Simple or weighted average: how to correctly calculate the average price and the average wage?

The most common mistake when calculating an "average price" or "average salary" figure is this: prices or salaries are simply added up and divided by their number. If the quantity behind each value is different, such an average distorts reality.

Key points

• Simple arithmetic mean = sum of the values ÷ number of values. It is correct when each value has the same "weight": for example, the daily prices of goods sold in the same quantity every day.
• Weighted average = Σ(value × weight) ÷ Σ weight. The weight is the quantity behind the value: kilograms, units, number of employees, asset amount.
• Check rule: a correct average always equals "total ÷ total". Average price = total amount paid ÷ total kilograms. Average wage = total payroll ÷ total number of employees.
• When deriving an overall average from group averages, a simple average is not taken: each group average is weighted by the number of units in the group.
• If an amount rather than a quantity is given as the weight (for example, how much money was spent at each price), the quantity is found first: amount ÷ price. Then the total amount is divided by the total quantity.
• Averages of percentages and ratios are also weighted: when averaging the defect rate in different branches, each percentage is weighted by the number of products in that branch.
• The average wage hides the distribution. A few high salaries pull the average up. So it is useful to show it together with the median.

Example

Apple purchase. The "Bog'bon Savdo" shop bought apples from three orchards (amounts hypothetical):

• Simple average: (8 000 + 6 500 + 7 000) ÷ 3 = 7 166.67 soum.
• Weighted average: 6 950 000 ÷ 1 000 = 6 950 soum.

Calculated with the simple average, 1 000 kg of apples would be valued at 7 166 667 soum, i.e. 216 667 soum more than actually paid. Because the cheapest apples were bought in the largest quantity, the correct average is lower.

If only the amounts are known: apples worth 4 000 000 soum were bought at 8 000 soum and apples worth 6 000 000 soum at 6 000 soum. The quantities are 500 kg and 1 000 kg, the average price is 10 000 000 ÷ 1 500 = 6 666.67 soum, not (8 000 + 6 000) ÷ 2 = 7 000 soum.

Wages in the workshop:

• Simple average of the three positions: (12 + 6 + 3) ÷ 3 = 7 million soum (wrong).
• Weighted average: 96 ÷ 20 = 4.8 million soum.
• Median: if the 20 employees are ranked by salary, apprentices are in 10th and 11th place, the median is 3 million soum. More than half of the employees (12 out of 20) receive 3 million soum, while the average of 4.8 million soum does not show this.