Using moving averages to smooth fluctuations in data

Sales data gathered at regular intervals, such as monthly, forms time-series data when arranged in chronological order. Time-series analysis helps uncover patterns in this data. However, fluctuations can obscure these patterns, making interpretation challenging.

Moving averages address this by averaging data over successive periods, reducing short-term variations and highlighting underlying trends. A trend represents the overall long-term direction of a variable, such as sales growth or decline.

Calculating three-period moving averages

A three-period moving average computes the average of three consecutive periods, shifting forward one period each time—for example, periods 1 to 3, then 2 to 4. Here, a period refers to any consistent time unit, like a month or week.

In a table, the three-period moving average is positioned in the row of the middle period that was used to calculate it.

For instance, if each period is a month:

• The first moving average (using months 1-3) appears in the row for month 2 in a table
• The second moving average (using months 2-4) appears in the row for month 3 in a table
• The third moving average (using months 3-5) appears in the row for month 4 in a table

This method smooths out irregularities, aiding in trend identification for sales forecasting.

Formula for a three-period moving average

$\text{Three-period moving average}=\frac{\text{value in period 1 }+\text{ value in period 2 }+\text{ value in period 3}}{3}$

Where:

• Value in period 1 = Sales figure for the first period
• Value in period 2 = Sales figure for the second period
• Value in period 3 = Sales figure for the third period

Worked example - Calculating three-period moving averages

A clothing retailer records monthly sales revenue over five months: £68,000; £77,000; £59,000; £92,000; £88,000. Calculate the three-period moving averages for the available periods.

Step 1: Identify the values

• Month 1 = £68,000
• Month 2 = £77,000
• Month 3 = £59,000
• Month 4 = £92,000
• Month 5 = £88,000

Step 2: Formula

$\text{Three-period moving average}=\frac{\text{value in period 1 }+\text{ value in period 2 }+\text{ value in period 3}}{3}$

Step 3: Calculate the first, second, and third moving averages

Calculating four-quarter moving averages

Four-quarter moving averages average sales over four consecutive quarters to eliminate seasonal variations, such as higher sales in certain seasons. Since there is no exact middle point, centering is used: calculate initial four-quarter averages, then average pairs of these to align with specific quarters, typically placing the centered value against the third quarter of the first average.

This approach reveals smoother trends by accounting for yearly cycles.

Steps for calculating four-quarter moving averages:

1. Compute the average of four consecutive quarters.
2. Shift forward and repeat.
3. Average two consecutive four-quarter averages for centering.

Worked example - Calculating four-quarter moving averages

A toy shop records quarterly sales (in £ thousands): 12, 28, 40, 16, 14, 32, 46, 20. Calculate the four-quarter moving averages and centered values for the available periods.

Step 1: Identify the values

• Q1 = 12
• Q2 = 28
• Q3 = 40
• Q4 = 16
• Q5 = 14
• Q6 = 32
• Q7 = 46
• Q8 = 20

Step 2: Calculate four-quarter moving averages and centered moving averages

Scatter graphs and lines of best fit for identifying trends

Sales data can be plotted on a scatter graph, with time on the x-axis and sales on the y-axis, to visualise patterns. A line of best fit is drawn to pass as closely as possible to all points, illustrating the general trend.

Scatter graphs also indicate correlation between variables, such as sales and advertising spend. A line closely aligned with points suggests strong correlation, while a scattered fit implies weaker links. However, correlation does not imply causation—other factors, like economic conditions, may influence outcomes.

Extrapolation for predicting future sales

Extrapolation extends identified trends from past data into the future to estimate upcoming sales. This can involve prolonging a line of best fit or continuing a curved trend line.

Managers use these forecasts to establish sales targets and evaluate performance. However, extrapolation assumes historical patterns will persist, which may not hold true due to influences like shifting consumer tastes, new competitors, technological advances, or internal changes such as pricing adjustments.

It is most reliable in stable markets with consistent conditions, and forecasts should be limited to short-term horizons, like the next few quarters, in volatile environments.

Variations from trends and cyclical variations

Sales often follow cyclical patterns, such as peaks tied to product release cycles every few years. Cyclical variation measures the difference between actual sales and the moving average at a given point.

To refine forecasts, businesses calculate average cyclical variations across similar cycle points from past data and adjust predictions accordingly. For example, if a cycle point typically shows higher sales, add the average variation to the trend-based forecast.

This method presumes variations stay consistent, which may not always apply as market conditions evolve.

Formula for cyclical variation

$\text{Cyclical variation}=\text{actual sales}-\text{moving average}$

Where:

• Actual sales = Recorded sales for the period (£ thousands)
• Moving average = Smoothed trend value for the period (£ thousands)

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Worked example - Adjusting forecasts with cyclical variations

A software firm releases new versions every three years and has a trend forecast of £520,000 for the upcoming year. This year falls two years after the last launch, and the average cyclical variation at this cycle point from prior years is -£45,000. Calculate the adjusted forecast.

Step 1: Identify the values

• Trend forecast = £520,000
• Average cyclical variation = -£45,000

Step 2: Apply the adjustment

Adjusted forecast = £520,000 + (-£45,000) = £475,000

Step 3: Interpretation

The adjusted figure accounts for typical lower sales at this cycle stage, providing a more realistic prediction.