While watching the news you might have noticed the reporter saying that the temperature of a particular city or a country has broken a record. The rainfall of some state or country has set a new bar. How can they know about it? What are the measures that they have taken and studied to say so? These are the time-series data. You all are familiar with time-series data and the various components of the time series. In this section, we will study how to calculate the trend in a set of data by the method of moving average.

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## A Trend in a Time Series

A time series is broadly classified into three categories of long-term fluctuations, short-term or periodic fluctuations, and random variations. A long-term variation or a trend shows the general tendency of the data to increase or decrease during a long period of time. The variation may be gradual but it is inevitably present.

**Browse more Topics under Time Series Analysis**

### Analysis of Time Series

Suppose you have a time series data. What will you do with it? How can you calculate the effect of each component for the resulting variations in it? The main problems in the analysis of time series are

- To identify the components and the net effect of whose interaction is shown by the movement of a time series, and
- To isolate, study, analyze and measure each component independently by making others constant.

## Measurement of Trend by the Method of Moving Average

This method uses the concept of ironing out the fluctuations of the data by taking the means. It measures the trend by eliminating the changes or the variations by means of a moving average. The simplest of the mean used for the measurement of a trend is the arithmetic means (averages).

### Moving Average

The moving average of a period (extent) *m* is a series of successive averages of *m* terms at a time. The data set used for calculating the average starts with first, second, third and etc. at a time and *m* data taken at a time.

In other words, the first average is the mean of the first *m* terms. The second average is the mean of the *m* terms starting from the second data up to (*m* + 1)^{th} term. Similarly, the third average is the mean of the *m* terms from the third to (*m* + 2) ^{th} term and so on.

If the extent or the period, *m* is odd i.e., *m* is of the form (2k + 1), the moving average is placed against the mid-value of the time interval it covers, i.e., t = k + 1. On the other hand, if *m* is even i.e., *m* = 2k, it is placed between the two middle values of the time interval it covers, i.e., t = k and t = k + 1.

When the period of the moving average is even, then we need to synchronize the moving average with the original time period. It is done by centering the moving averages i.e., by taking the average of the two successive moving averages.

### Drawbacks of Moving Average

- The main problem is to determine the extent of the moving average which completely eliminates the oscillatory fluctuations.
- This method assumes that the trend is linear but it is not always the case.
- It does not provide the trend values for all the terms.
- This method cannot be used for forecasting future trend which is the main objective of the time series analysis.

## Solved Example for You

**Problem:** Calculate the 4-yearly and 5-yearly moving averages for the given data of the increase *I _{i}* in the population of a city for the 12 years. Make a graphic representation of it.

**Solution:**

t | I_{i} |
5-yearly moving totals | 5-yearly moving averages | 4-yearly moving totals (not centered) | 4-yearly moving average (not centered) | 2-period moving total (centered) | 4-yearly moving average (centered) |

(1) | (2) | (3) | (4) = (3) ÷ 5 | (5) | (6) = (5) ÷ 4 | (7) | (8) = (7) ÷ 2 |

1 | 100 | ||||||

2 | 100 | ||||||

400 | 100 | ||||||

3 | 100 | 500 | 100 | 200 | 100 | ||

400 | 100 | ||||||

4 | 100 | 475 | 95 | 193.75 | 96.875 | ||

375 | 93.75 | ||||||

5 | 100 | 450 | 90 | 181.25 | 90.625 | ||

350 | 87.50 | ||||||

6 | 75 | 470 | 94 | 180 | 90 | ||

370 | 92.50 | ||||||

7 | 75 | 490 | 98 | 190 | 95 | ||

390 | 97.50 | ||||||

8 | 120 | 510 | 102 | 206.25 | 103.125 | ||

435 | 108.75 | ||||||

9 | 120 | 555 | 111 | 228.75 | 114.375 | ||

480 | 120 | ||||||

10 | 120 | 600 | 120 | 240 | 120 | ||

480 | 120 | ||||||

11 | 120 | ||||||

12 | 120 |

Here, the 4-yearly moving averages are centered so as to make the moving average coincide with the original time period. It is done by dividing the 2-period moving totals by two i.e., by taking their average. The graphic representation of the moving averages for the above data set is