Unit 6: Time-based Data - Free Study Resources | Boundless Maths

📊 Unit 6: Time-based Data

Master Moving Averages & Least Squares Method

🎯 6 Marks Weightage

📈 Method of Moving Averages

• Odd Duration Moving Averages
• Even Duration Moving Averages
• Trend Analysis
• Smoothing Time Series

📉 Method of Least Squares

• Straight Line Fit (y = a + bx)
• Normal Equations
• Trend Prediction
• Future Forecasting

🎥 Video Tutorial

6
Time-based Data
6 Marks · Complete One-Shot

Time-based Data — Complete One-Shot (31 min)

▶ Watch on YouTube →
✓ Index Numbers ✓ Time Series ✓ Moving Averages ✓ Least Squares
📝 MCQs & Assertion-Reason 📋 5 Marks Questions
🎯

Multiple Choice Questions

Question 1 Moving Averages

A 3-period moving average is calculated for the data: 10, 12, 15, 18, 20. What is the moving average for the third period?

(a) 12.33
(b) 13.67
(c) 15.00
(d) 16.33
✓ Correct Answer: (a) 12.33
Explanation:
For a 3-period moving average, we take the average of the first 3 values:
MA₃ = (10 + 12 + 15) ÷ 3 = 37 ÷ 3 = 12.33

This moving average is centered at the middle period (second period).
Question 2 Even Duration

When calculating a 4-year moving average, the centering process requires:

(a) Taking average of two consecutive moving averages
(b) Multiplying the average by 2
(c) Dividing the average by 2
(d) No centering is needed
✓ Correct Answer: (a) Taking average of two consecutive moving averages
Explanation:
For even duration moving averages (like 4-year), we need to center the values since they fall between time periods. This is done by taking the average of two consecutive 4-year moving averages, which is called a centered moving average.
Question 3 Least Squares

The normal equations for fitting y = a + bx using least squares method are:

(a) Σy = na + bΣx and Σxy = aΣx + bΣx²
(b) Σy = na - bΣx and Σxy = aΣx - bΣx²
(c) Σx = na + bΣy and Σxy = aΣy + bΣy²
(d) Σy = a + bΣx and Σxy = ax + bx²
✓ Correct Answer: (a) Σy = na + bΣx and Σxy = aΣx + bΣx²
Explanation:
The two normal equations for least squares fitting of y = a + bx are:
1) Σy = na + bΣx
2) Σxy = aΣx + bΣx²

These equations are solved simultaneously to find values of 'a' (intercept) and 'b' (slope).
Question 4 Trend Analysis

The main purpose of calculating moving averages in time series analysis is to:

(a) Increase the variability in data
(b) Smooth out short-term fluctuations and reveal trends
(c) Make predictions less accurate
(d) Eliminate all patterns in data
✓ Correct Answer: (b) Smooth out short-term fluctuations and reveal trends
Explanation:
Moving averages are used to smooth out short-term fluctuations and highlight longer-term trends or cycles in time series data. This makes it easier to identify the underlying pattern in the data.
Question 5 Least Squares

If Σx = 55, Σy = 350, Σxy = 3025, Σx² = 385, and n = 10, the value of 'b' in y = a + bx is:

(a) 5
(b) 6
(c) 7
(d) 8
✓ Correct Answer: (a) 5
Explanation:
Using the formula: b = [nΣxy - ΣxΣy] / [nΣx² - (Σx)²]
b = [10(3025) - 55(350)] / [10(385) - (55)²]
b = [30250 - 19250] / [3850 - 3025]
b = 11000 / 825
b ≈ 5
Question 6 Moving Averages

In a 5-period moving average, the first moving average value will be placed against:

(a) The first period
(b) The second period
(c) The third period
(d) The fifth period
✓ Correct Answer: (c) The third period
Explanation:
For an odd number of periods (like 5), the moving average is centered at the middle period. For a 5-period moving average calculated from periods 1-5, it will be placed against period 3 (the middle period).
Question 7 Assertion-Reason

Assertion (A): The method of least squares minimizes the sum of squares of vertical deviations.

Reason (R): This ensures the best fit line passes through all data points.

(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
✓ Correct Answer: (c) A is true but R is false
Explanation:
Assertion (A) is TRUE - The method of least squares does minimize the sum of squares of vertical deviations from the line.

Reason (R) is FALSE - The best fit line does NOT necessarily pass through all data points. It passes through the mean point (x̄, ȳ) and minimizes the overall deviation, but individual points may not lie on the line.
Question 8 Assertion-Reason

Assertion (A): Moving averages with larger periods produce smoother trends.

Reason (R): Larger periods include more data points in each average, reducing the impact of individual fluctuations.

(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
✓ Correct Answer: (a) Both A and R are true and R is the correct explanation of A
Explanation:
Both statements are TRUE and R correctly explains A. Moving averages with larger periods (e.g., 7-day vs 3-day) do produce smoother trends because they average over more data points, which reduces the impact of short-term random fluctuations and makes the underlying trend more visible.
Question 9 Trend Prediction

If the trend equation is y = 20 + 3x where x represents years starting from 2020 (x = 0), what is the predicted value for year 2025?

(a) 30
(b) 32
(c) 35
(d) 38
✓ Correct Answer: (c) 35
Explanation:
For year 2025, x = 2025 - 2020 = 5
y = 20 + 3(5) = 20 + 15 = 35

The predicted value for 2025 is 35.
Question 10 Moving Averages

The data shows quarterly sales: Q1=100, Q2=150, Q3=200, Q4=250. What is the 4-quarter moving average?

(a) 150
(b) 165
(c) 175
(d) 180
✓ Correct Answer: (c) 175
Explanation:
4-quarter moving average = (100 + 150 + 200 + 250) ÷ 4 = 700 ÷ 4 = 175
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5 Marks Questions

Question 1 5 Marks

The following data gives the number of effective accidents during various days of the week. Calculate 3-day moving averages and determine the trend values.

DayMonTueWedThuFriSatSun
Accidents12181520252228
📝 Complete Solution

Step 1: Understanding 3-day Moving Average
A 3-day moving average is calculated by taking the average of three consecutive days. Since 3 is odd, the moving average is centered at the middle day.

MA₃ = (Value₁ + Value₂ + Value₃) ÷ 3
DayAccidents (Y)3-Day Moving AverageTrend
Monday12--
Tuesday18(12+18+15)/3 = 15.0015.00
Wednesday15(18+15+20)/3 = 17.6717.67
Thursday20(15+20+25)/3 = 20.0020.00
Friday25(20+25+22)/3 = 22.3322.33
Saturday22(25+22+28)/3 = 25.0025.00
Sunday28--
📌 Key Observations:
• The moving averages show an upward trend from 15.00 to 25.00
• Accidents tend to increase throughout the week
• We cannot calculate moving averages for the first and last days with a 3-day period

Final Answer: The trend values are: 15.00, 17.67, 20.00, 22.33, and 25.00 for Tuesday through Saturday respectively.

Question 2 5 Marks

Calculate 4-year moving averages and centered moving averages for the following data:

Year2016201720182019202020212022
Sales (₹ Lakhs)80909283949992
📝 Complete Solution

Step 1: Understanding 4-Year Moving Average
Since 4 is even, we need to calculate both the 4-year moving average and then center it by taking the average of two consecutive moving averages.

4-Year MA = (Y₁ + Y₂ + Y₃ + Y₄) ÷ 4
Centered MA = (MA₁ + MA₂) ÷ 2
YearSales (Y)4-Year MACentered MA (Trend)
201680--
201790(80+90+92+83)/4 = 86.25-
201892(90+92+83+94)/4 = 89.75(86.25+89.75)/2 = 88.00
201983(92+83+94+99)/4 = 92.00(89.75+92.00)/2 = 90.88
202094(83+94+99+92)/4 = 92.00(92.00+92.00)/2 = 92.00
202199--
202292--
📌 Key Points:
• First Centered MA: (86.25+89.75)/2 = 88.00 (at 2018)
• The centering process places the moving average at the actual year
• Trend shows steady increase from 88.00 to 92.00

Final Answer: Centered MAs: 2018 = 88.00, 2019 = 90.88, 2020 = 92.00 lakhs. Steady increasing trend.

Question 3 5 Marks

Fit a straight line trend y = a + bx by the method of least squares for the following data and estimate the sales for the year 2024:

Year20182019202020212022
Sales (₹ Crores)3035384245
📝 Complete Solution

Step 1: Transforming the data
Take 2020 as origin (x = 0). Then: 2018→x=-2, 2019→x=-1, 2020→x=0, 2021→x=1, 2022→x=2

Normal Equations:
Σy = na + bΣx
Σxy = aΣx + bΣx²
Yearxyxyx²
2018-230-604
2019-135-351
202003800
2021142421
2022245904
TotalΣx=0Σy=190Σxy=37Σx²=10

From eq 1: 190 = 5a → a = 38
From eq 2: 37 = 10b → b = 3.7

Trend Equation: y = 38 + 3.7x (origin: 2020)

For 2024: x = 4 → y = 38 + 3.7(4) = 38 + 14.8 = ₹52.8 crores

Question 4 5 Marks

The production of a company (in thousand tonnes) for the years 2017–2023 is given below. Fit a linear trend equation and estimate the production for 2025.

Year2017201820192020202120222023
Production50556065707580
📝 Complete Solution

Step 1: Taking 2020 as origin (middle year, n=7).

Yearxyxyx²
2017-350-1509
2018-255-1104
2019-160-601
202006500
2021170701
20222751504
20233802409
TotalΣx=0Σy=455Σxy=140Σx²=28

a = 455/7 = 65  |  b = 140/28 = 5

Trend Equation: y = 65 + 5x (origin: 2020)

For 2025: x = 5 → y = 65 + 25 = 90 thousand tonnes

Question 5 5 Marks

From the following data, calculate 5-year moving averages and represent the trend:

Year201520162017201820192020202120222023
Production120135140150155165170180185
📝 Complete Solution
5-Year MA = (Y₁ + Y₂ + Y₃ + Y₄ + Y₅) ÷ 5
YearProductionCalculation5-Year MA
2015120--
2016135--
2017140(120+135+140+150+155)/5140
2018150(135+140+150+155+165)/5149
2019155(140+150+155+165+170)/5156
2020165(150+155+165+170+180)/5164
2021170(155+165+170+180+185)/5171
2022180--
2023185--
📊 Trend Analysis:
2017: 140 → 2018: 149 → 2019: 156 → 2020: 164 → 2021: 171
Consistent upward trend; average annual increase ≈ 7–9 units.

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