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Univariate Data Analysis Univariate Data Analysis

Frequency Distribution Cross-Tabulation

Agenda

1) Overview 2) 3) 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

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Internet Usage Data (File Internet)

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

Univariate Data Analysis

Examining Summary Statistics for Individual Variables

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Level of M easurem ent

• 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)

Sum m ary M easures

• For categorical data – From the menus choose: • Analyse – • Frequencies • For Scale Variables – Analyise • Descriptive Statistics – Explore

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Frequency Distribution

• 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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Frequency Distribution of Fam iliarity w ith the Internet

Valid Cumulative Value label Value Frequency (N) Percentage percentage percentage

Not so familiar 1 0 0.0 0.0 0.0 2 2 6.7 6.9 6.9 3 6 20.0 20.7 27.6 4 6 20.0 20.7 48.3 5 3 10.0 10.3 58.6 6 8 26.7 27.6 86.2 Very familiar 7 4 13.3 13.8 100.0 M issing 9 1 3.3

TOTAL 30 100.0 100.0

Frequency

8 7 6 y

c 5 n e

u 4 q

e 3 r F 2 1 0 2 3 4 5 6 7 Familiarity

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Statistics

Familiarity N Valid 29

Missing 1

Median 5,00 6

Percentiles 25 3,00

50 5,00

75 6,00

Familiarity

Cumulative Frequency Percent Valid Percent Percent Valid 2 2 6,7 6,9 6,9 3 6 20,0 20,7 27,6 4 6 20,0 20,7 48,3 5 3 10,0 10,3 58,6

6 8 26,7 27,6 86,2 Very Familiar 4 13,3 13,8 100,0 Total 29 96,7 100,0 Missing 9 1 3,3 Total 30 100,0

Internet Usage Group

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

Internet Usage Group

Light Users Heavy Users

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Statistics Associated with Frequency Distribution M easures of Location

• The , or average value, is the most commonly used measure of .

• 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.

Statistics Associated with Frequency Distribution M easures of Location

• The of a sample is the middle value when the data are arranged in ascending or descending order. • 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 .

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Statistics Associated with Frequency Distribution M easures of Variability

• The measures the spread of the data. It is simply the difference between the largest and smallest values in the sample. Range = Xlargest – Xsmallest.

• The 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.

Statistics Associated with Frequency Distribution M easures of Variability

• The is the mean squared deviation from the mean. The variance can never be negative. • The is the square root of the variance. n X X 2 s ( i - ) x = Σ n i =1 - 1

• The is the ratio of the standard deviation to the mean expressed as a percentage, and is a unitless measure of relative variability. CV = sx/X

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Statistics Associated with Frequency Distribution M easures of Shape

• . 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.

• 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 that the distribution is flatter than a normal distribution.

Skew ness of a Distribution Symmetric Distribution

Skewed Distribution

Mean Median Mode (a)

Mean Median Mode (b)

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Bivariate Data Analysis

To obtain Crosstabulations …

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Cross-Tabulation

• 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.

Gender and Internet

Us age

Gender

Row Internet Usage Male Female Total

Light (1) 5 10 15

Heavy (2) 10 5 15

Column Total 15 15

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Tw o Variables Cross- Tabulation

• 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.

Internet Usage by Gender

Gender

Internet Usage Male Female

Light 33.3% 66.7%

Heavy 66.7% 33.3%

Column total 100% 100%

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Gender by Internet Usage

Internet Usage

Gender Light Heavy Total

Male 33.3% 66.7% 100.0%

Female 66.7% 33.3% 100.0%

Purchase of Fashion Clothing by M arital Status

Purchase of Current Marital Status Fashion Clothing Married Unmarried High 31% 52% Low 69% 48% Column 100% 100% Number of 700 300 respondents

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Ow nership of Expensive Autom obiles by Education Level

Own Expensive Education Automobile College Degree No College Degree

Yes 32% 21%

No 68% 79%

Column totals 100% 100%

Number of cases 250 750

Desire to Travel Abroad by Age

Desire to Travel Abroad Age

Less than 45 45 or More

Yes 50% 50%

No 50% 50%

Column totals 100% 100%

Number of respondents 500 500

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Eating Frequently in Fast-Food Restaurants by Fam ily Size

Eat Frequently in Fast- Family Size Food Restaurants Small Large

Yes 65% 65%

No 35% 35%

Column totals 100% 100%

Number of cases 500 500

Statistics Associated with Cross-Tabulation Chi-Square

• 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 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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Chi-square Distribution

Do Not Reject

H0

Reject H0

Critical χ 2 Value

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Statistics Associated with Cross-Tabulation

Nominal Data

• The phi coefficient (φ ) 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 χ2 φ = n • 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 of rows and columns in a table

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• The contingency coefficient (C) can be used to assess the strength of association in a table of any size.

χ2 C = χ2 + n

• 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.

• Cramer's V is a modified version of the phi correlation coefficient, φ , and is ranges between 0 and 1. 2 φ V = min (r-1), (c-1)

or

χ2/n V = min (r-1), (c-1)

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Lam bda Coefficient

• 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.

Internet Banking * Sex Crosstabulation

Count Sex Male Female Total Internet Banking Yes 8 1 9 No 7 14 21 Total 15 15 30

Symmetric Measures

Value Approx. Sig. Nominal by Phi ,509 ,005 Nominal Cramer's V ,509 ,005 N of Valid Cases 30 a. Not assuming the null hypothesis. b. Using the asymptotic assuming the null hypothesis.

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Statistics Associated with Cross-Tabulation

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

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• Gamma – A symmetric measure of association between two ordinal 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 Familiarity 2 3 4 5 6 Very Familiar Total Internet Usage Light Users 2 4 5 3 1 0 15 Group Heavy Users 0 2 1 0 7 4 14 Total 2 6 6 3 8 4 29

Symmetric Measures

Asymp. a b Value Std. Error Approx. T Approx. Sig. Ordinal by Ordinal Gamma ,768 ,142 4,617 ,000 N of Valid Cases 29 a. Not assuming the null hypothesis. b. Using the asymptotic standard error assuming the null hypothesis.

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Cross-Tabulation in Practice

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.