Measures of Variability€¦ · THE RANGE •The range of a set of numerical data is the difference...

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Measures of Variability

For Ungrouped Data

The RangeMeasures of Variability for

Ungrouped Data

THE RANGE•The range of a set of numerical data is the difference

between the highest and the lowest values.

• It is the simplest possible measure of spread.

• It cannot be used with grouped data and it ignores the

distribution of intermediate values.

•A single very large or very small value would give a

misleading impression of the spread of the data.

FORMULA:

•𝑹𝒂𝒏𝒈𝒆 = 𝑯𝑶𝑽 − 𝑳𝑶𝑽Where:

𝑯𝑶𝑽 = 𝒉𝒊𝒈𝒉𝒆𝒔𝒕 𝒐𝒃𝒔𝒆𝒓𝒗𝒆𝒅 𝒗𝒂𝒍𝒖𝒆

𝑳𝑶𝑽 = 𝒍𝒐𝒘𝒆𝒔𝒕 𝒐𝒃𝒔𝒆𝒓𝒗𝒆𝒅 𝒗𝒂𝒍𝒖𝒆

Example 6.1:•The following data are the numbers of deaths of

army officers caused by horse kicks, for the

Prussian Army during the period 1875 to 1894.

In order of size the numbers are:

3, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10

11, 11, 11, 12, 14, 15, 15, 17, 18

What is the range?

Example 6.2:•One year the numbers of academic staff (including part-

time staff) in the various departments of the University

of Essex (a small, friendly university) were as follows:

19.0, 15.7, 25.3, 28.0, 15.0, 10.0

12.0, 10.3, 22.0, 24.8, 13.8, 25.9,

23.0, 21.3, 12.0, 11.0, 23.0

What is the range?

The Interquartile Range (IQR)

Measures of Variability for Ungrouped Data

The Interquartile Range (IQR)

•It is more useful than range because it

concentrates on the middle portion of the

distribution which is the difference between

the upper and the lower quartiles.

•It is also called as the H-spread because of

the use of box plots.

Formula:

•𝑰𝑸𝑹 = 𝑸𝟑 −𝑸𝟏

•𝑰𝑸𝑹 = 𝑷𝟕𝟓 − 𝑷𝟐𝟓

3, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10

11, 11, 11, 12, 14, 15, 15, 17, 18

What is the IQR?

Example 6.4:•One year the numbers of academic staff (including part-

time staff) in the various departments of the University

of Essex (a small, friendly university) were as follows:

19.0, 15.7, 25.3, 28.0, 15.0, 10.0

12.0, 10.3, 22.0, 24.8, 13.8, 25.9,

23.0, 21.3, 12.0, 11.0, 23.0

What is the IQR?

The Semi-Interquartile Range

•It is the half of the

difference between 𝑷𝟕𝟓(or

𝑸𝟑) and the 𝑷𝟐𝟓(or 𝑸𝟏) in

the distribution.

Formula:

•𝑺𝑰𝑸𝑹 =𝑸𝟐−𝑸𝟏

•𝑺𝑰𝑸𝑹 =𝑷𝟕𝟓−𝑷𝟐𝟓

The Mean Deviation (MD)

The Mean Deviation (MD)•It is the average distance between the

mean and the scores in the distribution.

•The technique provides a reasonably

stable estimate of variation.

•It is also called “Average Deviation”.

Formula:

•𝑴𝑫 = 𝑿−𝒙

3, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10

11, 11, 11, 12, 14, 15, 15, 17, 18

What is the value of MD?

Steps in Calculating the MD

Step 1:Construct a Frequency

Distribution/Table.

X (individual

scores)x (Mean) 𝐗 − 𝐱

X (individual

scores)x (Mean)

𝐗 − 𝐱

Step 2:Calculate the Mean using the

Simple Arithmetic Mean.

X (individual

scores)x (Mean)

𝐗 − 𝐱

3 9.8 10 9.8

4 9.8 11 9.8

5 9.8 11 9.8

6 9.8 12 9.8

6 9.8 14 9.8

7 9.8 15 9.8

8 9.8 15 9.8

9 9.8 17 9.8

9 9.8 18 9.8

Step 3:Calculate the deviation of each score from the mean.

X (individual

scores)x (Mean)

𝐗 − 𝐱

3 9.8 6.8 10 9.8 0.2

4 9.8 5.8 11 9.8 1.2

5 9.8 4.8 11 9.8 1.2

6 9.8 3.8 12 9.8 2.2

6 9.8 3.8 14 9.8 4.2

7 9.8 2.8 15 9.8 5.2

8 9.8 1.8 15 9.8 5.2

9 9.8 0.8 17 9.8 7.2

9 9.8 0.8 18 9.8 8.2

Step 4:Calculate the summation of deviation

of each score from the mean.

The value of 𝑿 − 𝒙 :

• 𝑿 − 𝒙 =𝟕𝟐

Step 5:Use the formula.

Formula:

•𝑴𝑫 = 𝑿−𝒙

Final Answer:

•𝑴𝑫 = 𝟑. 𝟔

The VarianceMeasures of Variability for

Ungrouped Data

The Variance• It defines how close the scores in the distribution

are to the middle of the distribution.

•The more variation there is in the x-values, the

larger will be the value of the variance.

•The variance is define as the average squared

differences of the scores from the mean.

The Issue:•Diving the squares of the mean

deviation 𝑿𝒊 − 𝒙𝟐 by 𝒏 would

seem natural, but (unfortunately)

there is a strong case for diving

instead by 𝒏 − 𝟏 .

Using the divisor n:•This is appropriate in two

cases (1) if the values

𝒙𝟏, … , 𝒙𝒏 represent an entire

population;

•If 𝒙𝟏, … , 𝒙𝒏 represent the

entire population then 𝝈𝟐 is

called the POPULATION

VARIANCE.

Using the divisor n:•and (2) if the values 𝒙𝟏, … , 𝒙𝒏

represent a sample from a

population and we are interested

in the variation within the sample

itself.

•If 𝒙𝟏, … , 𝒙𝒏 represent a

sample of the data then 𝝈𝟐

is called the SAMPLE

VARIANCE.

Example 6.6:•An example of a case where the x-

values refer to the entire population

is where 𝒙𝟏, … , 𝒙𝒏 represent the

heights of ALL the children in a

particular class in a school.

Measures of Variability€¦ · THE RANGE •The range of a set of numerical data is the difference...

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