Chapter SH, Problem 10.8EP

To determine

To describe the center and spread of the data using either the mean and standard deviation or the five number summary.

Answer to Problem 10.8EP

Minimum=12

Median, Q=18

Lower quartile, Q₁=16.5 Upper quartile, Q₂=19.5

Maximum=23

Explanation of Solution

Given information:

Data: 12, 18, 21, 18, 19, 18, 16, 23, 20, 15, 17, 18.

Calculation:

Ordered data: 12, 15, 16, 17, 18, 18, 18, 18,19, 20, 21, 23

A five-number summary is especially useful in descriptive analyses or during the preliminary investigation of a large data set. A summary consists of five values: the most extreme values in the data set (the maximum and minimum values), the lower and upper quartiles, and the median. These values are presented together and ordered from lowest to highest: minimum value, lower quartile (Q1), median value (Q), upper quartile (Q3), maximum value. Minimum=12

  n=12

nis even

Then mean of 6th and 7th number give media Q

Median, Q= $\frac{\left(18+18\right)}{2}$

On solving,

Q=18

Lower quartile

12, 15, 16, 17, 18, 18

Median, Q₁is mean of 3rd and 4th term

Median, Q₂= $\frac{\left(16+17\right)}{2}$

On solving,

Q₁=16.5

Upper quartile

18, 18,19, 20, 21, 23

Median, Q₁is mean of 3rd and 4th term

Median, Q₂= $\frac{\left(19+20\right)}{2}$

On solving,

Q₁=19.5

Maximum=23

Box plot

  

These values have been selected to give a summary of a data set because each value describes a specific part of a data set: the median identifies the centre of a data set; the upper and lower quartiles span the middle half of a data set; and the highest and lowest observations provide additional information about the actual dispersion of the data. This makes the five-number summary a useful measure of spread.