Obtain the trend value for the following data using 4-yearly centered moving averages: | Years | 1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1983 | 1984 | 1985 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | Index | 0 | 2 | 3 | 3 | 2 | 4 | 5 | 6 | 7 | 10 |

Obtain the trend value for the following data usin – Class 12 Mathematics And Statistics

Question 22 of 25

Obtain the trend value for the following data using 4-yearly centered moving averages:

Years	1976	1977	1978	1979	1980	1981	1982	1983	1984	1985
Index	0	2	3	3	2	4	5	6	7	10

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Take 4-yearly moving totals, add them in overlapping pairs, and divide by 8 to get centred trend values: $2.25, 2.75, 3.25, 3.875, 4.875, 6.25$ for 1978–1983.

Step 1 — 4-yearly moving totals (sum of four consecutive index values):

$0+2+3+3 = 8,\quad 2+3+3+2 = 10,\quad 3+3+2+4 = 12,$

$3+2+4+5 = 14,\quad 2+4+5+6 = 17,\quad 4+5+6+7 = 22,\quad 5+6+7+10 = 28.$

These totals fall between years, so they must be centred.

Step 2 — 2-item moving totals of the above (to centre on a year), then divide by $8$ (since $4 \times 2 = 8$ ):

$1978:\ (8+10)/8 = 18/8 = 2.25$
$1979:\ (10+12)/8 = 22/8 = 2.75$
$1980:\ (12+14)/8 = 26/8 = 3.25$ …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2026Set ANNUAL1 mark
$Q. The following table shows the production of pig-iron and ferroalloys ('000 metric tons) Years 197419751976197719781979198019811982 Production 0499854810 Find trend values for the above data by using 5 yearly moving averages.$
›Reveal solutionSolution
5-year moving totals $=30,35,35,34,35$ ; dividing by 5 gives trend values $6, 7, 7, 6.8, 7$ centred on 1976–1980.

Data (production in '000 metric tons):

Year 1974 1975 1976 1977 1978 1979 1980 1981 1982
Production 0 4 9 9 8 5 4 8 10

A 5-yearly moving average is found by summing five consecutive values and dividing by 5, and the result is written against the middle (3rd) year of that group.

5-year moving totals and averages:

$1974\text{–}1978$ : $0+4+9+9+8 = 30 \Rightarrow 30/5 = 6$ (centred 1976)

$1975\text{–}1979$ : $4+9+9+8+5 = 35 \Rightarrow 35/5 = 7$ (centred 1977)

$1976\text{–}1980$ : $9+9+8+5+4 = 35 \Rightarrow 35/5 = 7$ (centred 1978)

$1977\text{–}1981$ : $9+8+5+4+8 = 34 \Rightarrow 34/5 = 6.8$ (centred 1979) …
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CBSE 2025Set ANNUAL1 markMCQ
$Q. Moving averages are useful in identifying ______.$
(a) Seasonal component
(b) Irregular component
(c) Trend component
(d) Cyclical component
›Reveal solutionSolution

A moving average averages successive groups of values, cancelling short-term ups and downs and revealing the smooth long-term direction of the series — the trend component.

A time series has four components: trend, seasonal, cyclical and irregular. The method of moving averages computes the average of a fixed number of consecutive observations and slides this window along the series. Averaging tends to cancel the seasonal, cyclical and irregular fluctuations (which oscillate above and below the general level), leaving behind the …

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Years	1974	1975	1976	1977	1978	1979	1980	1981	1982
Production	0	4	9	9	8	5	4	8	10
Find trend values for the above data by using 5 yearly moving averages.

Year	1974	1975	1976	1977	1978	1979	1980	1981	1982
Production	0	4	9	9	8	5	4	8	10

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