display a quantitative variable with a dot plot, stem-and-leaf plot, histogram, box plot or time plot, and describe its distribution by shape, center, spread and unusual values in context.
Instruction
Statistics is the science of learning from data when there is variation. Every investigation in this course follows four steps: ask a question that data can answer, collect or obtain data suited to it, analyze the data with displays and numbers, and interpret the result in the context of the question. Most of the work in this unit is analysis, but the question and the data always come first.
Variables. A categorical variable places each case in a group (station name, month, whether it rained). A quantitative variable is a number you can average (inches of rain, degrees). The kind of variable decides the display: bar graphs for categorical data, and dot plots, stem-and-leaf plots, histograms and box plots for quantitative data.
A real distribution. NOAA’s National Centers for Environmental Information publish a Global Summary of the Year for thousands of stations. At Los Angeles International Airport (station USW00023174), calendar-year precipitation from 1950 to 2024 gives 75 values. Grouped in 5-inch bins, the counts are 4, 28, 21, 14, 5 and 3:
Describe a distribution with four features, always in context:
- Shape. This histogram has one peak (5-10 inches) and a long tail to the right: it is skewed right. Rainfall is often like this, because a few storm-heavy years pull far above a typical year, while no year can fall below zero.
- Center. The median is 11.05 inches: half the years were drier. The mean is 12.11 inches, pulled toward the long tail.
- Spread. Values run from 3.65 inches (2013, the driest year) to 29.47 inches (1983), a range of 25.82 inches. Most years fall between 5 and 20 inches.
- Unusual values. 1983, the strong El Niño winter, stands apart from the rest. In Lesson 1.2 you will test it with a rule rather than by eye.
Choosing a display. A dot plot shows every value and works for small data sets. A stem-and-leaf plot keeps the values while showing shape: LAX precipitation for 2015-2024 is 5.96, 10.3, 12.28, 7.8, 18.73, 9.04, 12.1, 6.41, 25.37 and 15.94, so with stems in whole inches, stem 12 carries leaves 1 and 3 (12.1, 12.28 rounded to tenths). A histogram suits large data sets, but its shape changes with bin width, so try more than one. A box plot shows the five-number summary and makes comparing groups easy, but it hides gaps and multiple peaks. A time plot (a line graph) is the right choice whenever data were recorded in order over time.
Time matters. The Bureau of Labor Statistics reports California’s seasonally adjusted unemployment rate each month. It was 4.4% in February 2020 and 16.1% in April 2020, when pandemic closures began. A histogram of these monthly rates would show a strange tail. Only a time plot shows when and why. The same series has no value for October 2025, because the lapse in federal appropriations stopped data collection. Real data have gaps, and a good report says so rather than filling them in.
Formative check
Work through these before moving on. They are not graded — they tell you, and your teacher, whether the standard below has landed yet.
Annual rainfall totals at a Southern California airport over 75 years: what shape do you expect the histogram to have?
You want to show how California’s monthly unemployment rate changed from 2016 to 2025. Which display fits best?
Simulation & tools
Import a NOAA station record into CODAP
Data: NOAA NCEI Global Summary of the Year, Los Angeles International Airport (station USW00023174), 1950-2024. Import this CSV into CODAP (drag the link onto the CODAP window, or use Import and paste the URL): https://www.ncei.noaa.gov/access/services/data/v1?dataset=global-summary-of-the-year&stations=USW00023174&startDate=1950-01-01&endDate=2024-12-31&dataTypes=TAVG,PRCP&units=standard&format=csv
- Make a graph, drag PRCP to the horizontal axis. You now have a dot plot. Use the ruler menu to show a box plot as well.
- Switch to a histogram (graph configuration, Group into bins) and set bin width to 5, then to 2. Record how the shape changes.
- Make a second graph with DATE on the horizontal axis and PRCP on the vertical axis to get a time plot.
- Write three sentences describing shape, center and spread, and one sentence naming the most unusual year.
Choose the data you work with. All three earn full credit, and each one ends in the same write-up: three sentences on shape, center and spread, and one sentence naming the most unusual value.
- Protocol A — the full station record. Import the CSV above into CODAP, or open the URL in a browser, save the CSV and build the displays in any free spreadsheet. Record the station, the years and the date you downloaded it. This is the route the steps above describe.
- Protocol B — a fixed sample from the printed record, no device needed. Use the ten LAX calendar-year precipitation totals for 2015-2024 printed in the mini-lecture: 5.96, 10.3, 12.28, 7.8, 18.73, 9.04, 12.1, 6.41, 25.37, 15.94 inches. Ten values are too few for a histogram, so make a dot plot, a stem-and-leaf plot with stems in whole inches, and a time plot of total against year, all on paper. Then write those four sentences. Note in your write-up that ten years is a sample of the 75, and say which feature of the 75-year distribution your ten years does and does not show.
- Protocol C — data you generate yourself, no internet needed. Count something, on this exact procedure: take any book or magazine you have, open it at page 50 (or the nearest page of ordinary prose), and record the number of letters in each of the first 60 words, skipping numerals and counting a hyphenated word as one word. Write down the title, the page and the rule you used, so someone else could repeat it. Make a dot plot and a histogram with bin width 2, then write those four sentences about your distribution. Expect a right skew for the same structural reason rainfall has one: word length cannot fall below one letter, but a few long words stretch the upper tail.
Open in a new tab ↗ · CODAP, Concord Consortium · MIT open source
Write a four-sentence description of the LAX precipitation distribution (shape, center, spread, unusual values), using units and years. Then write one statistical question about California rainfall that this dataset could NOT answer, and say what data you would need.
How confident are you that you can choose a suitable display for a quantitative variable and describe its distribution in context?
Practice
Work these on paper or in your notebook, then open Check your answer. Aim for all of Fluency and Application; try at least one Challenge.
Printable version: this unit’s practice workbook (PDF)
Fluency
Build speed and accuracy with the core skill.
- For the data 2, 5, 13, 17, 21, 28, 28, 36, 38: find the mean, median, range and interquartile range. (Find the quartiles as the medians of the lower and upper halves, leaving out the overall median when the count is odd.)
Check your answer
Answer: mean = 20.89; median = 21.0; range = 36; IQR = 23.0 (Q1 = 9.0, Q3 = 32.0) - For the data 3, 11, 21, 21, 23, 24, 28, 29, 29, 32: find the mean, median, range and interquartile range. (Find the quartiles as the medians of the lower and upper halves, leaving out the overall median when the count is odd.)
Check your answer
Answer: mean = 22.10; median = 23.5; range = 29; IQR = 8.0 (Q1 = 21.0, Q3 = 29.0) - Classify each variable as categorical or quantitative: (a) the NOAA station ID, (b) annual precipitation in inches, (c) the month of the wettest day, (d) whether a day had measurable rain.
Check your answer
Answer: (a) categorical, (b) quantitative, (c) categorical, (d) categorical.Quantitative variables are numbers it makes sense to average; the others place each case in a group. - Which display best shows how California’s monthly unemployment rate changed from 2016 to 2025: histogram, box plot or time plot?
Check your answer
Answer: A time plot.The data are ordered in time; only a time plot keeps the order. - The LAX annual-precipitation histogram (1950-2024) has bins 0-5, 5-10, 10-15, 15-20, 20-25 and 25-30 inches with counts 4, 28, 21, 14, 5 and 3. How many years had at least 15 inches?
Check your answer
Answer: 22 years.14 + 5 + 3. - Using the same histogram, what fraction of the 75 years had less than 10 inches?
Check your answer
Answer: 32/75 ≈ 0.427.4 + 28 = 32 years below 10 inches. - Using the same histogram, which bin contains the median year? (There are 75 years, so the median is the 38th value.)
Check your answer
Answer: The 10-15 inch bin.Cumulative counts: 4, 32, 53. The 38th value is past 32 and within 53. - Describe the shape of the LAX precipitation histogram in one phrase.
Check your answer
Answer: Single-peaked and skewed right.One peak at 5-10 inches, with a long tail toward the wet years.
Application
Use the skill in context. Show your reasoning.
- San Diego (Lindbergh Field) calendar-year precipitation, 1950-2024 (NOAA): mean 9.67 in, median 8.72 in, minimum 3.42 in, maximum 19.43 in. Predict the shape of its histogram and explain your reasoning from these four numbers.
Check your answer
Answer: Skewed right.The mean is 0.95 in above the median, and the maximum is 10.71 in above the median while the minimum is only 5.30 in below it: a long right tail. - BLS reports California’s seasonally adjusted unemployment rate as 4.4% in February 2020 and 16.1% in April 2020. (a) By how many percentage points did it rise? (b) By what factor? (c) Why would a histogram of the 2016-2025 monthly rates hide the most important fact about these data?
Check your answer
Answer: (a) 11.7 percentage points. (b) About 3.66 times. (c) A histogram discards time order, so it shows a few very high values but not that they came suddenly in spring 2020 and then declined.A time plot is the display that answers ‘when’. - California’s monthly unemployment rates for 2019 (BLS, %) were 4.3, 4.3, 4.2, 4.1, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.1, 4.1. Make a dot plot and describe the distribution.
Check your answer
Answer: Values run from 4.0 to 4.3 (range 0.3 points), median 4.05%; half the months sit exactly at 4.0%, with no outliers.A very narrow distribution: 2019 was a stable year, which is what makes the 2020 jump so striking. - Round LAX precipitation for 2015-2024 to whole inches: 6, 10, 12, 8, 19, 9, 12, 6, 25, 16. Make a stem-and-leaf plot with stems 0, 1 and 2 (tens of inches) and describe the shape.
Check your answer
Answer: 0 | 6 6 8 9; 1 | 0 2 2 6 9; 2 | 5. Most years fall below 20 inches, with one high value (2023) to the right: skewed right.Sort first: 6, 6, 8, 9, 10, 12, 12, 16, 19, 25.
Challenge
Stretch problems. Expect to think before you write.
- The BLS series has no value for October 2025 because of the 2025 lapse in federal appropriations. September and November 2025 were both 5.5%. A student fills in October as 5.5% and says nothing. Is that acceptable in a statistical report? What should the student do instead?
Check your answer
Answer: Filling the gap silently is not acceptable. The value may be shown as an estimate only if it is clearly labeled as imputed (not published by BLS), with the method stated; better, leave the gap visible in the time plot and note why it exists.Readers must be able to tell published data from the analyst’s guesses.
Review
Keep earlier skills sharp.
- A quantity starts at 2,500 and grows by 3% each year. Write a model and find its value after 8 years, to the nearest whole number.
Check your answer
Answer: A = 2,500(1.03)t; after 8 years ≈ 3,167The growth factor is 1 + 0.03 = 1.03. - (Prerequisite: percent) LAX received 25.37 inches in 2023; its 1950-2024 mean is 12.11 inches. By what percent did 2023 exceed the mean?
Check your answer
Answer: About 109.5% above the mean.(25.37 − 12.11) / 12.11.