Measures
of Central Tendency and Measures of Dispersion
Example
1
The following table gives the no of working
hours and the number of persons to complete a particular task. Calculate mean ,
median, mode, standard deviation, skewness and kurtosis.
|
Number of Working Hours
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
|
Number of Persons
|
10
|
12
|
21
|
15
|
10
|
7
|
4
|
Procedure
Step
1
Name the variables in Variable View and enter date in Data View.
Step
2 Click Data
in the main menu, select Weight Cases.
Step
3 Select Weight
Cases by radio, Weight cases.
Step 4
Transfer Number of Persons To Frequency
Variable, then click OK. Now the
display disappears from the screen.
Step 5
Choose Analyze from the main menu.
Click Descriptive Statistics and
Select Frequencies.
Step
6 The
Frequencies dialog box appears. Transfer the variable Number of Hours into the Variable(s)
box. Check the display Number of hours
under the Variable(s).
Step
7 Click
Statistics to open Frequencies:
Statistics dialog box and select mean, median and mode under Central Tendency. Standard Deviation
and SE mean under Dispersion.
Skewness and kurtosis under Distribution.
Step
8 Then
click Continue and then click OK to
run the analysis.
Step
9 The output appears with descriptive
statistics like mean, median and mode, standard deviation and SE mean, skewness
and kurtosis as shown in the output.
No
of Working Hours
|
Statistics
|
|||
|
|
|
hours
|
persons
|
|
N
|
Valid
|
79
|
79
|
|
Missing
|
0
|
0
|
|
|
Mean
|
7.51
|
13.61
|
|
|
Std. Error of Mean
|
.186
|
.590
|
|
|
Median
|
7.00
|
12.00
|
|
|
Mode
|
7
|
21
|
|
|
Std. Deviation
|
1.655
|
5.241
|
|
|
Skewness
|
.328
|
.180
|
|
|
Std. Error of Skewness
|
.271
|
.271
|
|
|
Kurtosis
|
-.584
|
-1.034
|
|
|
Std. Error of Kurtosis
|
.535
|
.535
|
|
Correlation
A correlation is useful when we want
to see the relationship between two (or more) normally distributed interval
variables.
Example
2
A
firm wants to know the correlation between the size of its sales force and to
yearly sles revenue. The records for the past 10 years were examined, and the
results are recorded in a table. Calculate a numerical descriptive measure of
correlation between size of sales force, x, and yearly revenue, y.
|
Year
|
1990
|
1991
|
1992
|
1993
|
1994
|
1995
|
1996
|
1997
|
1998
|
1999
|
|
No
of sales Persons
|
15
|
18
|
24
|
22
|
25
|
29
|
30
|
32
|
35
|
38
|
|
Sales
|
1.35
|
1.62
|
2.33
|
2.41
|
2.63
|
2.93
|
3.41
|
3.26
|
3.63
|
4.15
|
Procedure
1.
Open the Spss package.
2.
Click File -->New -->Data
3.
Click variables column
found at the bottom of the screen. Go to the first row. Type “saleper” in the
name cell.
4.
Go to label cell, type
“Number of sales persons”
5.
Go to second row. Type
“sales” in the name cell.
6.
Go to label cell, type
“Sales”
7.
Go to data view and
type data in their respective columns of saleper and sales.
8.
After completing it, go
to File -->New -->Syntax
9.
Type the command in the
white space as shown below
Correlations
/variables
= saleper sales
Execute
the command with Ctrl + R combination keys. We will get the following result.
|
Correlations
|
|||
|
|
|
no of sales
persons
|
Sales
|
|
no of sales persons
|
Pearson Correlation
|
1
|
.987
|
|
Sig. (2-tailed)
|
|
.000
|
|
|
N
|
10
|
10
|
|
|
Sales
|
Pearson Correlation
|
.987
|
1
|
|
Sig. (2-tailed)
|
.000
|
|
|
|
N
|
10
|
10
|
|
There is a high positive correlation between number
of sales persons and their sales.
t- Test
Example
3
12 students were given intensive coaching and 2
tests were conducted in a month. The scored of tests 1 and 2 are given below.
Does the score form Test1 and Test 2 show and improvement? Use 5% level of
significance.
|
No.
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
|
Test 1
|
50
|
42
|
51
|
26
|
35
|
42
|
60
|
41
|
70
|
55
|
62
|
38
|
|
Test 2
|
62
|
40
|
61
|
35
|
30
|
52
|
68
|
51
|
84
|
63
|
72
|
50
|
Procedure
1.
Open the SPSS package
2.
Click variable View and
go to first row. Type “test 1” in the name cell.
3.
Go to label cell, type
“Marks in 1”
4.
Go to second row. Type
“test 2” in the name cell
5.
Go to label cell, type
“Marks in 2”
6.
Click File -->New -->Data
7.
In the sheet enter the
data of the exercise as such
8.
Go to file menu. File
-->New -->Syntax
9.
Type the command in the
white space as shown below
t-test
pairs
=test1with test2(paired)
/criteria=cin(.95)
10.
Execute the command (Ctrl=R). we will get the following results.
|
Paired
Samples Statistics
|
|||||
|
|
|
Mean
|
N
|
Std. Deviation
|
Std. Error
Mean
|
|
Pair 1
|
Marks in 1
|
47.67
|
12
|
12.644
|
3.650
|
|
Marks in 2
|
55.67
|
12
|
15.790
|
4.558
|
|
|
Paired
Samples Correlations
|
||||||||||||||
|
|
|
N
|
Correlation
|
Sig.
|
||||||||||
|
Pair 1
|
Marks in 1 & Marks in
2
|
12
|
.944
|
.000
|
||||||||||
|
Paired
Samples Test
|
||||||||||||||
|
|
|
Paired
Differences
|
t
|
df
|
Sig. (2-tailed)
|
|||||||||
|
|
|
Mean
|
Std.
Deviation
|
Std. Error
Mean
|
95%
Confidence Interval of the Difference
|
|||||||||
|
|
|
Lower
|
Upper
|
|||||||||||
|
Pair 1
|
Marks in 1- Marks in 2
|
-8.000
|
5.673
|
1.638
|
-11.604
|
-4.396
|
-4.885
|
11
|
0.000
|
|||||
ANALYSIS OF VARIANCE
Example 4
In
a factory 15 workers were given questionnaire related to their job
satisficaton. Higher scoring means higher satisfaction. Their income level is
compared. The income level is classified into three: 1 for very low income, 2
for low income, 3 for high income. Test it at 5% significance level. Use one
way ANOVA
Procedure
- Open the SPSS
package
- Click File
-->New -->Data
- Click variable
View and go to first row. Type “jobsat” in the name cell.
- Go to label cell,
type “Job Satisfaction”
- Go to second row.
Type “Income” in the name cell and go to the values cell and click. A new
screen “Value label” will appear. In the label, type”1” and type “low
income” in the value label. Click now Add button. Also type “2” in the
value box and type “high income” in the value label. Also type “3” in the
value box and type “very high income” in the value label and close it.
- Go to label cell,
type “income”
- Go to data view
and type data in their respective columns of jobsat and income.
- After completing
it, go to File -->New -->Syntax
- Type the command
in the white space as shown below
oneway
jobsat by income
- Execute the command Ctrl=R. We will get
the following results.
|
ANOVA
|
|||||
|
Job Satisfaction
|
|
|
|
|
|
|
|
Sum of
Squares
|
df
|
Mean Square
|
F
|
Sig.
|
|
Between Groups
|
4475.044
|
2
|
2237.522
|
14.263
|
.001
|
|
Within Groups
|
1882.556
|
12
|
156.880
|
|
|
|
Total
|
6357.600
|
14
|
|
|
|
The null
hypothesis that there is difference between the level and their job
satisfaction is rejected. The significance level is 0.001, which is lower than
0.005 levels. However the mean of income you have to find mean values. To see
the mean of job satisfaction for each level of income, type
- Open the file again
and go to File --> Open --> the same file
- Go to file menu.
File -->New -->Syntax
- We will have the
white space. Type the following command
Means
tables=jobsat by income
- After typing the
command prompt, use Ctrl+R combination to execute the command
|
Report
|
|||
|
Job Satisfaction
|
|
|
|
|
Income
|
Mean
|
N
|
Std.
Deviation
|
|
low income
|
47.78
|
9
|
14.771
|
|
high income
|
78.50
|
4
|
6.557
|
|
very high income
|
90.00
|
2
|
2.828
|
|
Total
|
61.60
|
15
|
21.310
|
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