Measures of central tendency describe the location of the center of a frequency distribution. There are three different measures of central tendency: the mode, median and mean.

The mode is simply the value of the observation that occurs most frequently. It is useful when you want the prevailing or most popular characteristic or quality. In a survey of adults aged 18 or older, the question "What is your age?" was answered as follows:

What is your age?

Hence the mode is 46, since 47 respondents provided this answer, more than any other category. Since there is only one mode in this distribution, it is referred to as ‘unimodal’. If a distribution has two modes (or two values that have the same amount of responses that are also the highest), it is referred to as ‘bimodal’.

The median is the middle observation, where half the respondents have provided smaller values, and half larger ones. It is calculated by arranging all observations from lowest to highest score and counting to the middle value. In our example above, the median is 47. The cumulative percentage will tell you at a glance where the median falls. Since the median is not as sensitive as the mean to extreme values, it is used most commonly in cases where you are dealing with ‘outliers’ or extreme values in the distribution that would skew your data in some way. The median is also useful when dealing with ordinal data and you are most concerned with a typical score.

The mean is also known as the ‘arithmetic average’ and is symbolized by ‘X’. The formula for calculating the mean is ∑ X/n (The Greek letter sigma ∑ is the symbol for sum) This means that you total all responses (X) and then divide them by the total number of observations (n). In our example, you would have to multiply all the values (actual ages or ‘x’) by the number of respondents or frequencies (‘f’) for each (∑ x • f = X or. 18 • 17 + 19 • 14 + etc.) and then divide the total by the 1500 respondents who participated in the survey. The result is 47.04. Since the computer will follow the same steps, you must be sure that the values are real and not just codes for categories. For example, the computer would calculate a mean of 3.7367 for the same information as the frequency above but recoded into age categories, based on the assumption that the values under x are 1 to 6:

What is your age?

Another useful calculation is the range, the calculation of the spread of the numerical data. It is calculated by subtracting the lowest value (in our example 18) from the highest value (or 86) to give us a total of 68. This is particularly useful when dealing with rating scores, for instance, where you would like to determine how close people are in agreement or alternatively, how wide the discrepancies are. 

When you want to know how respondents answered on two or more questions at the same time, you will need to run a cross-tabulation. In order to do so, you must first determine which is your independent variable, and which your dependent variable, since the first is traditionally used as column headings and the latter are found in the row.

Independent variables explain or predict a response or an outcome, which is the dependent variable under study. As a basic rule, demographic information is usually considered independent, since characteristics such as gender, age, education etc. will normally determine the responses we make. If the variables being studied are not demographic, then the independent variable is determined by the study’s objectives. For instance, if the objective is to determine whether the level of satisfaction with the past holiday at a destination influences the likelihood of return, then level of satisfaction is our independent variable and the likelihood to return the dependent one.

This is the typical output of a simple cross-tabulation (of education levels and overall satisfaction with a holiday) as produced by SPSS, when we also ask that column percentage be calculated. Note that the title gives the two variables with the dependent one first separated by *. When producing this information in a table, we would reword it to read "Overall holiday satisfaction by highest level of education completed" ( see Table 1), removing all extraneous information and leaving it as a statement, not a question.

Overall satisfaction with your holiday * What is the highest level of education completed Crosstabulation

Obviously, you would not be able to use this table as is in a report. It requires ‘cleaning’. Your first consideration would be whether you want to keep all of the categories in your independent and dependent variable. This depends, of course, on what you are trying to illustrate and the responses in each cell. First of all, very few people have less than a high school degree, and we could therefore collapse the first two categories into ‘high school or less’. But that still leaves us with five categories or more detail than we would probably need. So we could collapse the categories ‘graduated from technical or vocational school’ and ‘some college/university’ into ‘some advanced education’ and the last two into ‘graduated from university or more’. Similarly, we notice that the level of satisfaction with the holiday is very high. Indeed, any rows with less than 5% of respondents in cells should be collapsed. At the very least we should only have one category ‘not at all or nor very satisfied’. This collapsing of categories is knows as recoding and is a way of changing existing variables or creating new variables based on existing data as explained by John Urbik, the Technical Marketing Specialist for SPSS.

The resultant cross-tabulation would look like this:

overall satisfaction with holiday * highest level of education Crosstabulation

We can now proceed to present this information in a more pleasing table format by giving it the appropriate table number and title, indicating the total number of respondents who answered this question, and cleaning the table, as follows:

Table 1: Overall holiday satisfaction by highest level of education

n=1243

Graphically, we would follow very similar rules: the graph is numbered (Figure 1) with the same title and the number of respondents indicated; the independent variable identifies the columns since we want to compare the satisfaction level of each of the three education categories. It is the column percentage that is used for comparison purposes. The type of graph below is called a clustered bar chart.

The chi-square (pronounced ‘kai’) distribution is the most commonly used method of comparing proportions. Its symbolized by the Greek letter chi or  χ²). This test makes it possible to determine whether the difference exists between two groups and their preference or likelihood of doing something is real or just a chance occurrence. In other words, it determines whether a relationship or association exists between being in one of the two groups and the behaviour or characteristic under study. If in a survey of 692 respondents we asked whether or not they are interested attending attractions and events that deal with history and heritage during their vacation, and we wanted to determine whether there is a difference in how men and women respond to this question, we could calculate a chi-square.

 χ² determines the differences between the observed (f_o) and expected frequencies (f_e). The observed frequencies are the actual survey results, whereas the expected frequencies refer to the hypothetical distribution based on the overall proportions between the two characteristics if the two groups are alike. For example, if we have the following survey results:

Then we can calculate our expected frequencies (f_e) based on the proportion of respondents who said ‘yes’ versus ‘no’. It can also be calculated for each cell by the row total with the column total divided by the grand total (e.g. 254 x 294 : 692 = 108).

This second table, where no relationship exists between the interest in attending history and heritage attractions and events and gender, also represents the null hypothesis or H_o. (Therefore, if a study says that it "fails to reject the null hypothesis", it means that no relationship was found to exist between the variables under study.)

Hence, the calculation is as follows:

The critical value for a level of significance of .05 (or 95% level of confidence, the normal level in this type of research) is 3.841. This means that you are confident that 95% of the distribution falls below this critical value. Since our result is above this value, we can:

• Reject the null hypothesis that no difference exists between interest in attending historical attractions and events and gender (in other words, there is a difference between genders); and
• Conclude that the differences in the groups are statistically significant (or not due to chance)

You will not need to memorize all the critical values since computer programs such as SPSS will not only calculate the  χ² values for you, but will also give you the precise level of observed significance (known as p value), which in our case is .039. If this level of significance is above the standard .05 level of statistical significance, you are dealing with a statistically significant relationship.

A correlation is used to estimate the relationship between two characteristics. If we are dealing with two ordinal or one ordinal and one numerical (interval or ratio) characteristic, then the correct correlation to use is the Spearman rank correlation, named after the statistician, also known as rho or r_s. Computer software programs such as SPSS will execute the tedious task of calculating Spearman’s rho very easily.

Correlations range from –1 to +1. At these extremes, the correlation between the two characteristics is perfect, although it is negative or inverse in the first instance. A perfect correlation is one where the two characteristics increase or decrease by the same amount. This is a linear relationship as illustrated below.

A correlation coefficient of 0 therefore refers to a situation where no relationship exists between the two characteristics. In other words, changes in one characteristic cannot be explained by the changes occurring in the second characteristic. A common way of interpreting the strength of a correlation is as follows:

However, in most tourism and hospitality related research, corrections between ± .26 to ± .5 are generally considered to be quite high, with strong or very strong correlations rarely found.

The variation in one characteristic can be predicted if we know the value of correlation, since we can calculate the coefficient of determination or r².

If we have a correlation of  ±.5, then  ±.5² = .25. We can therefore conclude that 25% of the variation in one characteristics can be predicted by the value of the second measure.

Let us take a look at an example to make these interpretations clearer. Let us assume we wish to determine whether pleasure travellers who are interested in staying in first class hotels are motivated because they wish to indulge in luxury or because of some other motivation, such as the identification of luxury hotels with big modern cities. By running a Spearman correlation on these three ordinal variables that asked respondents to rate the importance of each, we obtain the following output:

**Correlation is significant at the .01 level (2-tailed).

If we look at the first pair of variables (‘big modern cities’ by ‘first class hotel’) we note that the correlation coefficient is .229 or ‘weak’, while the statistical significance is very high (.000 is so small that the number itself is cut off) Indeed, as the footnote says, the results are significant at the .01 level. This can also be interpreted as "the level of confidence is 99%"(1 - .01 = .99). Finally, the output informs us that 1472 of our survey respondents answered both questions.

Similarly, the second pair of variables (‘indulging in luxury’ by ‘first class hotel’) show us a strong correlation (indeed, unusually high for this type of research at .615) that is statistically significant and where 1468 respondents answered both questions.

We can therefore conclude that ‘indulging in luxury’ is strongly correlated with the choice of first class hotel accommodation and that 38% (r² = .615 x .615) of the variance in first class hotels is determined by this factor. Furthermore, this finding is statistically significant, which means we can reject the null hypothesis that there is no relationship between the importance attributed to staying in first class hotels and indulging in luxury. We can further conclude that while ‘big modern cities’ are associated with luxury hotels, that correlation is weak with only 5% (r² = .229 x .229) of the variance in luxury hotels explained by it.