📊 Unit 6: Time-based Data
Master Moving Averages & Least Squares Method
🎯 15 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
Multiple Choice Questions
A 3-period moving average is calculated for the data: 10, 12, 15, 18, 20. What is the moving average for the third period?
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).
When calculating a 4-year moving average, the centering process requires:
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.
The normal equations for fitting y = a + bx using least squares method are:
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).
The main purpose of calculating moving averages in time series analysis is to:
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.
If Σx = 55, Σy = 350, Σxy = 3025, Σx² = 385, and n = 10, the value of 'b' in y = a + bx is:
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
In a 5-period moving average, the first moving average value will be placed against:
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).
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.
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.
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.
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.
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?
For year 2025, x = 2025 - 2020 = 5
y = 20 + 3(5) = 20 + 15 = 35
The predicted value for 2025 is 35.
The data shows quarterly sales: Q1=100, Q2=150, Q3=200, Q4=250. What is the 4-quarter moving average?
4-quarter moving average = (100 + 150 + 200 + 250) ÷ 4
= 700 ÷ 4 = 175
This single moving average covers all four quarters.
5 Marks Questions
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.
| Day | Mon | Tue | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|---|
| Accidents | 12 | 18 | 15 | 20 | 25 | 22 | 28 |
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.
Step 2: Calculating Moving Averages
| Day | Accidents (Y) | 3-Day Moving Average | Trend |
|---|---|---|---|
| Monday | 12 | - | - |
| Tuesday | 18 | (12+18+15)/3 = 15.00 | 15.00 |
| Wednesday | 15 | (18+15+20)/3 = 17.67 | 17.67 |
| Thursday | 20 | (15+20+25)/3 = 20.00 | 20.00 |
| Friday | 25 | (20+25+22)/3 = 22.33 | 22.33 |
| Saturday | 22 | (25+22+28)/3 = 25.00 | 25.00 |
| Sunday | 28 | - | - |
• 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. The data shows an increasing trend in accidents as the week progresses.
Calculate 4-year moving averages and centered moving averages for the following data:
| Year | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
|---|---|---|---|---|---|---|---|
| Sales (₹ Lakhs) | 80 | 90 | 92 | 83 | 94 | 99 | 92 |
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.
Centered MA = (MA₁ + MA₂) ÷ 2
Step 2: Calculating 4-Year Moving Averages
| Year | Sales (Y) | 4-Year MA | Centered MA (Trend) |
|---|---|---|---|
| 2016 | 80 | - | - |
| 2017 | 90 | (80+90+92+83)/4 = 86.25 | - |
| 2018 | 92 | (90+92+83+94)/4 = 89.75 | (86.25+89.75)/2 = 88.00 |
| 2019 | 83 | (92+83+94+99)/4 = 92.00 | (89.75+92.00)/2 = 90.88 |
| 2020 | 94 | (83+94+99+92)/4 = 92.00 | (92.00+92.00)/2 = 92.00 |
| 2021 | 99 | - | - |
| 2022 | 92 | - | - |
• First 4-year MA: (80+90+92+83)/4 = 86.25 (between 2017-2018)
• Second 4-year MA: (90+92+83+94)/4 = 89.75 (between 2018-2019)
• First Centered MA: (86.25+89.75)/2 = 88.00 (at 2018)
• The centering process places the moving average at the actual year
Final Answer: The centered moving averages (trend values) are:
• 2018: 88.00 lakhs
• 2019: 90.88 lakhs
• 2020: 92.00 lakhs
The trend shows a steady increase in sales from 2018 to 2020.
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:
| Year | 2018 | 2019 | 2020 | 2021 | 2022 |
|---|---|---|---|---|---|
| Sales (₹ Crores) | 30 | 35 | 38 | 42 | 45 |
Step 1: Transforming the data
Let's take 2020 as the origin (x = 0) and 1 year as the unit.
Then: 2018 → x = -2, 2019 → x = -1, 2020 → x = 0, 2021 → x = 1, 2022 → x = 2
Σy = na + bΣx
Σxy = aΣx + bΣx²
Step 2: Preparing the calculation table
| Year | x | y | xy | x² |
|---|---|---|---|---|
| 2018 | -2 | 30 | -60 | 4 |
| 2019 | -1 | 35 | -35 | 1 |
| 2020 | 0 | 38 | 0 | 0 |
| 2021 | 1 | 42 | 42 | 1 |
| 2022 | 2 | 45 | 90 | 4 |
| Total | Σx = 0 | Σy = 190 | Σxy = 37 | Σx² = 10 |
Step 3: Solving the normal equations
Given: n = 5, Σx = 0, Σy = 190, Σxy = 37, Σx² = 10
From equation 1: Σy = na + bΣx
190 = 5a + b(0)
190 = 5a
a = 38
From equation 2: Σxy = aΣx + bΣx²
37 = a(0) + b(10)
37 = 10b
b = 3.7
Where x = 0 corresponds to year 2020
Step 4: Estimating sales for 2024
For year 2024: x = 2024 - 2020 = 4
y = 38 + 3.7(4)
y = 38 + 14.8
y = 52.8 crores
Final Answer:
• Trend equation: y = 38 + 3.7x (with 2020 as origin)
• Estimated sales for 2024: ₹52.8 crores
• The sales are increasing at a rate of ₹3.7 crores per year
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.
| Year | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
|---|---|---|---|---|---|---|---|
| Production | 50 | 55 | 60 | 65 | 70 | 75 | 80 |
Step 1: Setting up the problem
Taking 2020 as origin (x = 0), since it's the middle year (n = 7).
Units: 1 year = 1 unit
| Year | x | y | xy | x² |
|---|---|---|---|---|
| 2017 | -3 | 50 | -150 | 9 |
| 2018 | -2 | 55 | -110 | 4 |
| 2019 | -1 | 60 | -60 | 1 |
| 2020 | 0 | 65 | 0 | 0 |
| 2021 | 1 | 70 | 70 | 1 |
| 2022 | 2 | 75 | 150 | 4 |
| 2023 | 3 | 80 | 240 | 9 |
| Total | Σx = 0 | Σy = 455 | Σxy = 140 | Σx² = 28 |
Step 2: Applying normal equations
Given: n = 7, Σx = 0, Σy = 455, Σxy = 140, Σx² = 28
Equation 1: Σy = na + bΣx
455 = 7a + b(0)
a = 455/7 = 65
Equation 2: Σxy = aΣx + bΣx²
140 = a(0) + b(28)
b = 140/28 = 5
(Origin: 2020, Unit: 1 year)
Step 3: Prediction for 2025
For 2025: x = 2025 - 2020 = 5
y = 65 + 5(5)
y = 65 + 25
y = 90 thousand tonnes
Interpretation:
• The base production in 2020 was 65 thousand tonnes
• Production increases by 5 thousand tonnes every year
• This represents a steady linear growth pattern
• Expected production in 2025: 90 thousand tonnes
Final Answer: The linear trend equation is y = 65 + 5x, and the estimated production for 2025 is 90 thousand tonnes.
From the following data, calculate 5-year moving averages and represent the trend:
| Year | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
|---|---|---|---|---|---|---|---|---|---|
| Production | 120 | 135 | 140 | 150 | 155 | 165 | 170 | 180 | 185 |
Step 1: Understanding 5-Year Moving Average
Since 5 is odd, we calculate the average of 5 consecutive years and place it at the middle (3rd) year of each group.
Step 2: Calculating Moving Averages
| Year | Production (Y) | Calculation | 5-Year MA (Trend) |
|---|---|---|---|
| 2015 | 120 | - | - |
| 2016 | 135 | - | - |
| 2017 | 140 | (120+135+140+150+155)/5 | 140 |
| 2018 | 150 | (135+140+150+155+165)/5 | 149 |
| 2019 | 155 | (140+150+155+165+170)/5 | 156 |
| 2020 | 165 | (150+155+165+170+180)/5 | 164 |
| 2021 | 170 | (155+165+170+180+185)/5 | 171 |
| 2022 | 180 | - | - |
| 2023 | 185 | - | - |
Step 3: Detailed Calculations
For 2017: (120+135+140+150+155)/5 = 700/5 = 140
For 2018: (135+140+150+155+165)/5 = 745/5 = 149
For 2019: (140+150+155+165+170)/5 = 780/5 = 156
For 2020: (150+155+165+170+180)/5 = 820/5 = 164
For 2021: (155+165+170+180+185)/5 = 855/5 = 171
• 2017: 140 units
• 2018: 149 units (↑ 9 units)
• 2019: 156 units (↑ 7 units)
• 2020: 164 units (↑ 8 units)
• 2021: 171 units (↑ 7 units)
The moving averages show a consistent upward trend with an average annual increase of approximately 7-9 units.
Step 4: Trend Representation
The 5-year moving averages smooth out short-term fluctuations and reveal:
• A steady increasing trend from 140 to 171
• Growth of approximately 31 units over 4 years
• Average annual growth: ~7.75 units
• The trend is linear and positive
Final Answer: The 5-year moving averages for years 2017 to 2021 are 140, 149, 156, 164, and 171 respectively, showing a consistent upward trend in production.
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