Weighted averages and changes in mix
Separate changes in individual prices or margins from changes in what customers buy.
Calculate a weighted average price or margin and distinguish changes within a category from changes in the sales mix.
Overview
The owner of Birch Print Shop sees that average revenue per poster rose from $12 to $16. She has not changed either product's price. Before calling this a pricing success, she checks which posters customers bought.
Both businesses and all figures in this lesson are fictional teaching assumptions.
Recall Lesson 023: What is the denominator when calculating gross margin, and how does it differ from markup? Keep that distinction in mind when choosing today's weights.
Give each observation its proper weight
An average, or arithmetic mean, adds values and divides by their count. If three purchases have prices of $10, $10, and $20, their average is $40 ÷ 3, or about $13.33.
Simply averaging the two listed prices, ($10 + $20) ÷ 2 = $15, would give each price equal importance. That answers a different question from the average price actually paid across the three purchases.
A weighted average gives each value influence according to an appropriate weight. The general calculation multiplies each value by its weight, adds those products, and divides by total weight. NIST gives this weighted-mean formula. Source
For average selling price, use units sold as weights: add each product's price × quantity, then divide by total units. The numerator is total revenue. The result is dollars per unit.
You can also use shares. If a product accounts for 80 of 100 units, its weight is 80 ÷ 100 = 0.80. Multiply each price by its unit share and add the results. Shares must sum to one, equivalent to 100%.
Worked example
Birch sells standard and premium posters. These are its completed sales for two equal-length weeks. Prices and complete product costs per poster stay unchanged. Assume no discounts, returns, or sales taxes in the figures.
| Poster | Price | Product cost | Week A units | Week B units |
|---|---|---|---|---|
| Standard | $10 | $6 | 80 | 40 |
| Premium | $20 | $14 | 20 | 60 |
Week A revenue is (80 × $10) + (20 × $20) = $1,200. There are 100 posters, so average selling price is $1,200 ÷ 100 = $12. Using shares gives the same result: (0.80 × $10) + (0.20 × $20) = $12.
Week B revenue is (40 × $10) + (60 × $20) = $1,600. Average selling price is $1,600 ÷ 100 = $16.
The sales mix describes the proportions of different products sold. Premium posters rose from 20% to 60% of units. This mix change raised the average price by $4, or $4 ÷ $12 × 100 ≈ 33.3%, without any increase in either listed price.
A within-category change would mean a change in the price of standard posters or premium posters themselves. Check category prices and quantities separately before attributing an overall average's movement to pricing.
Use revenue weights for gross margin
Gross profit per standard poster is $10 − $6 = $4, giving a 40% gross margin. Premium gross profit is $20 − $14 = $6, giving a 30% margin. Premium earns more gross-profit dollars per poster but has a lower margin percentage.
Week A gross profit is (80 × $4) + (20 × $6) = $440. Its overall gross margin is $440 ÷ $1,200 × 100 ≈ 36.7%.
Week B gross profit is (40 × $4) + (60 × $6) = $520. Its overall gross margin is $520 ÷ $1,600 × 100 = 32.5%.
Overall gross margin uses total revenue as its denominator, so category margins must be weighted by revenue shares. In Week B, standard posters supply $400 of $1,600 revenue, or 25%; premium supplies 75%. Thus (0.25 × 40%) + (0.75 × 30%) = 32.5%.
Using unit shares instead would give (0.40 × 40%) + (0.60 × 30%) = 34%, which is not overall gross margin. The weights must match what the denominator measures.
Birch's average selling price and total gross profit rose while overall gross margin fell. None of those facts alone establishes total business profit: other expenses remain to be deducted. The owner should examine demand, production effort, and total dollars before deciding which product to promote.
Exercise
Allow 8–10 minutes. You manage established software company ClearPath. It sells Basic and Pro licenses billed for a full month. Each license is one comparable billing unit, although the plans provide different features.
The table includes all licenses billed in each month. There are no partial months, discounts, refunds, or taxes in the figures. Treat each price as the actual revenue per license for that month.
| Plan | Month A licenses | Month A price | Month B licenses | Month B price |
|---|---|---|---|---|
| Basic | 60 | $50 | 20 | $55 |
| Pro | 40 | $100 | 80 | $110 |
A colleague says, “Average revenue per license rose, so we can report the entire increase as a price increase.”
- Calculate each month's revenue and weighted average revenue per license. Why is the simple average of the two plan prices unsuitable?
- Calculate a comparison using Month B quantities but Month A prices. This changes the mix while holding plan prices constant.
- Compare that result first with Month A's average, then with Month B's actual average. How much of the dollar increase comes from each step? Calculate each plan's actual percentage price increase.
- Write a two-sentence correction to the colleague. Name one fact you would investigate before recommending more emphasis on Pro.
Try the exercise before reading the answer.
Answer
Month A revenue is (60 × $50) + (40 × $100) = $7,000. Divide by 100 licenses to get $70 per license.
Month B revenue is (20 × $55) + (80 × $110) = $9,900. Divide by 100 to get $99 per license. The increase is $29, or $29 ÷ $70 × 100 ≈ 41.4%.
The simple price averages would be $75 and $82.50. They give Basic and Pro equal weight even though the license counts are unequal.
Using Month B quantities with Month A prices gives (20 × $50) + (80 × $100) = $9,000, or $90 per license.
The comparison moves from $70 to $90 because the mix shifts toward Pro, a $20 increase with prices held fixed. Moving from $90 to $99 adds $9 because plan prices rise at the Month B mix. The two steps reconcile: $20 + $9 = $29.
Basic's price rose by ($55 − $50) ÷ $50 × 100 = 10%. Pro's rose by ($110 − $100) ÷ $100 × 100 = 10%. The 41.4% overall increase therefore cannot be described as a 41.4% increase in each plan's price.
This calculation changes mix first and prices second. When both change, another comparison order can allocate the combined increase differently. State the chosen method; the comparison does not prove why customers selected Pro.
A correction could read: “Average revenue per license rose from $70 to $99, while each plan's price rose 10%. Using old prices with the new mix gives $90, so the shift toward Pro accounts for a substantial part of the overall increase.”
Investigate Pro's delivery and support costs, or whether customers expect to renew at the new prices. Revenue per license alone cannot establish profitability or future demand.
Recall answer: Gross margin divides gross profit by sales revenue; markup divides by product cost. That is why revenue shares, rather than unit shares, combine product gross margins correctly.
Applying this lesson
Before acting on an average, identify its weights and compare the underlying categories. Keep a column for quantities and another for prices or margins so changes in mix remain visible.
Sources
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