Wolfram Alpha:

```Survey Data - Likert Item and Scale
-----------------------------------

Likert questionnaires are commonly used in research.

With a Likert questionnaire, respondents specify their level of
agreement or disagreement with a particular item (Likert item).

The format of a typical 5-level Likert item is:

- Strongly disagree
- Disagree
- Neither agree nor disagree
- Agree
- Strongly agree

The level of agreement is normally coded to allow processing of
the data.  For example:

SD = 5, D = 4, N = 3, A = 2 and SA = 1

Under certain circumstance reverse coding may be applied.  For
example, consider the 2 Likert items:

Do you prefer SPSS over Excel?  SD  D  N  A SA

Do you prefer Excel over SPSS?  SD  D  N  A SA

Both are asking the same thing but without reverse coding applied
to the second one, the questions would be contradictory.

After the questionnaire is completed, each item may be analyzed
separately or summed to create a score for a group of items.
The sum of responses on several LIKERT ITEMS is referred to as
a LIKERT SCALE.  Hence, Likert scales are often called summative
scales.  Of course, the grouped items need to be different
measurements of an underlying trait.  The summed questions
must use the same Likert scale and the scale must be a reasonable
approximation to an interval scale.  If these requirements are met
the central limit theorem allows treatment of the data as interval
data and parametric statistical tests can be applied.  As a rule of
thumb generally these can only be applied when more than 5
Likert questions are summed.

Likert scales can be analyzed in several different ways:

- Average, standard deviation and frequencies (bar chart)
- t-test - independent samples
- t-test - paired/dependent samples
- One-way ANOVA

NOTE:  SUMMING ITEMS LIKE THIS IS SOMEWHAT CONTROVERTIAL.  MANY
STATISTICIANS BELIEVE THAT ADDING THEM IMPLIES THAT THE
INDIVIDUAL ITEMS ARE INTERVAL SCALED RATHEER THAN ORDINAL
SCALED.

Indidual Likert items are normally treated as ordinal data because
although the response levels do have relative position, it cannot
be presumed that participants perceive the difference between
adjacent levels to be equal (a requirement for interval data).
In statistics, ordinal data is generally analyzed using non-
parametric tests.  Non-parametric tests do not assume that the
underlying population follows a normal distribution.  Non-
parametric tests use mode/median data as opposed to means data
in their analysis.

Likert items can be analyzed in several different ways:

- Median, mode, interr-quartile rangs and frequencies (bar chart)
- Mann-Whitney U - independent samples
- Wilcoxon signed-rank - paired/dependent samples
- Kruskal-Wallis - one-way ANOVA
- Crosstabs (Χ2)

Example.

100 swimmers and 80 marthoners were asked if they liked apples.
The raw data might look like:

Resp.  S    M
--------------
1      5    3
2      2    3
3      3    2
.      .    .
80     4    4
81     3
.      .
100    4

Which can be arranged as follows:

| SD  D  N  A SA
------------+------------------
Swimmers | 10 10  5 25 50
------------+----------------
Marathoners | 30 15 20  5 10

Swimmer median: 1.5
Swimmer mode: 1
Marathoner median: 4
Marathoner mode: 5

And use a non-parametric test to accept or reject H0 for
the given item.

If the above table represented a summation of 6 questions
testing the trait "do you like fruit?" for example:

1.  Do you like apples?
2.  Do you like pears?
3.  Do you like oranges?
4.  Do you like grapes?
5.  Do you like strawberries?
6.  Do you like pineapples?

Then it would be possible to compute the means:

Swimmer mean:  (10*5 + 10*4 + 5*3 + 25*2 + 50*1)/100 = 2.05
Marathoner mean:  (30*5 + 15*4 + 20*3 + 5*2 + 10*1)/80 = 3.63

And use a parametric test to accept or reject H0 for the
given scale.```