Question

the numbers of customers served by a restaurant on 40 consecutive days is shown. (the numbers have been ranked lowest to highest.) find the 85th percentile.
46 48 48 49 50 51 52 56 56 57
58 61 61 62 63 66 66 67 68 70
70 70 70 71 71 73 74 76 76 78
79 81 82 84 85 87 87 88 88 88
the number of customers representing the 85th percentile is

Explanation:

Step1: Calculate the index

Use the formula $i = p/100\times n$, where $p = 85$ (the percentile) and $n=40$ (the number of data - points). So $i=\frac{85}{100}\times40=34$.

Step2: Determine the value

Since the index $i = 34$ is an integer, the $85^{th}$ percentile is the average of the $34^{th}$ and $35^{th}$ ordered data - values. Looking at the ordered data, the $34^{th}$ value is 84 and the $35^{th}$ value is 85. The average of 84 and 85 is $\frac{84 + 85}{2}=84.5$. But in the context of percentile for discrete data, we usually take the next higher value when the index is an integer. So the $85^{th}$ percentile is 85.

Answer:

85