A small restaurant chain in Tashkent has two branches. Every month its owner, Rustam, reads one number in the report: the average customer rating out of five.
In March the report said 4.0, the same as the month before. The new branch near the university had received 20 reviews with an average rating of 4.5. The old branch in the city centre had received 80 reviews with an average of 3.5. The analyst had added 4.5 and 3.5 and divided by two.
Rustam's daughter, Zarina, who was studying for a business degree, looked at the same figures and frowned. "Your customers are not split half and half," she said. "Four out of five reviews came from the old branch. The average has to be weighted by the number of reviews."
She worked it out on a napkin. Twenty reviews at 4.5 are worth 90 points. Eighty reviews at 3.5 are worth 280 points. Together that is 370 points from 100 reviews, an average of 3.7, not 4.0.
The difference looked small, but it changed the story. The simple average said the chain was doing well. The weighted average said that most customers were eating at the weaker branch and were not very happy there. The bright new branch had been hiding the problem.
Rustam sent an inspector to the old branch the next week. The kitchen had a new cook and the waiting time had doubled. By June, the weighted average had risen to 4.2, and the report now showed both branches on separate lines.