Chapter 10.4, Problem 8PPS

To determine

To describe the center and spread the data using five-number summary.

Answer to Problem 8PPS

The five number summary is 67, 76, 82, 88 and 96.

Explanation of Solution

Given information:

Data set: 86, 45, 91, 104, 95, 88, 127, 85, 79, 102, 98, 103

Formula used: Median is,

  $M=\frac{{(\frac{n}{2})}^{th}+{(\frac{n}{2}+1)}^{th}}{2}term$

Lower quartile is,

  $Q1=(n+1)\frac{1}{4}$

Upper quartile is,

  $Q3=(n+1)\frac{3}{4}$

Calculation:

Re-arranging the data in an ascending order,

67, 69, 70, 72, 73, 74, 74, 75, 76, 77, 78, 79, 80, 80, 80, 81, 82, 82, 82, 83, 83, 84, 84, 85, 85, 86, 87, 88, 88, 89, 90, 91, 92, 94, 96, 96.

For median substituting the values,

  $M=\frac{{(\frac{36}{2})}^{th}+{(\frac{36}{2}+1)}^{th}}{2}$

On solving,

  $M=\frac{{18}^{th}+{19}^{th}}{2}term$

On substituting the values,

  $M=\frac{82+82}{2}$

On solving,

  $M=82$

Hence, Median is 82.

For lower quartile substituting the values,

  $Q1=(36+1)\frac{1}{4}$

On solving,

  $Q1=9.25$

On rounding off,

  $\approx 9$

Substituting the value of 9^th observation,

  $Q1=76$

Hence, Lower quartile is 76.

For upper quartile substituting the values,

  $Q3=(36+1)\frac{3}{4}$

On solving,

  $Q3=27.75$

On rounding off,

  $\approx 28$

Substituting the value of 28^th observation,

  $Q3=88$

Hence, higher quartile is 88.

Minimum is the smallest value of dataset.

Hence, minimum= 67.

Maximum is the largest value of the dataset.

Hence, maximum= 96.

The five number summary is 67, 76, 82, 88 and 96.