OP Malhotra Class 11 Maths Solutions Chapter 31 Moving Average Exercise 31

Question 1. The table shows the number of students in a school getting at least a grade C in mathematics for the years 1994 to 2001.
(i) Represent this data as a time series.
(ii) Calculate the 3-point moving average and plot it on the same graph.
(iii) Are the school's maths results improving?
(iv) Explain why this is not a good way to work out whether the school's results are improving.
Answer:
(i) The original data for the number of students is given below:

(ii) To calculate the 3-point moving average, we sum the number of students for three consecutive years and then divide by 3. This helps smooth out short-term fluctuations to reveal the trend. The calculated 3-point moving average is shown in the table below and plotted on the graph:

(iii) Based on the graph, the school's math results show some ups and downs, but the 3-year moving average line is mostly stable or slightly increasing, indicating a generally consistent to slightly improving trend.
(iv) Using only the number of students getting a C grade or above might not be the best way to check for improvement. This is because if the school's total student population grows, the number of students getting a C could increase even if the overall percentage or quality of results stays the same or drops. To get a clearer picture, it would be better to look at percentages of students achieving certain grades or compare results with the total number of students taking the exam each year.
In simple words: The school's math results look a bit better over time according to the average. But just counting students with C grades might not show the real improvement if the school has more students each year. It's better to check percentages or compare to the total students.

🎯 Exam Tip: When asked to comment on trends, always refer to both the actual data and the moving average line. Remember that a moving average helps smooth out random changes to show the underlying pattern.

Question 2. The profits of a soft drink firm in thousands of litres during each month of a year were:
January 1.2
February 0.8
March 1.4
April 1.6
May 2.0
June 2.0
July 3.6
August 4.8
September 3.4
October 1.8
November 0.7
December 1.2
Calculate 3-monthly moving averages and illustrate graphically. (ISC 1992 type)
Answer: To find the 3-monthly moving average, we sum the profits for three consecutive months and then divide by three. This helps to smooth out monthly variations and highlight the seasonal trend in profits. The calculation is shown in the table below:

The graph below illustrates the trend of the 3-monthly moving averages, showing the seasonal pattern of the firm's profits throughout the year.

In simple words: We added up the profits for three months at a time, then divided by three to get an average. This helped us see the profit changes more smoothly, like a general trend over the year, which is highest in the summer months.

🎯 Exam Tip: When presenting moving averages, always include both the calculation table and a clear graph to visually represent the trend. Remember that 3-monthly averages are typically plotted at the center month of the three-month period.

Question 3. The number of traffic offences committed in a certain city over a period of 3 years is given in the following table:

Calculate 4-quarterly moving averages and illustrate these and original figures on one graph using the same axis for both. Comment briefly on a local politician's claim that traffic offences were on the increase.
Answer: We will calculate the 4-quarterly moving averages to observe the trend in traffic offences over time. A 4-quarterly moving average is useful for removing seasonal variations and revealing the long-term trend. The calculations for the 4-quarterly moving total, moving average, and centered moving average are shown in the table below:

The graph below displays both the actual traffic offences and the smoothed trend line (moving average). The centered 4-quarterly moving average column shows a general increase in offences from 1968 to 1970. This confirms the local politician's claim that traffic offences were on the increase. Regular monitoring of these trends can help city planners and law enforcement to address traffic issues proactively.

In simple words: We calculated averages of traffic offences over four quarters, which helps to see the bigger picture without monthly ups and downs. The graph shows that traffic offences generally went up over these years, agreeing with what the politician said.

🎯 Exam Tip: For 4-quarterly moving averages, remember that the initial moving averages need to be centered to align with the time periods, which often involves taking a second 2-period average of the first set of moving averages.

Question 4. Find the 4-quarterly moving averages in the following table which gives the quarterly index numbers of coal production (for the years 1936-1938). Also plot on the same graph the quarterly index numbers as well as the 4-quarterly moving average. Comment on the nature of the general trend.

The graph below plots both the raw quarterly index numbers and the calculated 4-quarterly moving averages. The moving average line clearly shows an upward trend in coal production from 1936 to 1938, indicating growth in this sector during that period, despite some quarterly fluctuations.

In simple words: We calculated the average coal production over four quarters to see the general change over three years. The graph shows that, even with some ups and downs, coal production generally increased during this time.

🎯 Exam Tip: When describing the trend, make sure to state whether it's increasing, decreasing, or stable, and support your statement by referring to the direction of the moving average line on your graph.

Question 5. The annual incomes of a firm were recorded every quarter for 4 years. The results are shown in this table.

(i) Work out the 4-point moving average for the data.
(ii) Plot the original data and the moving average on the same graph.
(iii) Comment on how the firm's incomes have changed over the 4-years.
Answer:
(i) We need to calculate the 4-point moving average to see the trend in the firm's income over four years, smoothing out any quarterly variations. The calculations for the 4-point moving total, moving average, and centered moving average are shown below:

(ii) The graph below shows both the actual quarterly incomes and the smoothed moving average trend.

(iii) The data has been smoothed out, showing a steady increase in income. The moving average line on the graph clearly indicates that the firm's income has been consistently increasing over the four-year period from 1999 to 2002. This is a positive trend for the business.
In simple words: We calculated averages of the company's money earned over four quarters to see the main pattern. The graph shows that the company's income has been growing steadily for all four years.

🎯 Exam Tip: When plotting income data with large numbers, simplify the Y-axis labels (e.g., in lakhs or millions) to keep the graph clear, and state the unit clearly in the axis label or legend.

Question 6. The following table shows the daily sales of milk at a local corner shop for a month.

Make a table showing the moving average using a 7-day span, and draw a graph to show the trend of milk sales over the month.
Answer: To find the 7-day moving average, we sum the milk sales for seven consecutive days and then divide by seven. This helps to smooth out daily fluctuations and reveal the underlying sales trend over the month. The detailed calculation is shown in the table below:

The graph below illustrates the daily milk sales and their 7-day moving average, helping to visualize the sales trend over the month.

In simple words: We calculated the average milk sales over seven days to see the general pattern. The graph shows that, after some ups and downs, milk sales tended to increase towards the end of the month.

🎯 Exam Tip: When working with daily data, a 7-day moving average is excellent for removing weekly cycles and highlighting the underlying trend. Ensure your graph clearly distinguishes between actual data and the moving average line.

Question 7. The following table gives the monthly expenditure on a motor car for a period of two years.

Calculate 12-monthly moving average for the two years and display them and the original table on the same graph.
Answer: We will calculate the 12-monthly moving average to smooth out seasonal variations in car expenditure over a full year, revealing the long-term trend. The calculations for the 12-monthly moving total, moving average, and centered moving average are shown in the table below:

The graph below illustrates the monthly expenditure on a motor car and the 12-monthly moving average, which shows the general trend. The trend appears quite stable with slight fluctuations after smoothing out the monthly differences. This shows how regular costs for a car tend to be consistent over time, even with occasional higher or lower spending in certain months.

In simple words: We calculated the average money spent on a car over 12 months for two years. The graph shows the actual monthly spending and how it generally stayed around the same level, even with some months having more or less spending. This helps to see the regular cost of owning a car clearly.

🎯 Exam Tip: When using 12-monthly moving averages, ensure your data covers at least a full year plus the initial months needed for centering to accurately capture the annual trend. The starting and ending points for the centered moving average line will be shorter than the actual data span.

Question 8. A new film was shown at a theatre and ran for six weeks. The attendances are shown in the table.