Class notes STAT 1 (statistic)

CHAPTER 3
DATA DESCRIPTION
LESSON 2 - Measures of Variability

INTRODUCTION:

In the previous lessons, you learned how to compute mean, median
and mode. These measures of centrality focus only on giving information
of what score could best represent the entire set of data. However, if you
want to determine the spread of scores, the measure of variability can
address that query. Thus, in this lesson you will be learning the different
kinds of measures of variability or sometimes called as measures of
dispersion.



OBJECTIVES:
:At the end of this lesson, you should be able to:
1. Compute and interpret the various measures of variability such as
the range, variance, standard deviation and coefficient of variation for a
set of ungrouped data.
2. Compute and interpret the various measures of variability such as
the range, variance, standard deviation and coefficient of variation for a
set of grouped data.


Range
Range is the crudest measure of dispersion. It is the difference
between the highest and the lowest scores in the data set. This means
that range considers only two scores, thus making it the most unstable
measure of dispersion. For ungrouped data range is
R = H−L
where:
R – range
H – highest score
L – lowest score

For example, find the range of the two sets of data below.
1. 3, 4, 5, 5, 6, 8, 10, 15
Solution:
R = 15 – 3 = 12

2. 89, 70, 37, 40, 30, 95, 77, 25, 53, 36
Solution:

,Range of Ungrouped Data Using MS Excel
The MS Excel has no built-in formula to compute for the range of
data. However, since range is just the difference between the highest and
the lowest scores, we can find the range of the data set using the max and
min functions of MS Excel. Thus, the steps will be as follow:
1. Open MS Excel.
2. Enter the data.
3. Insert “=max(data set)-min(data set)” in the formula bar.
4. Press “Enter”.




Insert the formula.


Enter the data.




After pressing the “Enter” key, the result will be 12, which is the same
result as in manual computation. MS Excel will be most helpful if you have
hundreds of entries.

Range of Grouped Data
The range of grouped data will be the difference between the highest
upper limit and the lowest lower limit among the class intervals. In formula,

R = HU − L L
where:
R – range
HU – highest upper limit
LL – lowest lower limit

For example, considering the frequency distribution table given above.

Class Intervals f
90 - 94 5
85 - 89 7
80 - 84 10
75 - 79 15

, Solution:
The highest upper limit among the class intervals is 94 and the lowest
lower limit is 60. Therefore, the range will be

R = 94 − 60 = 34

Interquartile Range and Quartile Deviation
Other measures of variability that use quartiles are the interquartile
range (IQR) and Quartile Deviation (QD).

Interquartile range (IQR) is a measure of dispersion of values in
the data set between the third quartile, Q3 and the first quartile, Q1. The
formula is given as follows:
IQR = Q3 − Q1



You will notice that IQR constitutes the middle 50% of the distribution.


The quartile deviation (QD) is also called as semi-interquartile range.
This is half the difference of third and first quartile. The formula is


Q 3 − Q1
QD =
2


For Ungrouped Data

Find the IQR and QD of the following scores:

2, 3, 3, 4, 5, 5, 6, 6, 7, 8.

Solution:
N= 10
n + 1 10 + 1 11
Find Q1: The position of Q1 is at = = = 2.75
4 4 4
Therefore: Q1 = 3 + .75(3 − 3) = 3
3(n + 1) 3(10 + 1) 33
Find Q3: The position of Q3 is at = = = 8.25
4 4 4
Therefore: Q 3 = 6 + .25(7 - 6) = 6.25

Since we computed already the values of Q1 and Q3, we can now plug in the
values in our formulas.

IQR = 𝑄3 − 𝑄1 = 6.25 − 3 = 3.25
𝑄3 − 𝑄1 6.25 − 3
QD = = = 1.63
2 2