**6.6 Measures of Dispersion**

**(A) Determine the range of a set of data**

**1.**For an ungrouped data,

Range = largest value – smallest value.

**For a grouped data,**

2.

2.

Range = midpoint of the last class – midpoint of the first class.

Example 1:

Example 1:

Determine the range of the following data.

**(a)**720, 840, 610, 980, 900

**(b)**

Time (minutes) |
1 – 6 |
7 – 12 |
13 – 18 |
19 – 24 |
25 – 30 |

Frequency |
3 |
5 |
9 |
4 |
4 |

*Solution:***(a)**

Largest value of the data = 980

Smallest value of data = 610

**Range = 980 – 610 = 370**

(b)

(b)

Midpoint of the last class

= ½ (25 + 30) minutes

= 27.5 minute

Midpoint of the first class

= ½ (1 + 6) minutes

= 3.5 minute

**Range = (27.5 – 3.5) minute = 24 minutes**

**(B) Medians and Quartiles**

**1.**The

**first quartile (**is a number such that 1 4 of the total number of data that has a value less than the number.

*Q*_{1})**2.**The

**median**is the

**second quartile**which is the value that lies at the centre of the data.

**3.**The

**third quartile (**is a number such that 3 4of the total number of data that has a value less than the number.

*Q*_{3})4. The interquartile range is the difference between the third quartile and the first quartile.

Interquartile range = third quartile – first quartile |

**Example 2:**

The ogive in the diagram shows the distribution of time (to the nearest second) taken by 100 students in a swimming competition. From the ogive, determine

**(a)**the median,

**(b)**the first quartile,

**(c)**the third quartile

**(d)**the interquartile range

of the time taken.

*Solution:*

$\begin{array}{l}\text{(a)}\frac{1}{2}\text{of 100 students}=\frac{1}{2}\times 100=50\\ \text{From the ogive, median,}M=50.5\text{second}\\ \\ \text{(b)}\frac{1}{4}\text{of 100 students}=\frac{1}{4}\times 100=25\\ \text{From the ogive, first quartile,}{Q}_{1}\text{=}44.5\text{second}\\ \\ \text{(c)}\frac{3}{4}\text{of 100 students}=\frac{3}{4}\times 100=75\\ \text{From the ogive, third quartile,}{Q}_{3}\text{= 5}4.5\text{second}\end{array}$

**(d)**

Interquartile range

= Third quartile – First quartile

= 54.5 – 44.5

**= 10.0 second**