Introduction to Statistics · Chapter 2 · Descriptive Statistics
Contiguous boxes over a number line: read a distribution's shape, center, and spread in one picture — and learn where a squeezed axis can turn an honest 2% into a lie.
bookSHelf · Introduction to Statistics · §2.2 · a self-paced section
Learning objectives — by the end of this section you will be able to
§2.2.1 — Anatomy of a Histogram
A histogram is a graph of contiguous boxes over a horizontal and a vertical axis. The horizontal axis is labeled with what the data represent; the vertical axis carries frequency or relative frequency — the shape is the same either way.
Like the stemplot before it, a histogram shows you three things at a glance: the shape of the data, the center, and the spread.
§2.2.1 — the central definition
Definition 2.2.1 — Histogram
A histogram is a graph made of contiguous (adjoining) boxes drawn on a horizontal and a vertical axis. The horizontal axis is labeled with what the data represent; the vertical axis is labeled with frequency, or with relative frequency — the graph has the same shape under either vertical label.
Definition 2.2.1: A histogram sorts values into touching intervals; each bar's height is the count inside it.
Notice the word contiguous. A bar graph leaves gaps between its bars because its categories are separate things; a histogram's boxes touch because the horizontal axis is a continuous number line, and every value on it belongs to some interval.
Insight Note — seeing the crowd from above
A histogram is a crowd photographed from above
Each person stands over their own value on the number line, and identical values stack up. The taller the pile, the more people share that measurement — you are looking at where the crowd bunches and where it thins out.
§2.2.1 — turning a count into a share
Definition 2.2.2 — Relative Frequency
The relative frequency of an observed value is its frequency divided by the total number of data values, n:
RF=nfDefinition 2.2.2: Relative frequency relabels each count as its share of the whole; the shape does not change.
Three of 40 students in a class earned 90–100%: RF=403=0.075 — 7.5% of the class. A relative frequency always lands between 0 and 1, and the relative frequencies of every value in a data set add up to 1.
Your turn — check that the column sums to 1
Try It Now 2.2.1 — highlighters purchased in one afternoon
Robin Ashcroft manages the campus bookstore with their wife Kate; one afternoon they recorded how many highlighters each customer bought.
| Highlighters | Frequency |
|---|---|
| 1 | 12 |
| 2 | 18 |
| 3 | 6 |
| 4 | 4 |
Find the relative frequency for each value, and check your work.
Step 1. n=12+18+6+4=40.
Step 2. Divide each frequency by 40: 0.300, 0.450, 0.150, 0.100.
Step 3 — check. 0.300+0.450+0.150+0.100=1.000. Answer: 0.300, 0.450, 0.150, 0.100 — run this check every time; if the column does not sum to 1, n was mis-added or a value was dropped.
Try it in rāSHio — building the frequency table
Open rāSHio, type the forty afternoon purchases into the spreadsheet — twelve 1s, eighteen 2s, six 3s and four 4s — then choose Graph → Frequency Table with Discrete values checked. The relative frequency column you just worked out by hand (0.300, 0.450, 0.150, 0.100) appears beside the counts, and the cumulative column comes free — including the check that the shares add to 1.
Figure 2.2.1
Graph → Frequency Table with Discrete values checked produces the relative and cumulative relative frequency columns in one pass. The walkthrough narrates a different data set — run the same steps on the highlighter counts above.
§2.2.2 — Choosing Bars, Boundaries, and Widths
To construct a histogram you first decide how many bars — five to fifteen is the usual range — then choose a starting point and a class width, both whole numbers.
Round the starting point down to the whole number at or below the smallest value; round the width up to the next whole number. The two roundings go in opposite directions on purpose — rounding down protects the smallest value, rounding up protects the largest.
§2.2.2 — the boxes on the axis
Definition 2.2.3 — Class Interval
A class interval (or bin) is one of the contiguous ranges the horizontal axis is divided into. Every data value falls in exactly one class interval, and the height of the bar drawn over an interval is the frequency of the values inside it.
Definition 2.2.3: Class intervals on whole-number boundaries — left included, right excluded, except the last, which closes on both ends.
The drawing hides this: bars touch at a boundary, but that shared edge belongs to the bar on the right — every interval includes its left boundary and excludes its right one, except the very last, which closes on both ends so the largest value is counted.
§2.2.2 — the six-step rule
A quick guideline for step 1: take the square root of the number of data values and round to the nearest whole number. With 150 values, 150≈12.2, so about 12 bars — a starting suggestion, not a law.
Context Pause — settling the tie
A value sitting on a boundary belongs to the bar on its right
Whole-number boundaries are easy to read and easy to plot, but they come at a price: data values now do land exactly on boundaries. Every interval includes its left boundary and excludes its right one — an interval of width 2 starting at 1 holds 1, 2, 2.9, and everything up to but not including 3.
Context Pause — up means up, not nearest
"Round up" is not "round to the nearest."
A width of 1.76 rounds up to 2 — but so does a width of 1.1; the decimal is ignored, not compared to a half. Rounding down could make the bars fall short of the largest value, invisibly; rounding up can only ever make them reach too far, which is visible and harmless.
Your turn — six bars from 50 shoe sizes
Try It Now 2.2.2 — shoe sizes of 50 students
Fifty students' shoe sizes (discrete, whole and half sizes) run from 9 to 14. Construct a histogram using six bars, and calculate the width of each.
Step 1. Smallest 9, largest 14.
Step 2. Starting point: 9 (already whole).
Step 3. Width: ⌈(14−9)/6⌉=⌈0.83⌉=1.
Step 4. Boundaries: 9, 10, 11, 12, 13, 14, 15.
Answer: width 1; heights 4, 14, 20, 11, 0, 1 — summing to the 50 students. The empty [13,14) bar is information, not a mistake; the single size-14 student is counted only because the last interval closes on both ends.
§2.2.2 — continuous data, eight bars
Example 2.2.1 — Coach Emilio Ramos's roster
The heights (inches, nearest half-inch) of 100 semiprofessional soccer players run from 60 to 74 — continuous data, since height is measured. Construct a histogram using eight bars, with relative frequency on the vertical axis.
Steps 1–2. Smallest 60 (start = 60), largest 74.
Step 3. Width: ⌈(74−60)/8⌉=⌈1.75⌉=2 — rounded up even though 1.75 is closer to 2 than 1; the rule ignores the decimal.
Step 4. Boundaries: 60, 62, 64, …, 76.
Answer: eight bars of width 2 running 60 to 76, tallest in [66,68) and [68,70). Because n=100, each relative frequency is just the count divided by 100.
§2.2.3 — Histograms for Discrete Data
Discrete data — counts of things — get one useful simplification: take a width of 1. Because the data are integers and the boundary rule already excludes the right edge, [3,4) contains only the value 3 — one bar, one count.
Nothing else changes: round the smallest value down (for integers, that is the value itself), and label each bar with the value it holds instead of reading a tick mark.
Insight Note — no half-unit shift needed
One bar, one value — no offset required
Older treatments shift the boundaries by half a unit, starting at 0.5 so that 3 sits dead center in the interval 2.5 to 3.5. That works, but it is a second convention to remember, and it puts fractions on an axis where the data have none. The left-inclusive rule already gives one value per bar — discrete data needs no special case at all.
Your turn — fill in the boundaries
Try It Now 2.2.3 — sports played by 50 student athletes
Twenty student athletes play one sport, twenty-two play two sports, eight play three. Since the data are 1, 2, 3, the starting point is 1 and a width of 1 puts each value alone in its own interval.
Step 1. Boundaries step up by one from 1: 1, 2, 3, 4 — so [1,2), [2,3), and the closed last interval [3,4].
Step 2. Heights: 20, 22, 8.
Step 3 — check. 20+22+8=50. Answer: three bars of width 1, tallest at the value 2 — playing two sports is the most common answer.
§2.2.3 — Dr. Claire Nesbitt's survey
Example 2.2.2 — Books bought this term
Dr. Claire Nesbitt, who teaches at ABC College alongside her wife, asked 50 part-time students how many books they bought this term: eleven bought 1, ten bought 2, sixteen bought 3, six bought 4, five bought 5, two bought 6 — discrete data, since books are counted.
Steps 1–3. Starting point 1 (already whole), width 1 — the boundaries are 1, 2, 3, 4, 5, 6, 7, so each one holds exactly one book count.
Answer: six bars of width 1 spanning [1,2) through [6,7], with heights 11, 10, 16, 6, 5, and 2.
Try it in rāSHio — checking the books-bought histogram
Enter the fifty students' book counts, then choose Graph → Histogram and set the bin start to 1 with a width of 1 — exactly the scheme derived above — and the six heights 11, 10, 16, 6, 5, 2 read straight off the chart. Nudge the bin start to 0.5 afterward: the bars slide half a unit and the counts stay put, the clearest way to see that the boundary convention moves the picture without changing the data.
§2.2.4 — When a Value Lands on a Boundary
Sometimes you inherit a data set and an interval scheme that were not designed together, and values do land on boundaries. The rule: a value counts in the interval on whose left boundary it falls, never the right.
Two analysts can pick different starting points and widths for the same data and produce genuinely different-looking histograms — neither is wrong. What makes one wrong is inconsistency. Write down your starting point, width, and convention before you count a single value.
Your turn — restaurant staff counts
Try It Now 2.2.4 — employees at New York City restaurants
22 restaurants report employee counts from 15 to 42. Using 10–19 as the first interval, create a histogram.
Step 1. Width 10, intervals 10–19, 20–29, 30–39, 40–49.
Step 2. Tally: frequencies 3, 11, 5, 3.
Step 3 — check. 3+11+5+3=22. Answer: a single strong peak in the 20s with thin tails on both sides — most of these restaurants employ twenty-something people.
§2.2.4 — Tou Xiong's weekend survey
Example 2.2.3 — hours spent gaming on weekends
Tou Xiong's 25 classmates reported weekend gaming hours running from 0 to 24 — several sitting exactly on a boundary: 10, 15, and 20.
Step 1. Width 5 gives five bars: 0–5, 5–10, 10–15, 15–20, 20–25.
Step 2 — the boundary rule. A value counts in the interval on whose left boundary it falls: 10 counts in 10–15, not 5–10.
Answer: the histogram above. There is more than one correct way to set up a histogram, provided the convention is stated and used consistently.
Try it in rāSHio — watching the boundary values move
Paste the twenty-five weekend hours (9.95; 10; 2.25; 16.75; 0; …) into File → Delimited List… — the semicolons parse as-is — then choose Graph → Histogram with a bin start of 0 and a width of 5. Now watch the three boundary values 10, 15, and 20: nudge the bin start to 0.5 and each of them moves a bar, and the picture changes. That is the whole reason this section asks you to write your convention down.
Collaborative Exercise
Collaborative Exercise
Count the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think is appropriate — you may want to experiment with the number of intervals.
§2.2.5 — How NOT to Lie with Statistics
A picture answers "what does the data look like" faster than a column of numbers ever will — and the same picture can mislead, on purpose or by accident. Pie charts lose their reader past five or six slices, and two pies of different totals drawn the same size hide the real change.
Histograms have their own version: widening one bar hands that category extra area it did not earn, and the eye reads area as probability. A time-series graph can lie through its axes alone — stretching time, or starting the vertical axis above zero — without changing a single value.
Insight Note — where the lie actually lives
The trick is almost never a wrong number
Every graph described above plots correct data. The lie lives in the axis, the bar width, or the fixed-size pie — which is why checking someone's arithmetic will not catch it, and looking at their axes will.
Your turn — name the trick
Try It Now 2.2.5 — a city council's budget chart
A budget chart's vertical axis runs $9,800,000 to $10,200,000. Over five years the budget rose from $9.9 million to $10.1 million — about a 2% increase — but the last bar looks roughly three times the height of the first. What has been done, and how would you redraw the chart honestly?
Step 1. The vertical axis does not start at zero — it starts at $9,800,000, so only the top $400,000 of a $10 million budget is on the page.
Step 2. Visible height stands for "how far above $9,800,000," not the whole budget: 9,900,000−9,800,00010,100,000−9,800,000=100,000300,000=3.
Answer: the axis is truncated. Redrawn from zero, the actual ≈2.0% rise makes the five bars nearly indistinguishable — the honest picture. Every number on the original chart was correct; only the axis lied.
Key Terminology — §2.2 in five terms
§2.2 — conclusions
A histogram sorts data into touching, whole-number intervals — left boundary in, right boundary out, except the last, which closes on both ends. Relative frequency turns each bar into a share of the whole, and that column always sums to 1.
A boundary convention only works if you state it and apply it consistently. And a picture built from correct numbers can still lie — through a squeezed axis, an uneven bar width, or a fixed-size pie — so read the axes as carefully as the data.
Next: §2.3 — Measures of the Location of the Data, where percentiles and quartiles turn a single value's position into a number you can compare across data sets.