# Basic Statistics and Data Analysis

#### Lecture notes, MCQS of Statistics

Statistics Software, SPSS, Eviews, Gretl, SAS, Stata, R Language, MS-Excel

## Cronbach’s Alpha Reliability Analysis of Measurement Scales

Reliability analysis is used to study the properties of measurement scales (Likert scale questionnaire) and the items (questions) that make them up. The reliability analysis method computes a number of commonly used measures of scale reliability. The reliability analysis also provides information about the relationships between individual items in the scale. The intraclass correlation coefficients can be used to compute the interrater reliability estimates.

Consider that you want to know that does my questionnaire measures the customer satisfaction in a useful way? For this purpose, you can use the reliability analysis to determine the extent to which the items (questions) in your questionnaire are correlated to each other. The overall index of the reliability or internal consistency of the scale as a whole can be obtained. You can also identify problematic items that should be removed (deleted) from the scale.

As an example open the data “satisf.save” already available in SPSS sample files. To check the reliability of Likert scale items follows the steps given below:

Step 1: On the Menu bar of SPSS, Click Analyze > Scale > Reliability Analysis… option

Step 2: Select two more variables that you want to test and shift them from left pan to right pan of reliability analysis dialogue box. Note, multiple variables (items) can be selected by holding down the CTRL key and clicking the variable you want. Clicking the arrow button between the left and right pan will shift the variables to the item pan (right pan).

Step 3: Click on the “Statistics” Button to select some other statistics such as descriptives (for item, scale and scale if item deleted), summaries (for means, variances, covariances and correlations), inter-item (for correlations and covariances) and Anova table (for none, F-test, Friedman chi-square and Cochran chi-square) statistics etc.

Click on the “Continue” button to save the current statistics options for analysis. Click the OK button in the Reliability Analysis dialogue box to get analysis to be done on selected items. The output will be shown in SPSS output windows.

The Cronbach’s Alpha Reliability ($\alpha$) is about 0.827, which is good enough. Note that, deleting the item “organization satisfaction” will increase the reliability of remaining items to 0.860.

A rule of thumb for interpreting alpha for dichotomous items (questions with two possible answers only) or Likert scale items (question with 3, 5, 7, or 9 etc items) is:

• If Cronbach’s Alpha is $\ge 0.9$, the internal consistency of scale is Excellent.
• If Cronbach’s Alpha is $0.90 > \alpha \ge 0.8$, the internal consistency of scale is Good.
• If Cronbach’s Alpha is $0.80 > \alpha \ge 0.7$, the internal consistency of scale is Acceptable.
• If Cronbach’s Alpha is $0.70 > \alpha \ge 0.6$, the internal consistency of scale is Questionable.
• If Cronbach’s Alpha is $0.60 > \alpha \ge 0.5$, the internal consistency of scale is Poor.
• If Cronbach’s Alpha is $0.50 > \alpha$, the internal consistency of scale is Unacceptable.

However, the rules of thumb listed above should be used with caution. Since Cronbach’s Alpha reliability is sensitive to the number of items in a scale. A larger number of questions can results in a larger Alpha Reliability, while a smaller number of items may result in smaller $\alpha$.

## Introduction

A t-test for independent groups is useful when the same variable has been measured in two independent groups and the researcher wants to know whether the difference between group means is statistically significant. “Independent groups” means that the groups have different people in them and that the people in the different groups have not been matched or paired in any way.

## Objectives

The independent t-test compares the means of two unrelated/independent groups measured on the Interval or ratio scale. The SPSS t-test procedure allows the testing of hypothesis when variances are assumed to be equal or when are not equal and also provide the t-value for both assumptions. This test also provide the relevant descriptive statistics for both of the groups.

## Assumptions

• Variable can be classified in two groups independent of each other.
• Variable is Measured on interval or ratio scale.
• Measured variable is approximately normally distributed
• Both groups have similar variances  (variances are homogeneity)

## Data

Suppose a researcher want to discover whether left and right handed telephone operators differed in the time it took them to answer calls. The data for reaction time were obtained (RT’s measured in seconds):

 Subject no. RTs (Left) Subject no. RTs (Right) 1 500 11 392 2 513 12 445 3 300 13 271 4 561 14 523 5 483 15 421 6 502 16 489 7 539 17 501 8 467 18 388 9 420 19 411 10 480 20 467 Mean 476.5 430.8 Variance Ŝ2 5341.167 5298.84

The mean reaction times suggest that the left-handers were slower but does a t-test confirm this?

## Independent Sample t Test using SPSS

Perform the Following step by running the SPSS and entering the data set in SPSS data view

1. Click Analyze > Compare Means > Independent-Samples T Test… on the top menu as shown below.

Menu option for independent sample t test

2. Select continuous variables that you want to test from the list.

Dialog box for independent sample t test

3. Click on the arrow to send the variable in the “Test Variable(s)” box. You can also double click the variable to send it in “Test Variable” Box.
4. Select the categorical/grouping variable so that group comparison can be made and send it to the “Grouping Variable” box.
5. Click on the “Define Groups” button. A small dialog box will appear asking about the name/code used in variable view for the groups. We used 1 for males and 2 for females. Click Continue button when you’re done. Then click OK when you’re ready to get the output.  See the Pictures for Visual view.

Define Group for Independent sample t test

## Output

Independent sample t test output

First Table in output is about descriptive statistics concerning your variables. Number of observations, mean, variance, and standard error is available for both of the groups (male and female)

Second Table in output is important one concerning testing of hypothesis. You will see that there are two t-tests. You have to know which one to use. When comparing groups having approximately similar variances use the first t-test. Levene’s test checks for this. If the significance for Levene’s test is 0.05 or below, then it means that the “Equal Variances Not Assumed” test should be used (second one), Otherwise use the “Equal Variances Assumed” test (first one).  Here the significance is 0.287, so we’ll be using the “Equal Variances” first row in the second table.

In output table “t” is calculated t-value from test statistics, in example t-value is 1.401

df stands for degrees of freedom, in example we have 18 degree of freedom

Sig (two tailed) means two tailed significance value (P-Value), in example sig value is greater than 0.05 (significance level).

## Decision

As the P-value 0.178 id greater than our 0.05 significance level we fail to reject the null hypothesis. (two tailed case)

As the P-value 0.089 id greater than our 0.05 significance level we fail to reject the null hypothesis. (one tail case with 0.05 significance level)

As the P-value 0.089 id smaller than our 0.10 significance level we reject the null hypothesis and accept the alternative hypothesis. (one tail case with 0.10 significance level). In this case, it means that left handler have slower reaction time as compared to right handler on average.