The method of moving averages is of two types:

- Simple Moving Averages
- Weighted Moving Averages

**Simple Moving Averages**

If the observed values of a variable $Y$ are $y_1, y_2,\cdots, y_n$ corresponding to the time periods $t_1, t_2,\cdots, t_n$, respectively, the $k$-period simple moving averages are defined as

\begin{align*}

a_1 &= \frac{1}{k} \sum_{i=1}^{k} y_i\\

a_2 &= \frac{1}{k} \sum_{i=2}^{k+1} y_i,\\

a_3 &= \frac{1}{k} \sum_{i=3}^{k+2} y_i \\

\vdots &= \quad \vdots\\

a_m &= \frac{1}{k} \sum_{i=m}^{n} y_i

\end{align*}

where $a_1, a_2, \cdots, a_m$ is the sequence of $k$-period simple moving averages. That is, the $k$-period simple moving averages are calculated by averaging the first $k$ observations and then repeating this process of averaging the $k$ observations by dropping each time the first observation and including the next one. This process is continued till the last $k$ observations have been averaged. For example, the 3-period simple moving averages are given as:

\begin{align*}

a_1 &= \frac{1}{3} (y_1+y_2+y_3) = \frac{1}{3} \sum_{i=1}^{3} y_i\\

a_2 &= \frac{1}{3} (y_2+y_3+y_4) = \frac{1}{3} \sum_{i=2}^{4} y_i\\

a_3 &= \frac{1}{3} (y_3+y_4+y_5) = \frac{1}{3} \sum_{i=3}^{5} y_i\\

\vdots &= \quad \vdots\\

\text{and so on}

\end{align*}

Each of these simple moving averages of the sequence $a_1, a_2, a_3,\cdots$ is placed against the middle of each successive group. The $k$-period moving successive totals $S_1, S_2, S_3, \cdots$ are obtained by the following relations

\begin{align*}

S_1 = \sum_{i=1}^{k} y_i\\

S_2 &= S1+ y_{k+1}-y_1\\

S_3 &= S_2 + y_{k+2} – y_2\\

\vdots &= \quad \vdots\\

\text{so on}

\end{align*}

The $k$-period simple moving averages are obtained by dividing these $k$-period moving successive totals ($S_1, S_2, S_3, \cdots$) by $k$, as given in the following relations

\begin{align*}

a_1 &= \frac{S_1}{k}\\

a_2 &= a_1 + \frac{y_{k_1}0y_1} {k}\\

a_3 &= a_2 + \frac{y_{k+2} -y_2}{k}\\

\vdots &= \quad \vdots\\

\text{so on}

\end{align*}

- When $k$ is odd, the sequence $a_1, a_2, a_3, \cdots$ will be placed against the middle of its time-period.
- When $k$ is even, the sequence $a_1, a_2, a_3, \cdots$ of simple moving averages will be placed in the middle of two time periods. It is necessary to centralize these averages. For centralization, further 2-period moving averages of the former $k$-period moving averages are computed which are called $k$-period centered moving averages.

**Weighted Moving Averages**

For observed values ($y_1, y_2, \cdots, y_n$) of a variable $Y$ corresponding to the time periods $t_1, t_2, \cdots, t_n$, respectively, the $k$-period weighted moving averages with weights $w_1, w_2, \cdots, w_k$ are defined as

\begin{align*}

a_1 &= \frac{1}{\sum w} \sum_{i=1}^{k} y_i w\\

a_2 &= \frac{1}{\sum w} \sum_{i=2}^{k+1} y_i w\\

a_3 &= \frac{1}{\sum w} \sum_{i=3}^{k+2} y_i w\\

\vdots &= \vdots\\

a_m &= \frac{1}{\sum w} \sum_{i=m}^{n} y_i w\\

\end{align*}

where $a_1, a_2, \cdots, a_m$ is a sequence of $k$-period weighted moving averages with weights $w_1, w_2, \cdots, w_k$, respectively. The $k$-period weighted moving averages are calculated by taking the weighted average of the first $k$ observed values with weights $w_1, w_2, \cdots, w_k$ and then repeating this process of averaging the $k$ observations by dropping each time the first observation and including the next one. This process is continued until the last $k$ observations have been averaged.

**Merits** (Method of Moving Averages)

- The method of moving averages is simple and easy.
- This method is appropriate to remove, seasonal variations, cyclical fluctuations, and irregular variations.

**Demerits** (Method of Moving Averages)

- Some values at the beginning and the end of the series are lost.
- Moving averages are greatly affected by extreme values.
- The method does not provide a mathematical formula for the trend.

**Example: **Calculate 3-year simple moving averages for the following time series. Also, plot actual data and moving averages on a graph. Also, find the 3-year weighted moving averages with weights 2, 2, and 1, respectively.

Year | 1970 | 1971 | 1972 | 1973 | 1974 | 1975 | 1975 | 1977 |

Production | 170.0 | 154.8 | 156.6 | 158.9 | 140.3 | 154.2 | 160.7 | 178.3 |

**Solution:**

Year | Production | 3-year Simple MT | 3-year simple MA | 3-year WMT | 3-year WMA |

1970 | 170.0 | ||||

1971 | 154.8 | 481.3 | 160.43 | 806.1 | 161.22 |

1972 | 156.5 | 470.2 | 156.73 | 781.5 | 156.30 |

1973 | 158.9 | 455.7 | 151.90 | 771.1 | 154.22 |

1974 | 140.3 | 453.4 | 151.13 | 752.6 | 150.52 |

1975 | 154.2 | 455.2 | 151.73 | 749.7 | 149.94 |

1976 | 160.7 | 493.2 | 164.40 | 808.1 | 161.62 |

1977 | 178.3 |

*MT=moving total, MA=moving averages, WMT=weighted MT, WMA=Weighted MA