Time Series Analysis Cheat Sheet

A printable reference covering trends, moving averages, seasonal effects, forecasting errors, autocorrelation, and simple time series models for grades 11-12.

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Time series analysis studies data collected in order over time, such as monthly sales, daily temperatures, or yearly population totals. This cheat sheet helps students recognize patterns, smooth noisy data, and make basic forecasts from past observations. It is useful because time order matters, and methods for ordinary unordered data can miss trends, cycles, and seasonality. The most important ideas are trend, seasonal variation, random variation, smoothing, and forecast accuracy. Students should know how to compute changes, moving averages, residuals, and common error measures such as mean absolute error. Simple models often describe a value as a combination of trend, seasonal, and irregular components. Graphs are essential because a time plot can reveal patterns before formulas are applied.

Key Facts

A time series is a set of observations ordered by time, often written as $y_1, y_2, y_3, \ldots, y_t$ .
The first difference measures period-to-period change and is calculated by $\Delta y_t = y_t - y_{t-1}$ .
The percent change from one period to the next is $\frac{y_t - y_{t-1}}{y_{t-1}} \times 100\%$ .
A simple moving average with window size $k$ is $MA_t = \frac{y_t + y_{t-1} + \cdots + y_{t-k+1}}{k}$ .
An additive time series model can be written as $y_t = T_t + S_t + E_t$ , where $T_t$ is trend, $S_t$ is seasonal effect, and $E_t$ is random error.
A multiplicative time series model can be written as $y_t = T_t \times S_t \times E_t$ , and it is often used when seasonal variation grows as the level increases.
A forecast error is actual minus predicted, so $e_t = y_t - \hat{y}_t$ .
Mean absolute error summarizes forecast accuracy with $MAE = \frac{1}{n}\sum_{t=1}^{n}|y_t - \hat{y}_t|$ .

Vocabulary

Time Series: A time series is a data set whose values are recorded in chronological order.
Trend: A trend is the long-term upward, downward, or steady movement in a time series.
Seasonality: Seasonality is a repeating pattern that occurs at regular time intervals, such as days, months, or quarters.
Moving Average: A moving average is a smoothing method that replaces a value with the average of nearby time-ordered values.
Forecast: A forecast is a predicted future value of a time series based on past data and a chosen model.
Residual: A residual is the difference between an observed value and a fitted or forecasted value, calculated as $e_t = y_t - \hat{y}_t$ .

Common Mistakes to Avoid

Ignoring the order of the data is wrong because time series values are connected by their sequence, and rearranging them can hide trends or cycles.
Using a moving average without checking the window size is wrong because a window that is too small may leave too much noise, while a window that is too large may hide real changes.
Confusing seasonal patterns with random variation is wrong because seasonality repeats at regular intervals, while random variation has no consistent timing.
Calculating forecast error as predicted minus actual when the class uses $e_t = y_t - \hat{y}_t$ is wrong because the sign of the error will be reversed.
Extrapolating far beyond the data is risky because a trend that fits past values may not continue under new conditions.

Practice Questions

1 The monthly sales values for four months are $120$ , $135$ , $150$ , and $144$ . Find the first differences $\Delta y_2$ , $\Delta y_3$ , and $\Delta y_4$ .
2 For the time series values $8$ , $10$ , $13$ , $15$ , and $14$ , calculate the 3-period moving average ending at the fifth value.
3 A model forecasts $\hat{y}_1 = 50$ , $\hat{y}_2 = 55$ , and $\hat{y}_3 = 60$ for actual values $y_1 = 52$ , $y_2 = 53$ , and $y_3 = 66$ . Find the forecast errors and the $MAE$ .
4 A company has higher sales every December and lower sales every February for several years. Explain why this pattern is likely seasonality rather than random variation.

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