# Find The Z-score with a percentile

Relatively basic question here ..

A Company wants to give gifts to the top 5% of customers. If the mean spend per customer is $135 - with a standard deviation of 55 - what balance should the company specify? However, at the end of the first month it was found that 8% of customers qualified for the free gift. What has happened? Assuming that the standard deviation hasn’t changed, calculate the new mean spend per customer. HOW do I calculate the Z Score for these questions ? The first part of the question's z score = 1.645 which makes sense if looking at the z-table. But according to the answer for the second part of the question the z score = 1.405. This I do not understand -- the closest percent on the z-table is 0.9207 with a z-score of 1.401 -- How do they calculate this ? and why is the closest percentile to 92% on the z-table not used ? ## 1 Answer You may be misreading the table. If it looks like the one below (from here), the closest value to$0.92$is$0.9207$, which is at the intersection of the row for$z=1.4$and the column$0.01$. This would correspond to a$z$-score of$1.4+0.01=1.41$, not$1.401$. However,$0.92$is nearly exactly between the values$0.9192$($z=1.40$) and$0.9207$($z=1.41$) in the table, so the$z$-score for$0.92$would be roughly the average of$1.40$and$1.41$, or$1.405$. The exact$z$-score above which$8\%$of the normal distribution lies is (rounded to$6$decimal places)$1.404071\$ (see this calculator, for example) 