Question

Using the following data: 203 176 185 193 174 179 188 190 201 204 188 171 170.

A) Determine the percentile for 179 to 2 decimal points.

b) Calculate Q1, Q2, Q3 and the interquartile range.

c) Draw a box-and - whisker plot/diagram for the following data. Label all points on the graph.

Answer 1

First arrange the data in ascending manner

we get

170
171
174
176
179
185
188
188
190
193
201
203
204

Percentile rank of x = [(number of values below x + 0.5*y)/(total number of data values)]*100

We have 4 numbers below 179, y is frequency of given number, i.e. frequency of 179 =1 and total number =13

so, percentile rank of 179 = ((4+0.5)/13)*100 = 0.3462*100 = 34.62%

(B)

We have 13 data values, i.e. odd number of data values

Median or Q2 for odd number of data values is the middle data value

here middle data value is 188

So, median = Q2 = 188

Now divide the data into lower half and upper half, we get

Lower half = 170 171 174 176 179 185 and upper half = 188 190 193 201 203 204

Q1 is the median of lower half. We have even number of data values in the lower half and median for even number of data values is halfway between the two center value. Center values are 174 and 176

So, Q1 =(174+176)/2 = 350/2 = 175

Q3 is the median of upper half. We have even number of data values in the upper half and median for even number of data values is halfway between the two center value. Center values are 193 and 201

So, Q3 =(193+201)/2 = 394/2 = 197

Interquartile range = Q3-Q1 = 197 - 175 = 22

(C) Minimum value = 170, maximum value = 204, Q1 = 175, Q2 = 188 and Q3 =197

Mean value = (sum of all values)/(total number of data values) = (170+171+...+203+204)/13

= 2422/13

= 188.31

Using excel, the box and whisker plot is given below