3.5: Time Series Plots and Trends Over Time
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Not all data lives in a moment but instead changes over time. The trend in this data can tell us what’s rising, falling, or staying stable. These types of patterns show up everywhere: prices, temperatures, populations, interest rates, and even social media trends.
In this section, we’ll explore how to visualize **time series data**, numerical observations tied to moments in time, using the simple and powerful time series plot.
What Is a Time Series Plot?
A time series plot (also called a time plot or time series graph) is a type of scatter plot where:
- The x-axis represents time, for example days, months, or years
- The y-axis represents a numerical variable that changes over time
- Points are often connected with lines to emphasize the direction or shape of change
Time plots help us capture patterns, cycles, and changes, and help us watch how data evolves so we can make better decisions in the future.
Example: U.S. Gas Prices Over the Last 12 Months
Suppose you collected weekly data on the average price of a gallon of regular gasoline in the United States over the last 12 months. Here are 12 monthly averages (for readability):
Gas Prices Dataset – Monthly Averages
| Month | Price (USD per gallon) |
|---|---|
| January | 3.45 |
| February | 3.50 |
| March | 3.60 |
| April | 3.55 |
| May | 3.48 |
| June | 3.51 |
| July | 3.45 |
| August | 3.40 |
| September | 3.42 |
| October | 3.55 |
| November | 3.61 |
| December | 3.75 |
To analyze this data, we’ll plot each month on the x-axis and the gas price (in dollars) on the y-axis to build a time plot.
Figure: Time series plot of U.S. gas prices over the past year. Time plots help reveal subtle changes and overall direction.
Observations from the Plot
Gas prices appear to have some fluctuation month to month, but generally increase over the year. A small dip occurs in late summer before rising again into winter. This kind of insight is harder to notice when only looking at a list of numbers, but visualizing it helps us see the big picture.
Student Activity: Make Your Own Time Series Plot
- Find or collect a small set of time-based data you care about (temperature, step count, spending, etc.)
- Record at least 7–14 time points and corresponding values
- Use paper, Excel, or Google Sheets to build a scatter plot or line graph (with x = time and y = measurement)
- Describe any trend, pattern, dip, or spike you notice
How This Connects to Everything Before
Time plots are just one more way to describe data — and every visualization we’ve used so far helps us see different truths:
- Bar charts help compare quantities across named categories
- Pie charts help show parts of a whole
- Histograms reveal distributions and shape
- Boxplots show spread, center, and outliers
- Time series plots focus on change over time
This isn’t the end of visualization — there are many more tools we can use, some of which we'll explore later in the course, like scatterplots between two variables, histograms with density overlays, trend lines, correlation matrices, and more.
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Chapter Reflection
In this chapter, we’ve connected raw data to images that help people understand it. Data is difficult to interpret without mathematics and computations, but graphics allow for intuitive understanding.
The next time you see a chart, such as in other classes, the news, or on social media, consider: What kind of data is this? What is the visualization trying to show? How would I interpret it using what I’ve learned?