Question

Lower limit = Q1 – 1.5Å~IQR Upper limit = Q3 + 1.5Å~IQR IQR = Q3 − Q1

1. The manager of a local fast-food chain wants to improve the service provided to customers

who use the drive-up window. As a first step in this process, the manager records how long it

takes from the time the customer first enters the drive-in window until they are served their

food at the final window. Below are the times (in minutes) for the 30 customers (in order):

3 4 5 6 6 7 9 10 11 11 12 12 12 12 13

15 16 16 17 17 18 18 18 20 21 22 24 25 33 40

(a) Calculate the median. [2 marks]

ANSWER:

(b) Calculate the first and third quartile. [2 marks]

Q1 Q3

(c) Calculate the interquartile range.

(d) Calculate the lower and upper limits [2 marks]

Lower Upper

(e) List all outliers. Explain why they would be considered outliers. [3 marks]

ANSWER:

(f) Where would the whiskers on a boxplot of the times extend to on each side? [2 marks]

Low Side: High Side:

Answer 1

a)

The median is the middle number in a sorted list of numbers. So, to find the median, we need to place the numbers in value order and find the middle number.

Ordering the data from least to greatest, we get:

3   4   5   6   6   7   9   10   11   11   12   12   12   12   13   15   16   16   17   17   18   18   18   20   21   22   24   25   33   40   

As you can see, we do not have just one middle number but we have a pair of middle numbers, so the median is the average of these two numbers:

Median = 13+15 /2 =14

b)

The first quartile (or lower quartile or 25th percentile) is the median of the bottom half of the numbers. So, to find the first quartile, we need to place the numbers in value order and find the bottom half.

3   4   5   6   6   7   9   10   11   11   12   12   12   12   13   15   16   16   17   17   18   18   18   20   21   22   24   25   33   40   

So, the bottom half is

3   4   5   6   6   7   9   10   11   11   12   12   12   12   13   

The median of these numbers is 10.

The third quartile (or upper quartile or 75th percentile) is the median of the upper half of the numbers. So, to find the third quartile, we need to place the numbers in value order and find the upper half.

3   4   5   6   6   7   9   10   11   11   12   12   12   12   13   15   16   16   17   17   18   18   18   20   21   22   24   25   33   40   

So, the upper half is

15   16   16   17   17   18   18   18   20   21   22   24   25   33   40   

The median of these numbers is 18.

c)

The interquartile range is the difference between the third and first quartiles.

The third quartile is 18.

The first quartile is 10.

The interquartile range = 18 - 10 = 8.

d)
LF = Q1 - 1.5 * IQR = 10 - 1.5 * 8 = -2
UF = Q3 + 1.5 * IQR = 18 + 1.5 *8 = 30

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