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Mathematics and Statistics · Class 12 Commerce

Ch 12Time Series — Class 12 Mathematics and Statistics, concept-first.

Many quantities in business and economics are recorded at regular intervals of time — the annual sales of a firm, the monthly output of a factory, the quarterly profit of a company, the daily closing price of a share.

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Key concepts

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Method of Moving Averages

A moving average of period replaces each value by the mean of consecutive values, the window sliding one period at a time. For an odd period the average is placed against the middle year directly.

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Chapter contents

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Meaning and Uses of a Time Series

Many quantities in business and economics are recorded at regular intervals of time — the annual sales of a firm, the monthly output of a factory, the quarterly profit of a company, the daily closing…

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Components of a Time Series

When we look at a series over a long stretch of time, its ups and downs are not random noise alone — they are the combined effect of four distinct forces called the components of a time series.

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Measurement of Trend: Graphical (Freehand Curve) Method

The simplest way to show the trend of a time series is to draw it. In the graphical (freehand curve) method we plot the given values against time and then draw a smooth curve or straight line, by hand…

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Measurement of Trend: Method of Moving Averages

The method of moving averages smooths a series by replacing each value with the average of itself and its neighbours, so that short-term fluctuations cancel out and the trend stands clear.

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Measurement of Trend: Method of Least Squares

The method of least squares fits an objective mathematical trend line to the data — the one straight line for which the sum of the squares of the vertical gaps between the actual points and the line i…

Exercises

+Show 5 questions5 questions
  1. Q7Compute the three-yearly moving averages of the following data. | Year | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | |---|---|---|---|-…Free
  2. Q8Fit a straight-line trend by the method of least squares to the data below and estimate the trend value for 2018. | Year | 2012 | 2013 | 201…Free
  3. Q9The regular fluctuations in a time series that complete themselves within one year are called: (a) secular trend (b) seasonal variation (c)…Preview
  4. Q10State two reasons why the method of least squares is generally preferred over the method of moving averages for measuring the trend of a tim…Preview
  5. Q11In fitting a straight-line trend $Y = a + bX$ by the method of least squares, the time deviations are chosen so that $\sum X = 0$. The value…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 14 questions14 questions
  1. Q1Obtain the trend values for the following data using 5 yearly moving averages: | Year | 2000 | 2001 | 2002 | 2003 | 2004 | | --- | --- | ---…Preview
  2. Q2Following table shows the amount of sugar production (in lakh tonnes) for the years 1931 to 1941: | Year | Production | Year | Production |…Preview
  3. Q3The complicated but efficient method of measuring trend of time series is ______. (a) graphical method (b) method of moving average (c) meth…Preview
  4. Q4The following table shows gross capital information (in Crore ₹) for years 1966 to 1975: | Years | 1966 | 1967 | 1968 | 1969 | 1970 | | ---…Preview
  5. Q5The publisher of a magazine wants to determine the rate of increase in the number of subscribers. The following table shows the subscription…Preview
  6. Q6Obtain the trend values for the data in using 4-yearly centered moving averages. | Year | 1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1…Preview
  7. Q7Following table shows the all India infant mortality rates (per '000) for years 1980 to 2010: | Year | 1980 | 1985 | 1990 | 1995 | 2000 | 20…Preview
  8. Q8Moving averages are useful in identifying ______. (a) Seasonal component (b) Irregular component (c) Trend component (d) Cyclical componentPreview
  9. Q9Cyclic variation can occur several times in a year. (a) True (b) FalsePreview
  10. Q10The simplest method of measuring trend of time series is ______.Preview
  11. Q11Obtain the trend value for the following data using 4-yearly centered moving averages: | Years | 1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1…Preview
  12. Q12Which of the following can’t be a component of a time series? (a) Seasonality (b) Cyclical (c) Trend (d) MeanPreview
  13. Q13The following table shows the production of pig-iron and ferroalloys ('000 metric tons) | Years | 1974 | 1975 | 1976 | 1977 | 1978 | 1979 |…Preview
  14. Q14Following data shows the number of boxes of cereal sold in years 1977 to 1984. | Year | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1983 | 198…Preview

More questions

+Show 6 questions6 questions
  1. Example 1Classify each of the following as an example of a secular trend, seasonal variation, cyclical variation or irregular variation: (i) a steady…Free
  2. Example 2Explain the freehand (graphical) method of measuring the trend of a time series, and state one merit and one limitation of the method.Free
  3. Example 3Compute the three-yearly moving averages of the following data on annual production (in thousand units). | Year | 2011 | 2012 | 2013 | 2014…Preview
  4. Example 4Find the four-yearly centred moving averages for the following data. | Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | |---|-…Preview
  5. Example 5Fit a straight-line trend by the method of least squares to the following data and write down the trend value for each year. | Year | 2013 |…Preview
  6. Example 6The number of members of a co-operative society (in hundreds) over seven years is given below. Fit a straight-line trend by the method of le…Preview