Basic Statistics and Data Analysis

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 $\alpha$.

Creating Formula in Excel: Operators Order of Precedence

Creating customized (user defined) formulas in Microsoft Excel is not too difficult. For creating formulas just combine the references of your data with the correct mathematical operator (such as -, +, /, * and ^).

Microsoft Order of Precedence

The order of mathematical operations determines in which order the mathematical operations are carried out. If more than mathematical operators are used in formula, there is a specific order (sequence) that Microsoft Excel will follow to perform (compute) these mathematical operations. However, to change the order of operations, brackets (parenthesis) are used in the Excel formula. The easy way to remember the order of operations (precedence) is to remember the acronym: BEDMAS (PEDMAS), that i.e.,

The order of operations (precedence) is:

Bracket or Parenthesis
Exponents (^)
Division (/)
Multiplication (*)
Addition (+)
Subtraction (-)

Suppose, following is the screenshot of an Excel sheet. The formula is also shown in formula bar. As an example, addition (+), division (/) and multiplication (*) operators are used.

The formula in screenshot performs the computation in the following order,

• E1/F1 will be computed (answer is 1.5),
• the answer of E1/F1 will be multiplied by value of G1 (answer is 1.5*2 = 3)
• the answer of E1/F1 * G1 will be added to D1 (answer is 7)

Any operation(s) enclosed in brackets (parenthesis) will be carried out first followed by any exponents. After that, Excel will consider division or multiplication operations to be of equal importance. The operations are performed in the order they occur left to right in the formula. Similar sequence is also performed for addition and subtraction. Both (addition and subtraction) are considered equal in the order of operations. The operator which appears first will be computed first.

For example, see the screenshot The sequence of operation is

• First bracket will be computed, that is, multiplication will be performed (2 *2 = 4)
• E1 will be divided by the answer from multiplication of F1 and G1 (3/4 = 0.75)
• Lastly D1 will be added to the answer 0.75 (4 + 0.75 = 4.75)

Now check the sequence in the following screenshot

For Creating formula in Excel, see the link Creating Excel Formula

Changing the data

Before writing your required formula, you need numeric data in different columns or rows of Excels’ sheet. Suppose you want to enter few numbers in a column. Before entry these number you should first confirm the cell reference where you need to enter the data. Let start by entry number in Excels’ cell A1 and A2. For this purpose follow steps given below

1. Click on the cell A1
2. Type 3 from keyboard
3. Press the ENTER or DOWN ARROW key on the keyboard. You will be in Cell A2
4. Now type say 2 from keyboard and press ENTER key

Suppose you want to add these number in Cell C1. You need to write a formula in cell C1. After writing correct formula the content of Cell C1 will immediately changes to addition of two numbers typed in A1 and A2 and used in C1 as formula content.

Creating Formula in MS-Excel

In Excel, each formula begins with a equal sign (=), see the picture below

Therefore, when creating formulas in Excel, ALWAYS start by typing the equal sign. Equal sign is typed in the Cell where you want the answer to appear. Like image above, follow these steps

1. Click on cell C1 with ARROW keys from keyboard or with mouse pointer.
2. Type the equal sign in cell C1.

After typing the equal sign in step 2, you have two choices for adding cell references to the spreadsheet formula. Note that cell reference is the name of cell you want to use in formula. A1 and A2 are cell references of numbers 3 and 2, respectively.

1. You can type these references in or,
2. You can use an Excel feature called Pointing

Pointing allows you to click with your mouse on the cell contain the data or approaching to a cell reference using keyboard ARROW keys containing your data to add. This will add cell reference to the formula.

After typing an equal sign in cell C3 in step 2:

1. Click on cell A1 with the mouse pointer to enter the cell reference into the formula
2. Type a plus (+) sign. You can also use other operators such as for multiplication use you have to use * symbol, for division / symbol and for subtraction use – etc.
3. Click on cell A2 with the mouse pointer to enter the cell reference into the formula
4. Press the ENTER key on the keyboard

The answer 5 should appear in cell C1.

Note if you have more than one row or column of data then you need to perform calculations on each row or column cell. It is often possible to copy the first formula to other cells. The easiest way to do this is to copy formulas with the file handle.

Writing Excel Formulas

Writing Excel formulas is a little different than the way it is done in mathematics class. All Excel formulas starts with equal sign (=), that is, the equal sign always goes in that cell where you want the answer to appear from formula. Therefore, the equal sign informs Excel that this is formula not just a name or number. Excel formula looks like

= 3 + 2

rather than

3+2 =

Cell References in Formula

The example of formula has one drawback. If you want to change the number being calculated (3, and 2), you need to edit it or re-write the formula. A better way is to write formula in such a way that you can change the numbers without changing or re-writing the formulas themselves. To do this, cell references are used, which tells Excel that data/ numbers are located in a cell. Therefore a cell’s location/ reference in the spreadsheet is referred to as its cell reference.

To find a cell reference, simply click the cell of which you need cell reference and from NAME BOX (shown in figure below), see the text, such as F2.

F2 represents the cell in F column (horizontal position) and row 2 (vertical position). It means cells reference can also be found by reading column heading (at the top most position) of the cells and row number (at the left most position). Therefore, cell reference is a combination of the column letter and row number such as A1, B2, Z5, and A106 etc. For previous formula example, instead of writing = 3 + 2 in cell suppose (C1), follow this way of cell reference and formula writing:

In cell A1 write 3, and in cell B2 write 2. In C1 cell write the formula such as,

= A1 + A2

Note that there is no gap between A & 1 and A & 2, they are simply A1 and A2. See the diagram for clear understanding.

Updating Excel Formula

Upon wrong cell references in Excel formula, the results from formula will be automatically updated, whenever the data value in relevant cells is changed. For example, if you want to change data in cell A1 to 8 instead of 3, you only need to change the content of A1 only. The result of formula in cell C1 will automatically be updated after the updation of data value in A1 or B1.

Note that the formula will not change because the cells references are being used instead of data values or numbers.

Creating Matrices in Mathematica

A matrix is an array of numbers arranged in rows and columns. In Mathematica matrices are expressed as a list of rows, each of which is a list itself. It means a matrix is a list of lists. If a matrix has n rows and m columns then we call it an n by m matrix. The value(s) in the ith row and jth column is called the i, j entry.

In mathematica, matrices can be entered with the { } notation, constructed from a formula or imported from a data file. There are also commands for creating diagonal matrices, constant matrices and other special matrix types.

Creating matrices in Mathematica

• Create a matrix using { } notation
mat={{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}
but output will not be in matrix form, to get in matrix form use command like
mat//MatrixForm
• Creating matrix using Table command
mat1=Table[b{row, column},
{row, 1, 4, 1}, {column, 1, 2, 1}]
];
MatrixForm[mat1]
• Creating symbolic matrix such as
mat2=Table[xi+xj , {i, 1, 4}, {j, 1, 3}]
mat2//MatrixForm
• Creating a diagonal matrix with nonzero entries at its diagonal
DiagonalMatrix[{1, 2, 3, r}]//MatrixForm
• Creating a matrix with same entries i.e. a constant matrix
ConstantArray[3, {2, 4}]//MatrixForm
• Creating an identity matrix of order n × n
IdentityMatrix[4]

Matrix Operations in Mathematica

In mathematica matrix operations can be performed on both numeric and symbolic matrices.

• To find the determinant of a matrix
Det[mat]
• To find the transpose of a matrix
Transpose[mat]
• To find the inverse of a matrix for linear system
Inverse[mat]
• To find the Trace of a matrix i.e. sum of diagonal elements in a matrix
Tr[mat]
• To find Eigenvalues of a matrix
Eigenvalues[mat]
• To find Eigenvector of a matrix
Eigenvector[mat]
• To find both Eigenvalues and Eigenvectors together
Eigensystem[mat]

Note that +, *, ^ operators all automatically work element-wise.

Displaying matrix and its elements

• mat[[1]]         displays the first row of a matrix where mat is a matrix create above
• mat[[1, 2]]     displays the element from first row and second column, i.e. m12 element of the matrix
• mat[[All, 2]]  displays the 2nd column of matrix

References