1) Overview 2) Frequency Distribution 3) Statistics - - PDF document

• Frequency Distribution

Cross-Tabulation

• 1) Overview

2) Frequency Distribution 3) Statistics Associated with Frequency Distribution i. Measures of Location ii. Measures of Variability

• iii. Measures of Shape

4) Introduction to Hypothesis Testing 5) A General Procedure for Hypothesis Testing 6) Cross-Tabulations 7) Statistics Associated with Cross-Tabulation

• Respondent

Sex Familiarity Internet Attitude Toward Usage of Internet Number Usage Internet Technology Shopping Banking 1 1.00 7.00 14.00 7.00 6.00 1.00 1.00 2 2.00 2.00 2.00 3.00 3.00 2.00 2.00 3 2.00 3.00 3.00 4.00 3.00 1.00 2.00 4 2.00 3.00 3.00 7.00 5.00 1.00 2.00 5 1.00 7.00 13.00 7.00 7.00 1.00 1.00 6 2.00 4.00 6.00 5.00 4.00 1.00 2.00 7 2.00 2.00 2.00 4.00 5.00 2.00 2.00 8 2.00 3.00 6.00 5.00 4.00 2.00 2.00 9 2.00 3.00 6.00 6.00 4.00 1.00 2.00 10 1.00 9.00 15.00 7.00 6.00 1.00 2.00 11 2.00 4.00 3.00 4.00 3.00 2.00 2.00 12 2.00 5.00 4.00 6.00 4.00 2.00 2.00 13 1.00 6.00 9.00 6.00 5.00 2.00 1.00 14 1.00 6.00 8.00 3.00 2.00 2.00 2.00 15 1.00 6.00 5.00 5.00 4.00 1.00 2.00 16 2.00 4.00 3.00 4.00 3.00 2.00 2.00 17 1.00 6.00 9.00 5.00 3.00 1.00 1.00 18 1.00 4.00 4.00 5.00 4.00 1.00 2.00 19 1.00 7.00 14.00 6.00 6.00 1.00 1.00 20 2.00 6.00 6.00 6.00 4.00 2.00 2.00 21 1.00 6.00 9.00 4.00 2.00 2.00 2.00 22 1.00 5.00 5.00 5.00 4.00 2.00 1.00 23 2.00 3.00 2.00 4.00 2.00 2.00 2.00 24 1.00 7.00 15.00 6.00 6.00 1.00 1.00 25 2.00 6.00 6.00 5.00 3.00 1.00 2.00 26 1.00 6.00 13.00 6.00 6.00 1.00 1.00 27 2.00 5.00 4.00 5.00 5.00 1.00 1.00 28 2.00 4.00 2.00 3.00 2.00 2.00 2.00 29 1.00 4.00 4.00 5.00 3.00 1.00 2.00 30 1.00 3.00 3.00 7.00 5.00 1.00 2.00

• Examining Summary

Statistics for Individual Variables

• Different summary measures are appropriate

for different types of data, depending on the level of measurement:

– Nominal (categorical data where there is no inherent order to the categories) – Ordinal (categorical data where there is a meaningful order of categories, but there isn't a measurable distance between categories) – Scale (data measured on an interval or ratio scale)

• For categorical data

– From the menus choose:

• Analyse

– Descriptive Statistics

• Frequencies
• For Scale Variables

– Analyise

• Descriptive Statistics

– Explore

• In a frequency distribution, one variable is

considered at a time.

• A frequency distribution for a variable produces a

table of frequency counts, percentages, and cumulative percentages for all the values associated with that variable.

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• Statistics

Familiarity 29 1 5,00 6 3,00 5,00 6,00 Valid Missing N Median Mode 25 50 75 Percentiles Familiarity

2 6,7 6,9 6,9 6 20,0 20,7 27,6 6 20,0 20,7 48,3 3 10,0 10,3 58,6 8 26,7 27,6 86,2 4 13,3 13,8 100,0 29 96,7 100,0 1 3,3 30 100,0 2 3 4 5 6 Very Familiar Total Valid 9 Missing Total Frequency Percent Valid Percent Cumulative Percent

Internet Usage Group 15 50,0 50,0 50,0 15 50,0 50,0 100,0 30 100,0 100,0 Light Users Heavy Users Total Valid Frequency Percent Valid Percent Cumulative Percent

Light Users Heavy Users

Internet Usage Group

• The mean, or average value, is the most commonly used

measure of central tendency.

• The mode is the value that occurs most frequently. It

represents the highest peak of the distribution. The mode is a good measure of location when the variable is inherently categorical or has otherwise been grouped into categories.

• The median of a sample is the middle value when

the data are arranged in ascending or descending

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• If the number of data points is even, the median is

usually estimated as the midpoint between the two middle values – by adding the two middle values and dividing their sum by 2.

• The median is the 50th percentile.

• The range measures the spread of the data. It is

simply the difference between the largest and smallest values in the sample. Range = Xlargest – Xsmallest.

• The interquartile range is the difference between

the 75th and 25th percentile. For a set of data points arranged in order of magnitude, the pth percentile is the value that has p% of the data points below it and (100 - p)% above it.

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• The variance is the mean squared deviation from the
• mean. The variance can never be negative.
• The standard deviation is the square root of the

variance.

• The coefficient of variation is the ratio of the

standard deviation to the mean expressed as a percentage, and is a unitless measure of relative variability.

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• Skewness. The tendency of the deviations from the

mean to be larger in one direction than in the other. It can be thought of as the tendency for one tail of the distribution to be heavier than the other.

• Kurtosis is a measure of the relative peakedness or

flatness of the curve defined by the frequency

• distribution. The kurtosis of a normal distribution is
• zero. If the kurtosis is positive, then the distribution is

more peaked than a normal distribution. A negative value means that the distribution is flatter than a normal distribution.

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• While a frequency distribution describes one variable

at a time, a cross-tabulation describes two or more variables simultaneously.

• Cross-tabulation results in tables that reflect the joint

distribution of two or more variables with a limited number of categories or distinct values, e.g., Table 15.3.

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• Since two variables have been cross classified,

percentages could be computed either columnwise, based on column totals, or rowwise, based on row totals.

• The general rule is to compute the percentages in

the direction of the independent variable, across the dependent variable.

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• To determine whether a systematic association

exists, the probability of obtaining a value of chi- square as large or larger than the one calculated from the cross-tabulation is estimated.

• An important characteristic of the chi-square statistic

is the number of degrees of freedom (df) associated with it. That is, df = (r - 1) x (c -1).

• The null hypothesis (H0) of no association between

the two variables will be rejected only when the calculated value of the test statistic is greater than the critical value of the chi-square distribution with the appropriate degrees of freedom.

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• Nominal Data
• The phi coefficient ( ) is used as a measure of the

strength of association in the special case of a table with two rows and two columns (a 2 x 2 table).

• The phi coefficient is proportional to the square root of

the chi-square statistic

• The value ranges between:

– 0 (indicating no association between the row and column variables and values) – And 1 (indicating a high degree of association between the variables)

• The maximum value possible depends on the number
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• The contingency coefficient (C) can be used to

assess the strength of association in a table of any size.

• The contingency coefficient varies between 0 and 1.
• The maximum value of the contingency coefficient

depends on the size of the table (number of rows and number of columns). For this reason, it should be used only to compare tables of the same size.

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• Cramer's V is a modified version of the phi

correlation coefficient, , and is ranges between 0 and 1.

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• Asymmetric lambda measures the percentage

improvement in predicting the value of the dependent variable, given the value of the independent variable.

• Lambda also varies between 0 and 1. A value of 0 means

no improvement in prediction. A value of 1 indicates that the prediction can be made without error. This happens when each independent variable category is associated with a single category of the dependent variable.

• Asymmetric lambda is computed for each of the variables

(treating it as the dependent variable).

• A symmetric lambda is also computed, which is a kind of

average of the two asymmetric values. The symmetric lambda does not make an assumption about which variable is dependent. It measures the overall improvement when prediction is done in both directions.

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Internet Banking * Sex Crosstabulation Count 8 1 9 7 14 21 15 15 30 Yes No Internet Banking Total Male Female Sex Total Symmetric Measures ,509 ,005 ,509 ,005 30 Phi Cramer's V Nominal by Nominal N of Valid Cases Value

• Approx. Sig.

Not assuming the null hypothesis. a. Using the asymptotic standard error assuming the null hypothesis. b.

• Ordinal Data
• Kendall´s Tau b

– is the most appropriate with square tables in which the number of rows and the number of columns are equal – Its value varies between +1 and -1 – The sign of the coefficient indicates the direction of the relationship – and its absolute value indicates the strength, with larger absolute values indicating stronger relashionships

• For a rectangular table in which the number of rows

is different than the number of columns, Kendall´s tau c should be used

• Gamma

– A symmetric measure of association between two

• rdinal variables that ranges between -1 and 1.
• Values close to an absolute 1 indicate a strong

relationship between the two variables

• Values close to zero indicate little or no relationship

– Gamma generally has a higher numerical value than tau b or tau c.

• Somers´d

– A measure of association between two ordinal variables that ranges from -1 to 1 – Somer’ s d is an asymmetric extension of Gamma (but a symmetric version of this statistic is also calculated)

Internet Usage Group * Familiarity Crosstabulation Count 2 4 5 3 1 15 2 1 7 4 14 2 6 6 3 8 4 29 Light Users Heavy Users Internet Usage Group Total 2 3 4 5 6 Very Familiar Familiarity Total

Symmetric Measures ,768 ,142 4,617 ,000 29 Gamma Ordinal by Ordinal N of Valid Cases Value Asymp.

• Std. Error

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Not assuming the null hypothesis. a. Using the asymptotic standard error assuming the null hypothesis. b.

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While conducting cross-tabulation analysis in practice, it is useful to proceed along the following steps.

• 1. Test the null hypothesis that there is no association between the

variables using the chi-square statistic. If you fail to reject the null hypothesis, then there is no relationship.

• 2. If H0 is rejected, then determine the strength of the association

using an appropriate statistic (phi-coefficient, contingency coefficient, Cramer's V, lambda coefficient, or other statistics), as discussed earlier.

• 3. If H0 is rejected, interpret the pattern of the relationship by

computing the percentages in the direction of the independent variable, across the dependent variable.

• 4. If the variables are treated as ordinal rather than nominal, use

tau b, tau c, or Gamma as the test statistic. If H0 is rejected, then determine the strength of the association using the magnitude, and the direction of the relationship using the sign of the test statistic.