bars would leave most of them nearly empty and the shape would dissolve into noise.
Your turn — round the width up, not to the nearest whole
Try It Now 2.8.3 — six intervals for a range of 2 to 32
The demonstration data runs from 2 to 32 pairs. Find an interval width that fits the whole data set into six intervals, and give the six boundaries. Show why a width of 5 does not work here.
Step 1. 632−2=630=5.
Step 2 — test it. Starting a half unit below the minimum, at 1.5, six intervals of width 5 run 1.5 to 31.5 — the largest value, 32, falls outside.
Step 3 — round up. A width of 6, starting at 1.5, gives boundaries at 1.5, 7.5, 13.5, 19.5, 25.5, 31.5, 37.5, which covers every value.
Answer: a width of 6. Dividing the range by the bar count gives the smallest width that could work — any rounding down leaves data outside the last bar.
Insight Note — testing a bin width fast
With your values in rāSHio, choose Graph → Histogram and set the bin width you just worked out — the tool bins and draws the bars for you, so you can try a width of 5 against a width of 6 and watch your largest value fall outside the last bar in one and inside it in the other.
Your turn — tally the sorted list once
Try It Now 2.8.4 — Ethan Doyle's frequency table
Ethan Doyle is building the frequency table for the demonstration data using the six intervals from Try It Now 2.8.3. Work it out alongside him and check his total.
| Interval | Frequency |
|---|---|
| 1.5–7.5 | 11 |
| 7.5–13.5 | 11 |
| 13.5–19.5 | 4 |
| 19.5–25.5 | 3 |
| 25.5–31.5 | 0 |
| 31.5–37.5 | 1 |
Table 2.8.3: Frequency table for the demonstration data, six intervals of width 6.
Check: 11+11+4+3+0+1=30. Do not delete the empty interval to tidy the picture — that gap is data, the first visual hint of the outlier at 32.
Insight Note — checking the tallies
To check the tallies you just made by hand, open Graph → Frequency Table in rāSHio and read the counts back — they still have to total 30, and an interval showing 0 is data about your class, not an error to bin away.
Figure 2.8.2
Graph → Frequency Table
The tool tallies every interval for you — the total still has to be 30, and a 0 in one interval is a real reading, not a mistake.
Figure 2.8.2: Reading the interval counts back off rāSHio: Graph → Frequency Table.
Your turn — the mean and the median disagree
Try It Now 2.8.5 — Nathan Reed compares the mean to the median
Nathan Reed wants the sample mean of the demonstration data so he can compare it to the median found in Try It Now 2.8.2. Find both for him.
Step 1 — add the 30 values. Row by row: 48+92+187=327.
Step 2 — divide by the count. xˉ=30327=10.9.
Answer: xˉ=10.9 pairs, against a median of 9. The mean sitting above the median is the arithmetic fingerprint of a right-skewed data set — the single value of 32 drags the mean up, while the median only cares which position it sits in.
Your turn — a spread nearly two-thirds of the mean
Try It Now 2.8.6 — the sample standard deviation
Find the sample standard deviation of the demonstration data. The sum of the squared values is ∑x2=4,967.
Step 1. ∑(x−xˉ)2=∑x2−n(∑x)2=4,967−303272=1,402.7.
Step 2. s2=291,402.7≈48.37.
Step 3. s=48.37≈6.95.
Answer: s≈6.95 pairs — nearly two thirds of a mean of 10.9. Quoting "the average student owns about 11 pairs" without this number beside it tells a reader almost nothing about any individual student.
Insight Note — one panel, both numbers
Rather than square thirty deviations by hand every time, choose Stats → Summary Statistics in rāSHio and read the mean and the sample standard deviation off one panel — work the arithmetic yourself once, as you just did, then let the tool carry it for the rest of the lab.
Figure 2.8.3
Stats → Summary Statistics
Read the mean beside the sample standard deviation — the same two numbers you just computed by hand.
Figure 2.8.3: The mean and the sample standard deviation on one panel in rāSHio: Stats → Summary Statistics.
Your turn — counting vs. measuring
Try It Now 2.8.7 — Cruz Aguilar sorts out discrete vs. continuous
Cruz Aguilar cannot decide whether their shoe counts are discrete or continuous data. Settle it, name the test you used, and say what would change if the survey had asked for each classmate's shoe size instead.
Step 1. Ask whether the values come from counting or from measuring. You count pairs of shoes; there is no value between 6 and 7 that anyone could report.
Step 2. Shoe size comes from measuring a foot — the half sizes on the box are a rounding convention, not a limit on the foot.
Answer: the shoe counts are discrete — whole-number counts with nothing in between. Shoe size would be continuous — a length that can take any value in a range. This is why the half-unit interval boundaries work: with discrete counts, no data point can ever sit on a boundary at 7.5.
Insight Note — shape is a sentence, not a word
Say where you saw it
"Skewed right" earns its keep only when you can say where you saw it — which side the long tail runs off, which bars are tall, where the gap is. A reader who cannot rebuild your picture from your sentence has not been told the shape.
§2.8.2 — item 5, done properly
Then check the sentence against your mean and median, which should be telling the same story in numbers.
Your turn — write the shape sentence
Try It Now 2.8.8 — Claire Whitfield describes the histogram
Using the frequencies 11, 11, 4, 3, 0, 1 from Try It Now 2.8.4, write the description Claire needs: name the shape and point at the evidence for it.
Step 1. The first two intervals hold 11 classmates each — 22 of 30, over two thirds, own 13 pairs or fewer.
Step 2. The bars fall away steadily after that: 4, then 3, then 0, then a single lone bar holding the value 32.
Answer: the histogram is skewed to the right. Two tall bars sit at the low end, the bars fall off through 4 and 3, an entire interval is empty, and one isolated classmate sits in the last interval — the mean of 10.9 lying above the median of 9 says the same thing in numbers.
§2.8.2 — item 6, the step students most often skip
Try It Now 2.8.9 — testing both end values against the fences
The demonstration data has Q1=6 and Q3=14. Test both end values, 2 and 32, against the fences of Definition 2.8.1 and say what you have shown.
Step 1. IQR=14−6=8.
Step 2. lower=6−1.5(8)=−6; upper=14+1.5(8)=26.
Step 3. 2 sits above −6 — inside the lower fence. 32 sits above 26 — outside the upper fence.
Answer: 2 is not a potential outlier; 32 is. The lower fence came out negative, which no shoe count can be — with a distribution pushed against zero, the rule can only ever flag values on the high side here.
§2.8.3 — analyze the data
Pull the five-number summary off your sorted list, draw a box plot from it, and read what its shape says about where your class's data is concentrated.
Your turn — read it off positions, not off a calculation
Try It Now 2.8.10 — the whole five-number summary
Produce the whole five-number summary for the demonstration data, plus the IQR, reading everything off Table 2.8.1 by position.
Step 1 — the ends. First value 2, thirtieth value 32: Min=2, Max=32.
Step 2 — the median. Positions 15 and 16 both hold 9: M=29+9=9.
Step 3 — quarter the halves. Position 8 (lower half's median) is 6; position 23 (upper half's median) is 14: Q1=6, Q3=14.
Step 4. IQR=14−6=8.
Answer: Min 2, Q1 6, M 9, Q3 14, Max 32, IQR=8. The gap from Min to Q1 is 4; from Q3 to Max it is 18 — the picture is already lopsided.
Insight Note — drawing the box plot
Choose Graph → Box Plot in rāSHio to draw the five-number summary you just read off the table — the tool marks the quartiles, runs each whisker out to the last value inside the fences, and plots anything past them as its own dot.
Figure 2.8.4
Graph → Box Plot
Quartiles marked, whiskers run to the last non-outlier value, anything past the fences plotted as its own dot.
Figure 2.8.4: Drawing the five-number summary in rāSHio: Graph → Box Plot.
§2.8.3 — step 2: construct the box plot
Find the box first — the middle half of your class lives inside it. Then compare the two whiskers: a long one beside a short one is the same right-skew you named from the histogram, seen from a different angle.
Insight Note — a box plot is a map with the scale printed on it
Width is crowding, not headcount
Every piece of the box covers exactly a quarter of your classmates, so a short piece means those people are packed tightly into a narrow range and a long piece means the same number of people are stretched thin across a wide one.
Your turn — read concentration off unequal widths
Try It Now 2.8.11 — describing the concentration
Using the five-number summary from Try It Now 2.8.10, describe the concentration of the demonstration data, and say how the drawn box plot lets a reader spot the outlier without doing any arithmetic.
Step 1 — measure the pieces. Left whisker 6−2=4; left half of box 9−6=3; right half 14−9=5; right whisker to 25, 25−14=11. The flagged 32 plots as its own dot beyond the whisker.
Step 2. Each piece holds about 7–8 classmates, yet the middle half is squeezed into 8 pairs (6–14) while the top quarter alone spans 11 pairs (14–25).
Answer: the data is concentrated at the low end. A reader spots the outlier because the whisker stops at 25 and a lone dot sits past it — the whisker is drawn only to the most extreme non-flagged value, so anything beyond it has already failed the 1.5×IQR test.
Your turn — the standard deviation confirms the shape
Try It Now 2.8.12 — Danielle Tallchief checks the 1.5-SD boundary
Danielle has xˉ=10.9 and s≈6.95. Find the values 1.5 standard deviations above and below the mean, and say what the lower one tells her about the shape.
Step 1. xˉ+1.5s=10.9+1.5(6.95)≈21.3.
Step 2. xˉ−1.5s=10.9−1.5(6.95)≈0.5.
Answer: about 21.3 and 0.5 pairs. Half a pair of shoes is impossible, yet that value lands only 1.5 standard deviations below the mean — the data cannot run that far down because it stops at zero, while nothing stops it running upward. That lopsidedness is the signature of a distribution skewed to the right, agreeing with the fences from Try It Now 2.8.9.
Key Terminology — the vocabulary this lab put to work
box plot — a graph that gives a quick picture of the middle 50% of the data.
first quartile — the median of the lower half of the ordered data set.
frequency — the number of times a value of the data occurs.
frequency polygon — a line graph that uses intervals to display ranges of large amounts of data.
frequency table — grouped data displayed with its corresponding frequencies.
histogram — a graph of x–y form where x is the data and y is the frequency, drawn as contiguous rectangles.
interquartile range — IQR, the range of the middle 50% of the data: Q3−Q1.
interval — a range of data used when displaying large data sets.
mean — the sum of all values divided by the count; xˉ for a sample, μ for a population.
median — the number that separates ordered data into halves.
midpoint — the mean of an interval in a frequency table.
mode — the value that appears most frequently in a data set.
outlier — an observation that does not fit the rest of the data.
paired data set — two equal-size data sets with a one-to-one relationship.
percentile — a number that divides ordered data into hundredths.
quartiles — the numbers that separate the data into quarters.
relative frequency — the ratio of a value's count to the count of all outcomes.
skewed — not symmetrical; chopped off on the right is skewed left, chopped off on the left is skewed right.
standard deviation — the square root of the variance; s for a sample, σ for a population.
variance — the mean of the squared deviations from the mean.
The headline result
Center, spread, and shape have to agree — or you have made an arithmetic mistake.
Demonstration data: xˉ=10.9, median =9, s≈6.95 — the mean sits above the median, and 1.5s below the mean lands on 0.5, essentially the floor of the variable.
Three independent tools — the shape sentence, the 1.5×IQR fences, and the 1.5-SD check — all pointed at the same lone value: 32.
† The lower fence for this data comes out negative — no shoe count can be negative — so the 1.5×IQR rule here can only ever flag values on the high side. That is a property of a distribution pinned at zero, not a flaw in the test.
§2.8 — conclusions
A lab turns the chapter's formulas into a real, once-collected data set. Every number you compute — the mean, the standard deviation, the five-number summary — has to be read against how you defined and gathered the variable, not treated as free-standing arithmetic.
A formula flags a potential outlier — it does not remove it, and a value quietly dropped is the easiest way to make honest data lie. When your shape sentence, your fences, and your box plot all point at the same value, that agreement is the finding, not a coincidence.
Next: Chapter 3 — Probability Topics, where the frequencies you just tallied by hand become the language of chance.